Background: Treatment verification with PET imaging in charged particle therapy is conventionally done by comparing measurements of spatial distributions with Monte Carlo (MC) predictions. However, decay curves can provide additional independent information about the treatment and the irradiated tissue. Most studies performed so far focus on long time intervals. Here we investigate the reliability of MC predictions of space and time (decay rate) profiles shortly after irradiation, and we show how the decay rates can give an indication about the elements of which the phantom is made up. Methods and Materials: Various phantoms were irradiated in clinical and near-clinical conditions at the Cyclotron Centre of the Bronowice proton therapy centre. PET data were acquired with a planar 16x16 cm$^2$ PET system. MC simulations of particle interactions and photon propagation in the phantoms were performed using the FLUKA code. The analysis included a comparison between experimental data and MC simulations of space and time profiles, as well as a fitting procedure to obtain the various isotope contributions in the phantoms. Results and conclusions: There was a good agreement between data and MC predictions in 1-dimensional space and decay rate distributions. The fractions of $^{11}$C, $^{15}$O and $^{10}$C that were obtained by fitting the decay rates with multiple simple exponentials generally agreed well with the MC expectations. We found a small excess of $^{10}$C in data compared to what was predicted in MC, which was clear especially in the PE phantom.
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