It was shown in the 1960s that the effect of radiotherapy could be determined by the duration of irradiation, the total dose, the fractionation dose, and the number of fractionations. The radiation dose (D) that achieves a certain effect is given as a function of the treatment period (T). This is determined by D _{∞} kT ^{c}, as proposed by L. Cohen ( Fig. 3 ). Subsequently, F. Ellis and C. G. Orton proposed a dose-fractionation relationship that provided an index for the tolerance dose of normal tissue. This relationship has been used for prescribing radiotherapy for a long time.

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After determining the dose-survival rate from the plating efficiency of cancer cells after radiation exposure, it was found that a linear quadratic model explained this relationship quite well: SF = Exp [−(αd + βd^{ 2} )] ( Fig. 4 ). Cells exposed to radiation can recover from sublethal damage (SLDR) with time, and this phenomenon is based on a repair mechanism. If it is thought that a cell recovered completely within each fractionation interval during fractionated radiotherapy, the survival rate after n rounds of irradiation should be SF = Exp [−n (αd + βd ^{2} )] ( Fig. 5 ). The biological effective dose is defined by BED = nd [1+d/(α/β)]. This was proposed as a formula to describe the biological effect of fractionated radiotherapy on the basis of this theory. This formula was changed so that the term in the exponent of the linear quadratic (LQ) model could be expressed in Gy.

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