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. Moreover, experimental and clinical data can be used to explore the key factors through mathematical analysis. 
Approach: 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.
Main results: 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 (CONV) irradiation to ensure the acquisition of effective experimental data across a wide dose range; pipelines of fitting parameters and predicting the FLASH effect.
Significance: This study establishes the quantitative relationship between the FLASH effect and key factors. 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 FLASH effect. Furthermore, this study indicates the key factors of FLASH effect and offers clues to further explore the FLASH mechanism.