Background: Two long-standing obstacles to accurately modeling solid tumor response to fractionated radiotherapy have been (1) non-uniformity in cellular radiosensitivity, the cells of the tumor interior, including quiescent cells, being less radiosensitive, and (2) the need to accurately model repopulation dynamics. Methods: A new model was developed to address these obstacles. It assumes that the effect of a given course of treatment is best described by means of the “next-generation number (NGN)” concept of mathematical biology. Results: The model estimates the expected number (Xf) of indefinitely clonogenic cancer cells in existence, given a starting number (X0), following W weeks of treatment using daily fraction size D. By Markov’s Inequality, the tumor control probability (TCP) satisfies TCP ≥1−Xf , which relation is useful in the Xf << 1 regime. The model provides a clear-cut criterion by which D could be selected to minimize effects in normal tissues. It also provides insight into which types of tumor are best treated using hypofractionation. Model parameters could be identified from clinical TCP data, as has been done for the linear-quadratic model, or identified from clonogenic assay data. Conclusions: Ionizing radiation works, in the mathematical sense, to reduce the number of cancer cells present by modulating the fruitfulness of the process of cellular replication. This process, which usually results in the replacement of one indefinitely clonogenic cancer cell with two at the conclusion of each cell cycle, is rendered less fruitful than normal, so that a cell does not even replace itself, on average, when it attempts to replicate.
DOI: 10.13140/RG.2.2.31262.50241
Publication Date: 2026-05