Radiobiological model for $\beta$-emitter radiopharmaceutical therapy in dynamic cell cultures in the framework of the ISOLPHARM project.

In medical physics and radiobiology, the most common method to probe the efficacy of radiation therapy approaches $in vitro$ is the cell survival trial. Recently, the traditional procedure for external beams has been extended by some groups to targeted radionuclides. In parallel, the bioengineering state of the art allows for the use of 3D tissue-mimicking scaffolds to obtain realistic cell cultures in dynamic conditions. The aim of this study is to implement a mathematical model for the assessment of $\beta$-emitting radiopharmaceuticals, considering their molecular kinetics $in vitro$ and how it affects the radiation delivery to cells. The molecular transitions will be assumed to fulfill the definition of Markov processes, while the cell survival will depend on the DNA damage, in competition with a logistic growth. The solutions of the resulting differential system will be evaluated and compared to numerical examples, $in silico$ simulations and literature data. This work belongs to the framework of the ISOLPHARM project, headed by INFN-LNL, which has the aim of developing innovative radiopharmaceuticals exploiting the Isotope Separation On-Line (ISOL) at the SPES facility.