Quantification of mean energy and photon contamination for accurate dosimetry of high-energy electron beams.

The scientific background of the standard procedure for determination of the mean electron energy at the phantom surface (E0) from the half-value depth (R50) has been studied. The influence of energy, angular spread and range straggling on the shape of the depth dose distribution and the R50 and Rp ranges is described using the simple Gaussian range straggling model. The relation between the R50 and Rp ranges is derived in terms of the variance of the range straggling distribution. By describing the mean energy imparted by the electrons both as a surface integral over the incident energy fluence and as a volume integral over the associated absorbed dose distribution, the relation between E0 and different range concepts, such as R50 and the maximum dose and the surface dose related mean energy deposition ranges, Rm and R0, is analysed. In particular the influence of multiple electron scatter and phantom generated bremsstrahlung on R50 is derived. A simple analytical expression is derived for the ratio of the incident electron energy to the half-value depth. Also, an analytical expression is derived for the maximum energy deposition in monoenergetic plane-parallel electron beams in water for energies between 2 and 50 MeV. Simple linear relations describing the relative absorbed dose and mass ionization at the depth of the practical range deposited by the bremsstrahlung photons generated in the phantom are derived as a function of the incident electron energy. With these relations and a measurement of the extrapolated photon background at Rp, the treatment head generated bremsstrahlung distribution can be determined. The identification of this photon contamination allows an accurate calculation of the absorbed dose in electron beams with a high bremsstrahlung contamination by accounting for the difference in stopping power ratios between a clean electron beam and the photon contamination. The absorbed dose determined using ionization chambers in heavily photon contaminated (10%) electron beams may be too low--by as much as 1.5%--without correction.

The role of phantom and treatment head generated bremsstrahlung in high-energy electron beam dosimetry.

An analytical expression has been derived for the phantom generated bremsstrahlung photons in plane-parallel monoenergetic electron beams normally incident on material of any atomic number (Be, H2O, Al, Cu and U). The expression is suitable for the energy range from 1 to 50 MeV and it is solely based on known scattering power and radiative and collision stopping power data for the material at the incident electron energy. The depth dose distribution due to the bremsstrahlung generated by the electrons in the phantom is derived by convolving the bremsstrahlung energy fluence produced in the phantom with a simple analytical energy deposition kernel. The kernel accounts for both electrons and photons set in motion by the bremsstrahlung photons. The energy loss by the primary electrons, the build-up of the electron fluence and the generation, attenuation and absorption of bremsstrahlung photons are all taken into account in the analytical formula. The longitudinal energy deposition kernel is derived analytically and it is consistent with both the classical biexponential relation describing the photon depth dose distribution and the exponential attenuation of the primary photons. For comparison Monte Carlo calculated energy deposition distributions using ITS3 code were used. Good agreement was found between the results with the analytical expression and the Monte Carlo calculation. For tissue equivalent materials, the maximum total energy deposition differs by less than 0.2% from Monte Carlo calculated dose distributions. The result can be used to estimate the depth dependence of phantom generated bremsstrahlung in different materials in therapeutic electron beams and the bremsstrahlung production in different electron absorbers such as scattering foils, transmission monitors and photon and electron collimators. By subtracting the phantom generated bremsstrahlung from the total bremsstrahlung background the photon contamination generated in the treatment head can be determined to allow accurate dosimetry of heavily photon contaminated electron beams.

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.

An accurate energy-range relationship for high-energy electron beams in arbitrary materials.

A general analytical energy-range relationship has been derived to relate the practical range, Rp, to the most probable energy, Ep, of incident electron beams in the range 1 to 50 MeV and above, for absorbers of any atomic number. The expression is cubic in energy and requires as input parameters the total stopping power, So, the ratio of the scattering power and the total specific stopping power, T0/epsilon0, both taken at 10 MeV, and the radiation length for the material involved, X0. In addition to these parameters, five of the derived parameters are used to 'fine tune' the equation and minimize the mean square deviation from experimental and/or Monte Carlo data by means of non-linear regression. In the present study only Monte Carlo data determined with the new ITS.3 code have been employed. The standard deviations of the mean deviation from the Monte Carlo data at any energy are about 0.10, 0.12, 0.04, 0.11, 0.04, 0.03, 0.02 mm for Be, C, H2O, Al, Cu, Ag and U, respectively, and the relative standard deviation of the mean is about 0.5% for all materials. The fitting program gives some priority to water-equivalent materials, which explains the low standard deviation for water. A small error in the fall-off slope can give a different value for Rp. We describe a new method which reduces the uncertainty in the Rp determination, by fitting an odd function to the descending portion of the depth-dose curve in order to accurately determine the tangent at the inflection point, and thereby the practical range. An approximate inverse relation is given expressing the most probable energy of an electron beam as a function of the practical range. The resultant relative standard error of the energy is less than 0.7%, and the maximum energy error deltaEp is less than 0.3 MeV.

Energy distributions from a racetrack microtron measured with a magnetic spectrometer.

Energy spectra of accelerated electron beams from a racetrack microtron were measured using a magnetic spectrometer. The spectrometer utilized a 90 degrees dipole magnet. A ray-tracing program was developed to determine the slit positions of the spectrometer based on a detailed map of the magnetic field measured at field strengths corresponding to about 20 and 50 MeV. The total a priori uncertainty (previously often called systematic or class B uncertainty) of the measured most probable energy Ep is 0.22% (one approximate standard deviation) and the a posteriori uncertainty (previously often called random or class A uncertainty) is 0.04% (1 sigma). The estimated energy resolution (delta E/E) of the spectrometer is 4 x 10(-4). Spectral energy distributions of the electron beam were measured at a Ep = 21.1 and 51.6 MeV, and the obtained full width at half-maximum of the energy distributions were 53 and 34 keV, respectively. All the measurements were performed in vacuum to minimize the influence of electron energy loss and scatter.