We study the spatial distributions of $beta^+$-activity produced by therapeutic beams of $^3$He and $^{12}$C ions in various tissue-like materials. The calculations were performed within a Monte Carlo model for Heavy-Ion Therapy (MCHIT) based on the GEANT4 toolkit. The contributions from $^{10,11}$C, $^{13}$N, $^{14,15}$O, $^{17,18}$F and $^{30}$P positron-emitting nuclei were calculated and compared with experimental data obtained during and after irradiation. Positron emitting nuclei are created by $^{12}$C beam in fragmentation reactions of projectile and target nuclei. This leads to a $beta^+$-activity profile characterised by a noticeable peak located close to the Bragg peak in the corresponding depth-dose distribution. On the contrary, as the most of positron-emitting nuclei are produced by $^3$He beam in target fragmentation reactions, the calculated total $beta^+$-activity during or soon after the irradiation period is evenly distributed within the projectile range. However, we predict also the presence of $^{13}$N, $^{14}$O, $^{17,18}$F created in charge-transfer reactions by low-energy $^3$He ions close to the end of their range in several tissue-like media. The time evolution of $beta^+$-activity profiles was investigated for both kinds of beams. Due to the production of $^{18}$F nuclide the $beta^+$-activity profile measured 2 or 3 hours after irradiation with $^{3}$He ions will have a distinct peak correlated with the maximum of depth-dose distribution. We found certain advantages of low-energy $^{3}$He beams over low-energy proton beams for reliable PET monitoring during particle therapy of shallow located tumours. In this case the distal edge of $beta^+$-activity distribution from $^{17}$F nuclei clearly marks the range of $^{3}$He in tissues.
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