A proposed alternative to phase-space recycling using the adaptive kernel density estimator method.

We have implemented a nonparametric density estimation technique, the adaptive kernel density estimator (AKDE), to generate additional phase space (PS) variables in the vicinity of simulated PS points in Monte Carlo linear accelerator simulation. The method involves the placement of kernels at simulated PS points that have a "window width" that depends on the density of simulated PS points. This method has been tested on known one-dimensional (1-D) and two-dimensional (2-D) probability density functions (PDFs) and has been used to sample (photons only) from PS files generated from accelerator simulations. The original simulated PS vector (x, y, u, v, E) was reduced to a rotationally invariant PS vector (r, theta, alpha, E) that takes advantage of the azimuthal symmetry (phi) above the collimating jaws. The new PS vector (r', theta', alpha', E') is sampled in the vicinity of the sampled PS vector (r, theta, alpha, E). The first step in assessing the accuracy of the method was a correlation analysis among the AKDE generated PS variables compared with correlations among the original PS variables. "In-air" particle fluence distributions between AKDE samples and the original PS distribution showed agreement within 2% (-8.8% to 6.8%) across the entire phase space plane. Central axis energy distributions and angular distributions agreed on average to within 1.5% (range = -1.5% to 6.6%) and 0.1% (range = 0 to 3.0%), respectively. Dose profiles were calculated for field sizes 3 x 3 cm2, 10 x 10 cm2, and 30 x 30 cm2 for AKDE and compared against calculations performed with PS recycling. AKDE calculated depth doses and profiles were within 2% and 2%/1 mm, respectively, of those computed using PS recycling.

Description and dosimetric verification of the PEREGRINE Monte Carlo dose calculation system for photon beams incident on a water phantom.

PEREGRINE is a three-dimensional Monte Carlo dose calculation system written specifically for radiotherapy. This paper describes the implementation and overall dosimetric accuracy of PEREGRINE physics algorithms, beam model, and beam commissioning procedure. Particle-interaction data, tracking geometries, scoring, variance reduction, and statistical analysis are described. The BEAM code system is used to model the treatment-independent accelerator head, resulting in the identification of primary and scattered photon sources and an electron contaminant source. The magnitude of the electron source is increased to improve agreement with measurements in the buildup region in the largest fields. Published measurements provide an estimate of backscatter on monitor chamber response. Commissioning consists of selecting the electron beam energy, determining the scale factor that defines dose per monitor unit, and describing treatment-dependent beam modifiers. We compare calculations with measurements in a water phantom for open fields, wedges, blocks, and a multileaf collimator for 6 and 18 MV Varian Clinac 2100C photon beams. All calculations are reported as dose per monitor unit. Aside from backscatter estimates, no additional, field-specific normalization is included in comparisons with measurements. Maximum discrepancies were less than either 2% of the maximum dose or 1.2 mm in isodose position for all field sizes and beam modifiers.

An apparatus for applying strong longitudinal magnetic fields to clinical photon and electron beams.

Monte Carlo studies have recently renewed interest in the use of the effect of strong transverse and longitudinal magnetic fields to manipulate the dose characteristics of clinical photon and electron beams. A 3.5 T superconducting solenoidal magnet was used to evaluate the effect of a longitudinal field on both photon and electron beams. This note describes the apparatus and demonstrates some of the effects on the beam trajectory and dose distributions for measurements in a homogeneous phantom. The effects were studied using film in air and in phantoms which fit in the magnet bore. The magnetic field focused and collimated the electron beams. The converging, non-uniform field confined the beam and caused it to converge with increasing depth in the phantom. Due to the field's collecting and focusing effect, the beam flux density increased, leading to increased dose deposition near the magnetic axis, especially near the surface of the phantom. This study illustrates some benefits and challenges associated with the use of non-uniform longitudinal magnetic fields in conjunction with clinical electron and photon beams.

DPM, a fast, accurate Monte Carlo code optimized for photon and electron radiotherapy treatment planning dose calculations.

