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The well-known Bragg-Kleeman rule RCSDA = A dot E0p has become a pioneer workin radiation physics of charged particles and is still a useful tool toestimate the range RCSDA of approximately monoenergetic protons with initialenergy E0 in a homogeneous medium. The rule is based on thecontinuous-slowing-down-approximation (CSDA). It results from a generalized(nonrelativistic) Langevin equation and a modification of the phenomenologicalfriction term. The complete integration of this equation provides informationabout the residual energy E(z) and dE(z)/dz at each position z (0 <= z <=RCSDA). A relativistic extension of the generalized Langevin equation yieldsthe formula RCSDA = A dot (E0 +E02/2M dot c2)p. The initial energy oftherapeutic protons satisfies E0 << 2M dot c2 (M dot c2 = 938.276 MeV), whichenables us to consider the relativistic contributions as correction terms.Besides this phenomenological starting-point, a complete integration of theBethe-Bloch equation (BBE) is developed, which also provides the determinationof RCSDA, E(z) and dE(z)/dz and uses only those parameters given by the BBEitself (i.e., without further empirical parameters like modification offriction). The results obtained in the context of the aforementioned methodsare compared with Monte-Carlo calculations (GEANT4); this Monte-Carlo code isalso used with regard to further topics such as lateral scatter, nuclearinteractions, and buildup effects. In the framework of the CSDA, the energytransfer from protons to environmental atomic electrons does not account forlocal fluctuations.

Theoretical methods for the calculation of Bragg curves and 3D distributions of proton beams