Depth distributions of positron-emitting nuclei in PMMA phantoms are calculated within a Monte Carlo model for Heavy-Ion Therapy (MCHIT) based on the GEANT4 toolkit (version 8.0). The calculated total production rates of $^{11}$C, $^{10}$C and $^{15}$O nuclei are compared with experimental data and with corresponding results of the FLUKA and POSGEN codes. The distributions of e$^+$ annihilation points are obtained by simulating radioactive decay of unstable nuclei and transporting positrons in surrounding medium. A finite spatial resolution of the Positron Emission Tomography (PET) is taken into account in a simplified way. Depth distributions of $beta^+$-activity as seen by a PET scanner are calculated and compared to available data for PMMA phantoms. The calculated $beta^+$-activity profiles are in good agreement with PET data for proton and $^{12}$C beams at energies suitable for particle therapy. The MCHIT capability to predict the $beta^+$-activity and dose distributions in tissue-like materials of different chemical composition is demonstrated.
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