An analytical model is presented that describes radiation-induced cellular inactivation in the presence of sublethal damage repair, cellular repopulation and redistribution in the mitotic cycle (the 3 Rs). The parameters of the model are measurable experimentally. Also taken into account are the initial age distribution of the cell population, the fact that subgroups of cells progress through the cycle at different speeds, the effects of a dose of radiation on the duration of the four phases of the cycle ( G1, S, G2, M), the possibility that a certain fraction of the cells are quiescent, and cell loss and/or cell removal from the proliferating population. Survival probabilities are expressed as linear-quadratic functions of dose where the coefficients α and β as well as the recovery constant (t0) are taken to depend on the position of the cell in the mitotic cycle. Explicit analytical expressions for inactivation probability are given for clonogenic cells exposed to continuous or fractionated radiation. Two model calculations are used to illustrate the formalism: in one, the redistribution of cells during fractionated therapy is examined. In the other calculation, it is shown that it is sufficient to take into account differences in proliferation rates and the change in the ratio α/β within the generation cycle for cells that may have otherwise equal response to acute exposures to explain that in a fractionated treatment protocol late-responding cells are more sensitive to the dose per fraction than early-responding cells. It is not necessary to invoke differences in radiosensitivity between these two classes of cells.

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