A theoretical model for event statistics in microdosimetry. I: Uniform distribution of heavy ion tracks

In this work we describe a novel approach to solving microdosimetry problems using conditional probabilities and geometric concepts. The intersection of a convex site with a field of randomly oriented straight track segments is formulated in terms of the relative overlap between the chord associated with the action line of the track and the track itself. This results in a general formulation that predicts the contribution of crossers, stoppers, starters, and insiders in terms of two separate functions: the chord length distribution (characteristic of the site geometry and the type of randomness) and an independent set of conditional probabilities. A Monte Carlo code was written in order to validate the proposed approach. The code can represent the intersection between an isotropic field of charged particle tracks and a general ellipsoid of unrestricted geometry. This code was used to calculate the event distribution for a sphere as well as the expected mean value and variance of the track length distribution and to compare these against the deterministic calculations. The observed agreement was shown to be very good, within the precision of the Monte Carlo approach. The formulation is used to calculate the event frequency, lineal energy, and frequency mean specific energy for several monoenergetic and isotropic proton fields in a spherical site, as a function of the site diameter, proton energy, and the event type.