Analytical theory for the fluence, planar fluence, energy fluence, planar energy fluence and absorbed dose of primary particles and their fragments in broad therapeutic light ion beams.

The purpose of the present work is to develop analytical expressions for the depth variation of the fluence, planar fluence, the energy fluence, planar energy fluence, the mean energy and absorbed dose of primary ions and their associated fragments in tissue-like media with ranges of clinical interest. The analytical expressions of the primary ions and associated fragments take into account nuclear interactions, energy losses, range straggling and multiple scattering. The analytical models of the radiation field quantities were compared with the results of the modified Monte Carlo (MC) code SHIELD-HIT(+). The results show that the shape of the depth absorbed dose distribution of the primary particles is characterized by an increasingly steep exponential fluence decrease with depth as the charge and atomic weight increase. This is accompanied by a compensating increased energy loss towards the Bragg peak as the charge of the ion increases. These largely compensating mechanisms are the main reason that the depth absorbed dose curve of all light ions is surprisingly similar. In addition, a rather uniform dose in the plateau region is obtained since the increasing fragment production almost precisely compensates the loss of primaries. The dominating light fragments such as protons and alpha particles are characterized by longer ranges than the primaries and their depth dose curves to some extent coincide well with the depth fluence curves due to a rather slow variation of mean stopping powers. In contrast, the heavier fragments are characterized by the build up of a slowing down spectrum similar to that of the primaries but with initially slightly shorter or longer ranges depending on their mass to atomic number ratio. The presented analytical theory for the light ion penetration in matter agree quite well with the MC and experimental data and may be very useful for fast analytical calculations of quantities like mean energy, fluence, energy fluence, absorbed dose, and LET.