A new Monte Carlo (MC) algorithm, the 'dose planning method' (DPM), and its associated computer program for simulating the transport of electrons and photons in radiotherapy class problems employing primary electron beams, is presented. DPM is intended to be a high accuracy MC alternative to the current generation of treatment planning codes which rely on analytical algorithms based on an approximate solution of the photon/electron Boltzmann transport equation. For primary electron beams, DPM is capable of computing 3D dose distributions (in 1 mm3 voxels) which agree to within 1% in dose maximum with widely used and exhaustively benchmarked general-purpose public-domain MC codes in only a fraction of the CPU time. A representative problem, the simulation of 1 million 10 MeV electrons impinging upon a water phantom of 128(3) voxels of 1 mm on a side, can be performed by DPM in roughly 3 min on a modern desktop workstation. DPM achieves this performance by employing transport mechanics and electron multiple scattering distribution functions which have been derived to permit long transport steps (of the order of 5 mm) which can cross heterogeneity boundaries. The underlying algorithm is a 'mixed' class simulation scheme, with differential cross sections for hard inelastic collisions and bremsstrahlung events described in an approximate manner to simplify their sampling. The continuous energy loss approximation is employed for energy losses below some predefined thresholds, and photon transport (including Compton, photoelectric absorption and pair production) is simulated in an analogue manner. The delta-scattering method (Woodcock tracking) is adopted to minimize the computational costs of transporting photons across voxels.

150-250 meV electron beams in radiation therapy.

High-energy electron beams in the range 150-250 MeV are studied to evaluate the feasibility for radiotherapy. Monte Carlo simulation results from the PENELOPE code are presented and used to determine lateral spread and penetration of these beams. It is shown that the penumbra is comparable to photon beams at depths less than 10 cm and the practical range (Rp) of these beams is greater than 40 cm. The depth dose distribution of electron beams compares favourably with photon beams. Effects caused by nuclear reactions are evaluated, including increased dose due to neutron production and induced radioactivity resulting in an increased relative biological effectiveness (RBE) factor of < 1.03.

Towards the elimination of Monte Carlo statistical fluctuation from dose volume histograms for radiotherapy treatment planning.

The Monte Carlo calculation of dose for radiotherapy treatment planning purposes introduces unavoidable statistical noise into the prediction of dose in a given volume element (voxel). When the doses in these voxels are summed to produce dose volume histograms (DVHs), this noise translates into a broadening of differential DVHs and correspondingly flatter DVHs. A brute force approach would entail calculating dose for long periods of time--enough to ensure that the DVHs had converged. In this paper we introduce an approach for deconvolving the statistical noise from DVHs, thereby obtaining estimates for converged DVHs obtained about 100 times faster than the brute force approach described above. There are two important implications of this work: (a) decisions based upon DVHs may be made much more economically using the new approach and (b) inverse treatment planning or optimization methods may employ Monte Carlo dose calculations at all stages of the iterative procedure since the prohibitive cost of Monte Carlo calculations at the intermediate calculation steps can be practically eliminated.

A thin target approach for portal imaging in medical accelerators.

A new thin-target method (patent pending) is described for portal imaging with low-energy (tens of keV) photons from a medical linear accelerator operating in a special mode. Low-energy photons are usually produced in the accelerator target, but are absorbed by the target and flattening filter, both made of medium- or high-Z materials such as Cu or W. Since the main contributor to absorption of the low-energy photons is self-absorption by the thick target through the photoelectric effect, it is proposed to lower the thickness of the portal imaging target to the minimum required to get the maximum low-energy photon fluence on the exit side of the target, and to lower the atomic number of the target so that predominantly photoelectric absorption is reduced. To determine the minimum thickness of the target, EGS4 Monte Carlo calculations were performed. As a result of these calculations, it was concluded that the maximum photon fluence for a 4 MeV electron beam is obtained with a 1.5 mm Cu target. This value is approximately five times less than the thickness of the Cu target routinely used for bremsstrahlung production in radiotherapeutic practice. Two sets of experiments were performed: the first with a 1.5 mm Cu target and the second with a 5 mm Al target (Cu mass equivalent) installed in the linear accelerator. Portal films were taken with a Rando anthropomorphic phantom. To emphasize the low-energy response of the new thin target we used a Kodak Min-R mammographic film and cassette combination, with a strong low-energy response. Because of its high sensitivity, only 1 cGy is required. The new portal images show a remarkable improvement in sharpness and contrast in anatomical detail compared with existing ones. It is also shown that further lowering of the target's atomic number (for example to C or Be) produces no significant improvement.

Fluence non-uniformity effects in air kerma determination around brachytherapy sources.

In ionization chamber dosimetry close to brachytherapy sources, the non-uniform photon fluence in the vicinity of the air cavity of the ionization chamber must be corrected for. In this paper, experimentally determined relative non-uniformity correction factors are determined for a PTW23331 chamber and a number of Farmer-type chambers. It is concluded that the 'anisotropic' theory of Bielajew agrees better with the experimental results than the 'isotropic' theory of Kondo and Randolph. The experiments show that neither the material choice nor the radius of the central electrode have any significant effect on the non-uniformity correction factors. Experiments on the wall-material dependence in the non-uniformity correction factor, which is predicted by the anisotropic theory, were inconclusive. Parameters for the Farmer-type chambers for calculation of theoretical non-uniformity correction factors are given. Data to derive anisotropic fluence non-uniformity correction factors are given in tabular form. An error in the Kondo and Randolph equations is reported and the corrected equations presented.

