Mathematical models of radiation action on living cells: From the target theory to the modern approaches. A historical and critical review

• Several mathematical models were proposed to describe the survival curves of irradiated cells.
• The Linear-Quadratic model is the most used but its biological meaning is unknown.
• We revisit literature by providing clues for resolving the Linear-Quadratic model.

Cell survival is conventionally defined as the capability of irradiated cells to produce colonies. It is quantified by the clonogenic assays that consist in determining the number of colonies resulting from a known number of irradiated cells. Several mathematical models were proposed to describe the survival curves, notably from the target theory. The Linear-Quadratic (LQ) model, which is to date the most frequently used model in radiobiology and radiotherapy, dominates all the other models by its robustness and simplicity. Its usefulness is particularly important because the ratio of the values of the adjustable parameters, α and β, on which it is based, predicts the occurrence of post-irradiation tissue reactions. However, the biological interpretation of these parameters is still unknown.Throughout this review, we revisit and discuss historically, mathematically and biologically, the different models of the radiation action by providing clues for resolving the enigma of the LQ model.