Energy-range relation and mean energy variation in therapeutic particle beams

Analytical expressions for the mean energy and range of therapeutic light ion beams and low- and high-energy electrons have been derived, based on the energy dependence of their respective stopping powers. The new mean energy and range relations are power-law expressions relevant for light ion radiation therapy, and are based on measured practical ranges or known tabulated stopping powers and ranges for the relevant incident particle energies. A practical extrapolated range, Rp, for light ions was defined, similar to that of electrons, which is very closely related to the extrapolated range of the primary ions. A universal energy-range relation for light ions and electrons that is valid for all material mixtures and compounds has been developed. The new relation can be expressed in terms of the range for protons and alpha particles, and is found to agree closely with experimental data in low atomic number media and when the difference in the mean ionization energy is low. The variation of the mean energy with depth and the new energy-range relation are useful for accurate stopping power and mass scattering power calculations, as well as for general particle transport and dosimetry applications.