Mixture model for fast estimation of positron range

We present a mixture model of exponential distributions to describe the variation of the positron end-point in tissue. The physics of positron trajectories through tissue was simulated by a Monte-Carlo simulator based on elastic scattering from the nucleus, inelastic collisions with atomic electrons, hard elastic collisions producing delta electrons, and the positron emission energy spectra. Data from this comprehensive physics based Monte Carlo simulation was fed into the Expectation Maximization (EM) algorithm, and adapted to a binary mixture of exponential distributions. This binary mixture distribution provides a fast and accurate way to estimate positron-range for PET Monte Carlo simulation packages. For ¹⁸F and ¹⁵O point source simulations, the root mean square (rms) deviations within 2xFWHM between this mixture model and the full Monte Carlo simulation of positron endpoint probabilities was 4 and 7%, respectively.