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.
Highlights
The unique physical and biological properties of light ion beams make them ideal for use in radiation therapy when treating radiation resistant tumors using conventional low linear energy transfer (LET) beams [1]
To take full advantage of the clinical properties of light ion beams, the full three-dimensional spatial distribution is of vital importance in terms of fluence, absorbed dose, and radiation quality
An accurate 3D description of the primary ion transport is highly desirable, for light ion beams such as He, Li, Be, and B, where the greater part of the dose and practically all the high LET dose components are delivered by the primary particles and the absorbed dose due to their fragments is generally of minor importance
Summary
The unique physical and biological properties of light ion beams make them ideal for use in radiation therapy when treating radiation resistant tumors using conventional low LET beams [1]. More accurate and generally applicable analytical expressions, for the transport of the primary particles and their associated fragments, are still missing They are needed not least to allow fast calculation of radiation quality and absorbed dose distributions in the patient during light ion radiation therapy. The analytical solutions are combined with recently developed accurate energy loss, range relations, and multiple scattering expressions for the primary ions and their generated fragments [16,17]. An accurate 3D description of the primary ion transport is highly desirable, for light ion beams such as He, Li, Be, and B, where the greater part of the dose and practically all the high LET dose components are delivered by the primary particles and the absorbed dose due to their fragments is generally of minor importance. The results of the analytical calculations as well as the Monte Carlo ones are compared for clinically relevant cases and found to be in good agreement
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