The Frequency Distribution of the Number of Ion Pairs in Irradiated Tissue

The statistical distribution of the number of ion pairs per ionizing event in a small volume simulating a tissue sphere was obtained by applying the Expectation-Maximization (EM) algorithm to experimental spectra measured by exposing a Rossi-type spherical proportional counter to γ radiation. The normalized experimental spectrum, r(x), which is the distribution of the number of ion pairs per event from both the primary track and the subsequent electron multiplication, can be represented as$\sum_{{\rm n}}p_{{\rm n}}\cdot f(n,x),\ \text{where the}\ f(n,x)\text{'}{\rm s}\ \text{for}\ n=1,2,3,\ldots $, n are the normalized spectra for exactly 1, 2, 3,..., n primary ion pairs and are calculated by convoluting the single-electron spectrum. The coefficients$p_{{\rm n}}$ represent the mixing proportions of the spectra corresponding to 1, 2, 3,..., n ion pairs in forming the experimental spectrum. The single-electron spectrum used in our calculations is the distribution of the number of ion pairs due to the multiplication process, and it is represented in analytical form by the Gamma distribution$f(1,x)=a\cdot x^{{\rm b}}\cdot {\rm e}^{{\rm cx}}$, where x is energy, usually in eV, and a, b and c are constants. The EM algorithm is an iterative procedure for computing the maximum likelihood or maximum a posteriori estimates of the mixing proportions$p_{{\rm n}}$, which we also refer to as the primary distribution of ion pairs in a microscopic spherical tissue-equivalent volume. The experimental and primary spectra are presented for simulated tissue spheres ranging from 0.25 to 8 μm in diameter exposed to⁶⁰ Co γ radiation.