A general scheme of short- and long-term processes governing the total number of pre-malignant cells. Details are discussed throughout the main text. As the individual ages, the number of viable pre-malignant cells grows, but the curve may turn over and decrease at very old age (blue line in the main graph). This pattern parallels background cancer incidence, since cancer risk is assumed to be proportional to the number of pre-malignant cells (discussed in the text). Radiation exposure (e.g., radiotherapy for an existing cancer) initially causes the number of pre-malignant cells to decrease due to cell killing (red line). After exposure stops, the irradiated tissues recover, allowing pre-malignant cells to repopulate and reach a number higher than was present before irradiation, i.e., a net excess radiogenic cancer risk is produced. Fluctuations in the number of pre-malignant cells throughout the irradiation and recovery periods (i.e., the short-term processes) are shown in the inset graph

Fits of the long-term part of our model to spontaneous incidence data for female breast and male lung cancers from SEER (the data for young ages were excluded because cancer incidence in these age groups is likely to be dominated by genetic predisposition). Error bars represent 95% confidence intervals. The different curves represent our default two-stage formalism and its multi-stage extensions described in the main text. These extended versions of the formalism do not alter the fit substantially

The effect of dose-fractionation on predicted excess relative cancer risk (ERR): as the same total radiation dose is split into fractions (one fraction per day, with gaps on weekends), thereby protracting it over a longer time, cancer risk predicted by the model increases. This occurs because cell repopulation during prolonged exposure partially compensates for cell killing by radiation. The doses refer to a given organ, such as the lungs or female breast, and not to whole body exposures. The following parameter values were used: b = 0.25 y⁻¹, X = 10.0 y × Gy⁻¹, