Theoretical methods for the calculation of Bragg curves and 3D distributions of proton beams
Abstract
The wellknown BraggKleeman rule R_{CSDA} = A ṡ E has become a pioneer work in radiation physics of charged particles and is still a useful tool to estimate the range R_{CSDA} of approximately monoenergetic protons with initial energy E_{0} in a homogeneous medium. The rule is based on the continuousslowingdownapproximation (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 ≦ R_{CSDA}). A relativistic extension of the generalized Langevin equation yields the formula R_{CSDA} = A ṡ (E_{0} + E/2M ṡ c^{2})^{p}. The initial energy of therapeutic protons satisfies E_{0} ≪ 2M ṡ c^{2}(M ṡ c^{2} = 938.276 MeV), which enables us to consider the relativistic contributions as correction terms. Besides this phenomenological startingpoint, a complete integration of the BetheBloch equation (BBE) is developed, which also provides the determination of R_{CSDA}, 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 MonteCarlo calculations (GEANT4); this MonteCarlo 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. Based on statistical quantum mechanics, an analysis of the Gaussian convolution and the LandauVavilov distribution function is carried out to describe these fluctuations. The Landau tail is derived as Hermite polynomial corrections of a Gaussian convolution. It is experimentally confirmed that proton Bragg curves with E_{0} ≧ 120 MeV show a buildup, which increases with the proton energy. This buildup is explained by a theoretical analysis of impinging proton beamlets. In order to obtain a complete dose calculation model for proton treatment planning, some further aspects have to be accounted for: the decrease of the fluence of the primary protons due to nuclear interactions, the transport of released secondary protons, the dose contribution of heavy recoil nuclei, the inclusion of lateral scatter of the primary and secondary protons based on Molière's multiplescatter theory, and the scatter contributions of collimators. This study also presents some results which go beyond proton dose calculation models; namely, the application of the relativistic generalization of the BraggKleeman rule to electrons and, in an appendix, a method to determine inelastic crosssections of therapeutic protons in media of therapeutic interest.
 Publication:

European Physical Journal Special Topics
 Pub Date:
 December 2010
 DOI:
 10.1140/epjst/e2010013357
 arXiv:
 arXiv:1008.3645
 Bibcode:
 2010EPJST.190....1U
 Keywords:

 Physics  Medical Physics;
 Nuclear Theory;
 Physics  Computational Physics
 EPrint:
 97 pages review paper