Development and validation of an analytical model allowing accurate predictions of gamma and electron beam dose distributions in a water medical phantom

Abstract

The aim of this work was to study photon and electron dose distributions in a phantom filled with water using the Monte Carlo Geant4 tool for electron energies ranging from 1 to 21 MeV and for photon energies ranging from 1.25 MeV to 25 MeV, corresponding to conventional radiotherapy Linac energies. The results of the Geant4 calculations were validated based on the relevant experimental data previously published. The results obtained were fitted and analytical models of dose distributions were developed for gamma radiation and electrons. For each of these models, one-dimensional (including dose depth profiles as a function of the depth inside the phantom) and two-dimensional (including the dose distribution as a function of depth and lateral position inside the phantom) dose distributions have been considered. Results are presented for photons and electrons of various energies. The coefficient of determination $R^{2}$ illustrates an excellent match between the developed analytical model and the Geant4 results. It is demonstrated that the analytical models developed in the present study can be applied in various fields such as those used for calibration applications and radiation therapy. It is concluded that the analytical models developed allow for quick, easy and reliable clinical dose estimates and offer promising alternatives to the standard tools and methods used in radiotherapy for treatment planning.

Appendices

Appendix

Methods of calculation of the buildup factor B

There are currently several methods for calculating the buildup factor B. Among these methods, one can use:

One of the simplest and least accurate approximations, the linear formula given in Eq. (A1) (Turbey 1968):

$$B\left( {\mu x} \right) = 1 + k\left( {\mu x} \right),$$

(A1)

where x is the depth in the material of interest.

The linear formula is generally valid only over very limited distances. The value of k is readily derived from the original data by means of formulae. A2:

$$k = B\left( 1 \right) - 1,$$

(A2)

A good compromise between accuracy and computational complexity is offered by the Berger formula (Eq. A3) (Taylor 1954):

$$B\left( {\mu x} \right) = 1 + a\mu xe^{b\mu x} ,$$

(A3)

where coefficients a and b are described also in the reference (Taylor 1954).

The formula of Capo is given in Eq. (A4) (Capo 1958):

$$B\left( {E_{0} ,\mu x} \right) = \mathop \sum \limits_{i}^{3} \beta_{i} \left( {\mu x} \right)^{i} .$$

(A4)

Capo has derived values of the coefficients in Eq. (A4). The results given by Capo match Goldstein and Wilkins’s point source buildup factors (Goldstein et al. 1954) very closely over the whole range of distances involved (out to 15 or 20 mean free paths), and for energies from 0.5 to 10 MeV. In Eq. (A4), the coefficients β_i are also given Capo (1958).

The most precise formula, which is used in most cases, is the so-called Taylor formula (Eq. A5) (Berger 1956):

$$B = Ae^{{ - \mu x\alpha_{1} }} + \left( {1 - A} \right)e^{{ - \mu x\alpha_{2} }} ,$$

(A5)

where the coefficients A, α₁ and α₂ are not dependent on depth x. These coefficients do not have any physical significance. The coefficients A, α₁ and α₂ are described in Jaeger et al. (1968).