The monitoring of dose localization at carbon ion tumour therapy at GSI Darmstadt by means of BASTEI [1] requires the comparison between the measurement and a model calculation of the positron activity distributions [2]. For this a realistic description of the positron range distribution in tissue is required. For that purpose a new model is developed and an improved numerical implementation to the code is used.
Up to now the probability distribution of the positron emitter range is supposed to be a bilinear exponential function [3] with three parameters depending on the maximum positron energy. These parameters have been estimated for b⁺-endpoint energies up to 3.5 MeV, which is sufficient for PET applications to nuclear medicine. However, in the nuclear fragmentation reactions between the therapeutic carbon ion beam and the atomic nuclei of the tissue b⁺-emitters of much higher endpoint energy (up to 16.7 MeV) are produced for which the parametrization of [3] is not proved. Therefore, the positron range distributions have been calculated by means of GEANT [4] simulations for all positron emitting isotopes that may be produced by the fragmentation of ¹²C ions in tissue. The geometry used in the GEANT calculations consists of an isotropic point-like source in the centre of a water cube being sufficiently large for stopping all the positrons. The initial positron energy values were deduced from the b⁺-energy spectra. In Fig. 1 the projection of the 3D spatial distribution on an arbitrary oriented axis (denoted with x) obtained by GEANT is compared with those of Hasch [2] and Derenzo [3]. A much more realistic description of the positron transport in the calculation of the b⁺-activity distribution is expected if the 3D positron range distribution (Fig. 2) is applied. Now the distribution function method is used for sampling the range values. A cumulated distribution function (cdf) is constructed from the positron range distribution, which was derived from the GEANT results. The inverse function of cdf is stored in a lookup-table. This database is used for the sampling of the random quantity positron range by means of chosing equally distributed random numbers within the interval [0,1].

Fig. 1 Projected point spread function by Derenzo (dashed), Hasch (dotted) and GEANT (solid) for 11C		Fig. 2 Positron range distribution of random quantities implemented by Hasch (dotted) and by GEANT (solid) for 15O.

¹ TU Dresden, Institut für Kern- und Teilchenphysik

References

[1] W. Enghardt et al., Annual Report 1999, FZR-271 (1999) 89
[2] B. G. Hasch et al., Annual Report 1996, FZR-179 (1997) 87
[3] S. E. Derenzo, IEEE Trans. on Nucl. Science, Vol. 33, No.1 (1986) 565
[4] GEANT-Detector Description and Simulation Tool, CERN Prog. Lib. W 5013 (1994)

 IKH 06/25/01 © F. Pönisch