Alpha dose modeling in diffusing alpha-emitters radiation therapy—Part I: single-seed calculations in one and two dimensions

Background: Diffusing alpha-emitters Radiation Therapy (“DaRT”) is a new method, presently in clinical trials, which allows treating solid tumors by alpha particles. DaRT relies on interstitial seeds carrying μCi-level ²²⁴Ra activity below their surface, which release a chain of short-lived alpha emitters that spread throughout the tumor volume primarily by diffusion. Alpha dose calculations in DaRT are based on describing the transport of alpha emitting atoms, requiring new modeling techniques. Purpose: A previous study introduced a simplified framework, the “Diffusion-Leakage (DL) model”, for DaRT alpha dose calculations, and employed it to a point source, as a basic building block of arbitrary configurations of line sources. The aim of this work, which is divided into two parts, is to extend the model to realistic seed geometries (in Part I), and to employ single-seed calculations to study the properties of DaRT seed lattices (Part II). Such calculations can serve as a pragmatic guide for treatment planning in future clinical trials. Methods: We derive a closed-form asymptotic solution for an infinitely long cylindrical source, and extend it to an approximate time-dependent expression that assumes a uniform temporal profile at all radial distances from the source. We then develop a finite-element one-dimensional numerical scheme for a complete time-dependent solution of this geometry and validate it against the closed-form expressions. Finally, we discuss a two-dimensional axisymmetric scheme for a complete time-dependent solution for a seed of finite diameter and length. Different solutions are compared over the relevant parameter space, providing guidelines on their usability and limitations. Results: We show that approximating the seed as a finite line source comprised of point-like segments significantly underestimates the correct alpha dose, as predicted by the full two-dimensional calculation. The time-dependent one-dimensional solution is shown to coincide to sub-percent-level with the two-dimensional calculation in the seed midplane, and maintains an accuracy of a few percent up to ∼2 mm from the seed edge. Conclusions: For actual treatment plans, the full two-dimensional solution should be used to generate dose lookup tables, similarly to the TG-43 format employed in conventional brachytherapy. Given the accuracy of the one-dimensional solution up to ∼2 mm from the seed edge it can be used for efficient parametric studies of DaRT seed lattices.