  Clinical applications of FLASH radiotherapy require formulas to describe how
the FLASH radiation features and other related factors determine the FLASH
effect. Mathematical analysis of the models can connect the theoretical
hypotheses with the radiobiological effect, which provides the foundation for
establishing clinical application models. We abstract the complex models of the
oxygen depletion hypothesis and radical recombination-antioxidants hypothesis
into concise mathematical equations. The equations are solved to analyze how
the radiation features and other factors influence the FLASH effect. Then we
propose methodologies for determining the parameters in the models and
utilizing the models to predict the FLASH effect.The formulas linking the
physical, chemical and biological factors to the FLASH effect are obtained
through mathematical derivation of the equation. The analysis indicates that
the initial oxygen concentration, radiolytic oxygen consumption and oxygen
recovery are key factors for the oxygen depletion hypothesis and that the level
of antioxidants is the key factor for the radical recombination-antioxidants
hypothesis. According to the model derivations and analysis, the methodologies
for determining parameters and predicting the FLASH effect are proposed: the
criteria for data filtration, the strategy of hybrid FLASH and conventional
dose rate irradiation to ensure the acquisition of effective experimental data
across a wide dose range, the pipelines of fitting parameters and predicting
the FLASH effect. The derived formulas can be used to calculate the FLASH
effect in future clinical FLASH radiotherapy. The proposed methodologies guide
to obtain sufficient high-quality datasets and utilize them to predict the
FLASH effect. Furthermore, this study indicates the key factors of the FLASH
effect and offers clues to further explore the FLASH mechanism.
