Solution of the Boltzmann equation for primary light ions and the transport of their fragments

Abstract

The Boltzmann equation for the transport of pencil beams of light ions in semi-infinite uniform media has been calculated. The equation is solved for the practically important generalized 3D case of Gaussian incident primary light ion beams of arbitrary mean square radius, mean square angular spread, and covariance. The transport of the associated fragments in three dimensions is derived based on the known transport of the primary particles, taking the mean square angular spread of their production processes, as well as their energy loss and multiple scattering, into account. The analytical pencil and broad beam depth fluence and absorbed dose distributions are accurately expressed using recently derived analytical energy and range formulas. The contributions from low and high linear energy transfer (LET) dose components were separately identified using analytical expressions. The analytical results are compared with SHIELD-HIT Monte Carlo (MC) calculations and found to be in very good agreement. The pencil beam fluence and absorbed dose distributions of the primary particles are mainly influenced by an exponential loss of the primary ions combined with an increasing lateral spread due to multiple scattering and energy loss with increasing penetration depth. The associated fluence of heavy fragments is concentrated at small radii and so is the LET and absorbed dose distribution. Their transport is also characterized by the buildup of a slowing down spectrum which is quite similar to that of the primaries but with a wider energy and angular spread at increasing penetration depths. The range of the fragments is shorter or longer depending on their nuclear mass to charge ratio relative to that of the primary ions. The absorbed dose of the heavier fragments is fairly similar to that of the primary ions and also influenced by a rapidly increasing energy loss towards the end of their ranges. The present analytical solution of the Boltzmann equation accurately accounts for the loss of primary particles as well as their energy losses and multiple scattering. At the same time these quantities for the fragments are also accurately derived as based on the generalized Gaussian solution of the primaries and compared both with Monte Carlo and experimental data. The results are useful for fast transport calculations and biologically optimized therapy planning with light ion beams.

Analytical Solutions for the Pencil-Beam Equation with Energy Loss and Straggling

“…The analysis performed in Section 2.2 shows that the results by Kempe and Brahme (2010) are essentially valid. They introduced a factor ε(E), which was then assumed to be zero without further motivation.…”

“…Therefore, there have been attempts at deriving correction factors for the Fermi-Eyges solution to increase the accuracy of predictions. One such factor was derived by Kempe and Brahme (2010) to include absorption and energy loss in the model. However, the derivation by Kempe and Brahme was somewhat unclear mathematically and the underlying assumptions were not apparent.…”

Ion transport in inhomogeneous media based on the bipartition model for primary ions

“…[28]. Therefore although, the fact that, in this model we do not consider any absorption term, as given in [30,31], the bipartition model for ion transport, under continuously slowing down assumption, can in the first approximation be related to transport of a therapeutic proton beam. The bipartition model was therefore applied in a case of a therapeutic 70 MeV/u and 202 MeV/u proton beam.…”

“…X -ray) and charged (electron and ion) particle beams are extensively used in radiation therapy both for early cancer detection and dose computations/algorithms see, e.g. [24][25][26][27][28][29][30][31][32][33][34][35].…”

Spherical Harmonics and a Semidiscrete Finite Element Approximation for the Transport Equation

“…In this regard, for example, Prinja and Pomraning (1992) considered asymptotic scaling for forward-peaked transport, Börgers and Larsen (1996) derived the Fermi pencil beam equation, Asadzadeh and colleagues (2010b) studied Galerkin methods for broad beam transport, Asadzadeh and associates (2010a) extended the bipartition model for high energy electrons by Luo and Brahme (1992) to high energy ions and inhomogeneous media, and finally Kempe and Brahme (2010) studied the solution of the Boltzmann equation for light ions. In all these studies ion particles are considered to be normally incident at the boundary of a semi-infinite medium.…”