An improved energy-range relationship for high-energy electron beams based on multiple accurate experimental and Monte Carlo data sets.

A theoretically based analytical energy-range relationship has been developed and calibrated against well established experimental and Monte Carlo calculated energy-range data. Only published experimental data with a clear statement of accuracy and method of evaluation have been used. Besides published experimental range data for different uniform media, new accurate experimental data on the practical range of high-energy electron beams in water for the energy range 10-50 MeV from accurately calibrated racetrack microtrons have been used. Largely due to the simultaneous pooling of accurate experimental and Monte Carlo data for different materials, the fit has resulted in an increased accuracy of the resultant energy-range relationship, particularly at high energies. Up to date Monte Carlo data from the latest versions of the codes ITS3 and EGS4 for absorbers of atomic numbers between four and 92 (Be, C, H2O, PMMA, Al, Cu, Ag, Pb and U) and incident electron energies between 1 and 100 MeV have been used as a complement where experimental data are sparse or missing. The standard deviation of the experimental data relative to the new relation is slightly larger than that of the Monte Carlo data. This is partly due to the fact that theoretically based stopping and scattering cross-sections are used both to account for the material dependence of the analytical energy-range formula and to calculate ranges with the Monte Carlo programs. For water the deviation from the traditional energy-range relation of ICRU Report 35 is only 0.5% at 20 MeV but as high as -2.2% at 50 MeV. An improved method for divergence and ionization correction in high-energy electron beams has also been developed to enable use of a wider range of experimental results.

Measurements of the electron dose distribution near inhomogeneities using a plastic scintillation detector.

PURPOSE: Accurate measurement of the electron dose distribution near an inhomogeneity is difficult with traditional dosimeters which themselves perturb the electron field. We tested the performance of a new high resolution, water-equivalent plastic scintillation detector which has ideal properties for this application. METHODS AND MATERIALS: A plastic scintillation detector with a 1 mm diameter, 3 mm long cylindrical sensitive volume was used to measure the dose distributions behind standard benchmark inhomogeneities in water phantoms. The plastic scintillator material is more water equivalent than polystyrene in terms of its mass collision stopping power and mass scattering power. Measurements were performed for beams of electrons having initial energies of 6 and 18 MeV at depths from 0.2-4.2 cm behind the inhomogeneities. RESULTS: The detector reveals hot and cold spots behind heterogeneities at resolutions equivalent to typical film digitizer spot sizes. Plots of the dose distributions behind air, aluminum, lead, and formulations for cortical and inner bone-equivalent materials are presented. CONCLUSION: The plastic scintillation detector is suited for measuring the electron dose distribution near an inhomogeneity.

Constraints of the multiple-scattering theory of Molière in Monte Carlo simulations of the transport of charged particles.

The breakdown of Molière's multiple-scattering theory for short pathlengths occurring during Monte Carlo simulations with charged particles is demonstrated. It has been found that in certain conditions where the theory is assumed to be valid, significant distortions of the angular distribution occur that make the sampling of the polar angle questionable in numerous steps of Monte Carlo simulations. The limits of the theory have been investigated, both using a large number of terms in the Molière's series and using steps of Molière's theory where 1/B expansions are not involved. At B = 4.5 the commonly accepted 3-term series expansion yields differences up to +/- 6% compared with the evaluation of the complete Molière angular distribution, and up to 7 terms in the series are needed in order to achieve agreement within +/- 2%. One percent agreement requires B = 10. Numerical values of the full distribution are given in terms of Molière's parameters B and reduced angle theta. By using the general dependence of the distribution results are valid for both electron and proton Monte Carlo simulations in any material.

The effect of strong longitudinal magnetic fields on dose deposition from electron and photon beams.

The use of strong, uniform, longitudinal magnetic fields for external electron and photon beam irradiation is considered. Using the EGS4 Monte Carlo code modified to account for the presence of magnetic fields, dramatic narrowing of penumbra for photon and electron irradiations is demonstrated. In the vicinity of heterogeneities, "hot" and "cold" spots due to multiple scattering in electron beams are reduced substantially. However, in the presence of strong magnetic fields, the effect of inhomogeneities can be observed far from the location of the inhomogeneity due to reduced "washout" caused by lateral multiple scattering. The enhanced "Bragg peak," proposed or calculated by other authors, is not observed on the central axis of broad beams, owing to lateral equilibrium. It is proven that for broad parallel beams, the central axis depth-dose curve is independent of the strength of the external longitudinal magnetic field, as long as it is uniform. However, strong longitudinal magnetic fields can induce enhancements by redirection of the electron fields coming from point sources. Strong uniform longitudinal magnetic fields provide a way of controlling the spreading of electron beams due to multiple scattering, making the electron beams more "geometrical" in character, simplifying dose-deposition patterns, possibly allowing electron beams to be used in new ways for radiotherapy. Photon therapy also benefits from strong uniform longitudinal magnetic fields since the penumbra or other lateral disequilibrium effects associated with lateral electron transport can be eliminated.

