The Study of Field Equivalence Determined by the Modeled Percentage Depth Dose in Electron Beam Radiation Therapy

Abstract

Introduction. This study presents an empirical method to model the curve of electron beam percent depth dose (PDD) by using the primary-tail function in electron beam radiation therapy. The modeling parameters $N$ and $n$ can be used to predict the minimal side length when the field size is reduced below that required for lateral scatter equilibrium (LSE) in electron radiation therapy. Methods and Materials. The electrons’ PDD curves were modeled by the primary-tail function in this study. The primary function included the exponential function and the main parameters of $N$ and $\mu$ , while the tail function was composed of a sigmoid function with the main parameter of $n$ . The PDD of five electron energies was modeled by the primary and tail function by adjusting the parameters of $N$ , $\mu$ , and $n$ . The $R_{50}$ and $R_{\text{p}}$ can be derived from the modeled straight line of 80% to 20% region of PDD. The same electron energy with different cone sizes was also modeled by the primary-tail function. The stopping power of different electron energies in different depths can also be derived from the parameters $N$ , $\mu$ , and $n$ . Results. The main parameters $N$ and $n$ increase but $\mu$ decreases in the primary-tail function for characterizing the electron beam PDD when the electron energy increased. The relationship of parameter $n$ , $N$ , and $\ln \left(-\mu \right)$ with electron energy are $n=31.667E_0-88$ , $N=0.9975E_0-2.8535$ , and $\ln \left(-\mu \right)=-0.1355E_0-6.0986$ , respectively. Percent depth dose was derived from the percent reading curve by multiplying the stopping power relevant to the depth in water at a certain electron energy. The stopping power of different electron energies can be derived from $n$ and $N$ with the following equation: $\text{stopping}\text{power}=\left(-0.042\ln \left(N_{E_0}\right)+1.072\right)e^{\left(-{\text{n}}_{{\text{E}}_0}·5·{10}^{-5}+0.0381\right)·x}$ , where $x$ is the depth in water. The lateral scatter equivalence (LSE) of the clinical electron beam can be described by the parameters $E_0$ , $n$ , and $N$ in the equation of ${\text{S}}_{\text{eq}}={\left(n_{E_0}-N_{E_0}\right)}^{0.288}/{\left(E_0/n_{E_0}\right)}^{0.0195}$ . The LSE was compared with the root mean square scatter angular distribution method and shows the agreement of depth dose distributions within ±2%. Conclusions. The PDD of the electron beam at different energies and cone sizes can be modeled with an empirical model to deal with what is the minimal field size without changing the percent depth dose when approximate LSE is given in centimeters of water.