Effective particle energies for stopping power calculation in radiotherapy treatment planning with protons and helium, carbon, and oxygen ions

The stopping power ratio (SPR) of body tissues relative to water depends on the particle energy. For simplicity, however, most analytical dose planning systems do not account for SPR variation with particle energy along the beam's path, but rather assume a constant energy for SPR estimation. The range error due to this simplification could be indispensable depending on the particle species and the assumed energy. This error can be minimized by assuming a suitable energy referred to as an 'effective energy' in SPR estimation. To date, however, the effective energy has never been investigated for realistic patient geometries. We investigated the effective energies for proton, helium-, carbon-, and oxygen-ion radiotherapy using volumetric models of the reference male and female phantoms provided by the International Commission on Radiological Protection (ICRP). The range errors were estimated by comparing the particle ranges calculated when particle energy variations were and were not considered. The effective energies per nucleon for protons and helium, carbon, and oxygen ions were 70 MeV, 70 MeV, 131 MeV, and 156 MeV, respectively. Using the determined effective energies, the range errors were reduced to. 0.3 mm for respective particle species. For SPR estimation of multiple particle species, an effective energy of 100 MeV is recommended, with which the range error is <= 0.5 mm for all particle species.