A mathematical approach to aspects of LET, micro- and nano-dosimetry in radiation therapy with photons and charged particles

TL;DR

This paper introduces a mathematical model based on reaction-diffusion equations to describe LET and RBE in micro- and nano-dosimetry, improving understanding of radiation effects in particle therapy.

Contribution

It generalizes the linear-quadratic model to account for dense ionizing radiation effects using a nonlinear reaction-diffusion approach, linking micro- and nano-dosimetry to therapy planning.

Findings

The model accurately describes RBE variations in proton and carbon ion therapy.

Parameters derived from Co60 data can be applied to dense ionizing particles.

Mathematical descriptions clarify nano-dosimetry effects in therapy planning.

Abstract

The radiation effects induced by Co60 serve as a reference system for the consideration of LET and RBE in normal and tumor tissue dose-effect relations are usually handled by the linear-quadratic model (LQ). This approximation excellently works up to the shoulder domain. In particle therapy we have strictly differ between RBE in the initial plateau and environment of the Bragg peak. Thus for protons LET and RBE of the initial plateau agree with Co60, whereas in the Bragg peak domain both properties are increased,, but RBE of SOBP only varies between 1.1 and 1.17. The RBE of carbon ions is increased once again Their dose-effect curves are much steeper with a rather small shoulder domain due to dense ionizing radiation effect. Thus protons are also dense ionizing in the Bragg peak region, but with rather smaller magnitude compared to carbon ions. A generalization of the LQ-model based on the nonlinear reaction-diffusion model is proposed to describe LET and RBE of dense ionizing particles, which accounts for properties of micro- and nano-dosimetry. A linear term of a reaction diffusion formula describes the destroy of cells, the nonlinear term is related to repair and the diffusion term accounts for the density of the radiation damages. Based on dose-effect properties of Co60 the parameters of dense ionizing particles can be determined and compared with measurement data. The local dense of radiation effects and their consequences in RBE and dose effect curves provide a key of understanding modern therapy planning with different modalities and properties of nano-dosimetry are interpreted by mathematical descriptions. The irradiation of spheroids is a feature of micro-dosimetry, whereas intracellular exposure refers to nano-dosimetry.