  The well-known Bragg-Kleeman rule RCSDA = A dot E0p has become a pioneer work
in radiation physics of charged particles and is still a useful tool to
estimate the range RCSDA of approximately monoenergetic protons with initial
energy E0 in a homogeneous medium. The rule is based on the
continuous-slowing-down-approximation (CSDA). It results from a generalized
(nonrelativistic) Langevin equation and a modification of the phenomenological
friction term. The complete integration of this equation provides information
about the residual energy E(z) and dE(z)/dz at each position z (0 <= z <=
RCSDA). A relativistic extension of the generalized Langevin equation yields
the formula RCSDA = A dot (E0 +E02/2M dot c2)p. The initial energy of
therapeutic protons satisfies E0 << 2M dot c2 (M dot c2 = 938.276 MeV), which
enables us to consider the relativistic contributions as correction terms.
Besides this phenomenological starting-point, a complete integration of the
Bethe-Bloch equation (BBE) is developed, which also provides the determination
of RCSDA, E(z) and dE(z)/dz and uses only those parameters given by the BBE
itself (i.e., without further empirical parameters like modification of
friction). The results obtained in the context of the aforementioned methods
are compared with Monte-Carlo calculations (GEANT4); this Monte-Carlo code is
also used with regard to further topics such as lateral scatter, nuclear
interactions, and buildup effects. In the framework of the CSDA, the energy
transfer from protons to environmental atomic electrons does not account for
local fluctuations.