The application of correlated sampling to the computation of electron beam dose distributions in heterogeneous phantoms using the Monte Carlo method.

Although the Monte Carlo method is capable of computing the dose distribution in heterogeneous phantoms directly, there are some advantages to computing a heterogeneity correction factor. If this approach is adopted there are savings in time using correlated sampling. This technique forces histories to have the same energy, position, direction and random number seed as incident on both the heterogeneous and homogeneous water phantom. This ensures that a history that has, by chance, travelled through only water in the heterogeneous phantom will have the same path as it would have through the homogeneous phantom, resulting in a reduced variance when a ratio of heterogeneous dose to homogeneous dose is formed. Metrics to describe the distributions of uncertainty, efficiency, and degree of correlation are defined. EGS4 Monte Carlo calculation of the dose distribution from a 20 MeV electron beam on water phantoms containing aluminum or air slab heterogeneities illustrate that this technique is the most efficient when the heterogeneity is deep within the phantom, but that improved efficiency can be realized even when the heterogeneity is at or near the surface. This is because some correlation between the two histories is retained despite passage through the heterogeneity.

Wall-correction and absorbed-dose conversion factors for Fricke dosimetry: Monte Carlo calculations and measurements.

For megavoltage radiotherapy photon beams, EGS4 Monte Carlo calculations show, and experimental measurements confirm with an accuracy of 0.2%, that glass or quartz-walled vials used in Fricke dosimetry increase the dose in the Fricke solution. This is mainly caused by increased electron scattering from the glass which increases the dose to the Fricke solution. The dose perturbation is shown to vary from nothing in a 60Co beam up to 2% in a 24-MV beam. For plastic vials of similar shapes, calculations demonstrate that the effect is in the opposite direction and even at high energies it is much less (0.2% to 0.5%).

A standard timing benchmark for EGS4 Monte Carlo calculations.

A Fortran 77 Monte Carlo source code built from the EGS4 Monte Carlo code system has been used for timing benchmark purposes on 29 different computers. This code simulates the deposition of energy from an incident electron beam in a 3-D rectilinear geometry such as one would employ to model electron and photon transport through a series of CT slices. The benchmark forms a standalone system and does not require that the EGS4 system be installed. The Fortran source code may be ported to different architectures by modifying a few lines and only a moderate amount of CPU time is required ranging from about 5 h on PC/386/387 to a few seconds on a massively parallel supercomputer (a BBN TC2000 with 512 processors).

Calculation of water/air stopping-power ratios using EGS4 with explicit treatment of electron-positron differences.

Using the EGS4 Monte Carlo simulation program, a general purpose code has been written to calculate Bragg-Gray and Spencer-Attix stopping-power ratios for use in radiation dosimetry. The stopping-power ratios can be calculated in any material in any region in a general cylindrical geometry with a large number of source geometries possible. The calculations take into account for the first time the differences between the stopping powers and the inelastic scattering of positrons and electrons. The results show that previous calculations ignoring these effects were accurate. The present results agree, typically within 0.1%, with the Spencer-Attix water-to-air stopping-power ratios for broad parallel beams of electrons given in the AAPM and IAEA protocols except at the surface where the present calculations follow the buildup of secondary electrons in more detail and see a 2% reduction in the stopping-power ratios.

On the technique of extrapolation to obtain wall correction factors for ion chambers irradiated by photon beams.

Wall correction factors, which correct ion chamber response for photon attenuation and scatter, can differ by as much as 1.0% for spherical chambers depending on whether they are obtained experimentally by extrapolation measurements or by Monte Carlo simulation. This difference is not explained by experimental or calculational statistics which lie in the range 0.05%-0.2%. In this paper it is demonstrated that linear extrapolation of experimental data for spherical chambers is inappropriate, owing to the curvature of the chamber walls. A simple nonlinear theory is constructed that resolves the difference. The Monte Carlo calculations and the nonlinear theory are compared with extrapolation measurements for the NIST (formerly NBS) spherical chambers. It is concluded that wall correction factors should be obtained by Monte Carlo calculation for spherical chambers and that linear-extrapolation techniques should be regarded with suspicion for all chambers.