  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.
