A finite element method for modeling diffusion of alpha-emitting particles in tissue.

Abstract

Diffusing alpha-emitters Radiation Therapy ("DaRT") is a promising new modality for the treatment of solid tumors. Interstitial sources containing 224 Ra are inserted into the tumor, producing alpha particles via the decay of 224 Ra and its daughters. The alpha particles are able to produce a "kill region" of several mm due to the diffusion of the alpha-emitting atoms. The Diffusion-Leakage (D-L) model has been proposed to describe the movement of the alpha-emitters used in DaRT in tumor tissue. To date, estimating the dose delivered under the D-L model has been accomplished with numerical solutions based on finite difference methods, namely DART1D and DART2D, as well as with asymptotic expressions for the long time limit. The aim of this work is to develop a flexible method of finite elements for solving the D-L model and to validate prior solutions of the D-L model. We develop a two-dimensional finite element solution to the D-L model implemented using the FEniCS software library. Our approach solves the variational formulation of the D-L equationson an unstructured mesh of triangular Lagrangian elements. We calculate the local dose in the mid- and axial planes of the source and validate our results against the one- and two-dimensional solutions obtained using the previously proposed numerical scheme, DART1D and DART2D. We use our model to estimate the change in dose in the source midplane as a function of the physical parameters used in the D-Lmodel. The local dose at the end of a 30 day treatment period estimated by our numerical method differs from DART1D and DART2D by less than 1% in the source midplane and less than 3% along the source axis over clinically relevant distances, with the largest discrepancies in high gradient areas where the Finite Element Method (FEM) mesh has a higher element density. We find that within current experimentally estimated ranges for D-L model parameters, the dose in the source midplane at a distance of 2mm can vary by over a factor of 3. The 2D finite element model reproduces the calculated dose obtained with DART1D and DART2D under the assumptions D-L model. The variation in predicted dose within current experimental ranges for model parameters suggests the necessity of further studies to better determine their statistical distributions. Finally, the FEM model can be used to calculate dose from DaRT in a variety of realistic 2D geometries beyond the D-Lmodel.