Inclusion of electron range straggling in the Fermi-Eyges multiple-scattering theory

Feasibility of a multigroup Boltzmann-Fokker-Planck solution for electron beam dose calculations

Legacy nuclear-reactor Boltzmann solvers start clinical deployment as an alternative to Monte Carlo (MC) codes and Fermi-Eyges semiemprical models in radiation oncology treatment planning. Today's certified clinical solvers are limited to photon beams. In this paper, ELECTR, a state-of-the-art multigroup electron cross sections generation module in NJOY is presented and validated against Lockwood's calorimetric measurements, EGS-nrc and GEANT-4 for 1-20 MeV unidirectional electron beams. The nuclear-reactor DRAGON-5 solver is upgraded to access the library and solve the Boltzmann-Fokker-Planck (BFP) equation. A variety of heterogeneous radiotherapy and radiosurgery phantom configurations were used for validation purpose. Case studies include a thorax benchmark, that of a typical breast Intra-Operative Radiotherapy and a high-heterogeneity patient-like benchmark. For all beams, [Formula: see text] of the water voxels satisfied the American Association of Physicists in Medicine accuracy criterion for a BFP-MC dose error below [Formula: see text]. At least, [Formula: see text] of adipose, muscle, bone, lung, tumor and breast voxels satisfied the [Formula: see text] criterion. The average BFP-MC relative error was about [Formula: see text] for all voxels, beams and materials combined. By irradiating homogeneous slabs from [Formula: see text] (hydrogen) to [Formula: see text] (einsteinium), we reported performance and defects of the CEPXS mode [US. Sandia National Lab., SAND-89-1685] in ELECTR for the entire periodic table. For all Lockwood's benchmarks, NJOY-DRAGON dose predictions are within the experimental data precision for [Formula: see text] of voxels.

A comparison of Monte Carlo and Fermi-Eyges-Hogstrom estimates of heart and lung dose from breast electron boost treatment

PURPOSE

Electrons are commonly used in the treatment of breast cancer primarily to deliver a tumor bed boost. We compared the use of the Monte Carlo (MC) method and the Fermi-Eyges-Hogstrom (FEH) algorithm to calculate the dose distribution of electron treatment to normal tissues.

METHODS AND MATERIALS

Ten patients with left-sided breast cancer treated with breast-conservation therapy at the University of California, San Francisco, were included in this study. Each patient received an electron boost to the surgical bed to a dose of 1,600 cGy in 200 cGy fractions prescribed to 80% of the maximum. Doses to the left ventricle (LV) and the ipsilateral lung (IL) were calculated using the EGS4 MC system and the FEH algorithm implemented on the commercially available Pinnacle treatment planning system. An anthromorphic phantom was irradiated with radiochromic film in place to verify the accuracy of the MC system.

RESULTS

Dose distributions calculated with the MC algorithm agreed with the film measurements within 3% or 3 mm. For all patients in the study, the dose to the LV and IL was relatively low as calculated by MC. That is, the maximum dose received by up to 98% of the LV volume was < 100 cGy/day. Less than half of the IL received a dose in excess of 30 cGy/day. When compared with MC, FEH tended to show reduced penetration of the electron beam in lung, and FEH tended to overestimate the bremsstrahlung dose in regions well beyond the electron practical range. These differences were clinically likely to be of little significance, comprising differences of less than one-tenth of the LV and IL volume at doses > 30 cGy and differences in maximum dose of < 35 cGy/day to the LV and 80 cGy/day to the IL.

CONCLUSIONS

From our series, using clinical judgment to prescribe the boost to the surgical bed after breast-conserving treatment results in low doses to the underlying LV and IL. When calculated dose distributions are desired, MC is the most accurate, but FEH can still be used.

Electron dose calculation using multiple-scattering theory: energy distribution due to multiple scattering

In 1951, Yang derived formulas for computing the pathlength distribution of particles traversing foils, considering only the multiple-scattering process. We here improve upon the accuracy of that work, by using our second-order small-angle approximation. We derive the general solution for a broad parallel beam, and find simple formulas for Yang's two special cases: the pathlength distribution of all the particles at a particular point, taken together; and the pathlength distribution at a particular point of only those particles with zero net angular deflection. From the pathlength (or excess pathlength) distribution, residual range and energy distributions can immediately be deduced. All this work assumes relatively small energy loss, and we consider 5 MeV electrons penetrating lead, which provides considerable scattering without major energy loss. The second-order energy distribution is found to differ considerably from the (first-order) Yang energy distribution, and to agree more closely with EGS4 Monte Carlo calculations.

Electron dose calculations using the Method of Moments

The Method of Moments is generalized to predict the dose deposited by a prescribed source of electrons in a homogeneous medium. The essence of this method is (i) to determine, directly from the linear Boltzmann equation, the exact mean fluence, mean spatial displacements, and mean-squared spatial displacements, as functions of energy; and (ii) to represent the fluence and dose distributions accurately using this information. Unlike the Fermi-Eyges theory, the Method of Moments is not limited to small-angle scattering and small angle of flight, nor does it require that all electrons at any specified depth z have one specified energy E(z). The sole approximation in the present application is that for each electron energy E, the scalar fluence is represented as a spatial Gaussian, whose moments agree with those of the linear Boltzmann solution. Numerical comparisons with Monte Carlo calculations show that the Method of Moments yields expressions for the depth-dose curve, radial dose profiles, and fluence that are significantly more accurate than those provided by the Fermi-Eyges theory.

Dose calculations in proton beams: range straggling corrections and energy scaling

Three-dimensional dose planning systems employing accurate proton transport algorithms are essential for calculating absorbed dose distributions in proton therapy. In this paper, a pencil beam algorithm for the transport of protons in materials of interest for radiation therapy is developed. The Fermi-Eyges multiple-scattering theory is used to derive transport equations for calculating proton fluence and absorbed dose distributions. The multiple-scattering theory of Molière is used to predict mean square scattering angles and to develop an expression for calculating the root mean square (RMS) radial spread of a proton pencil beam, as a function of depth, in an arbitrary scattering material. A correction factor is suggested to account for the decrease in the radial spread at the end of the range due to range straggling. The effects of neglecting large-angle scattering events and the possibility of incorporating such events into the pencil beam algorithm are discussed. An energy scaling technique for determining the water-equivalent surface energy at a given depth in a heterogeneous scattering medium is developed. The water-equivalent energy, giving the same Molière scattering parameter B in water, is determined and the 1/e angle in water is scaled to the appropriate width in the scattering material. By using stored analytically or Monte Carlo calculated pencil beam distributions in water, the large-angle single-scattering events may be incorporated by approximating the scattering in an arbitrary material by the scattering in water for protons of the appropriate water-equivalent surface energy.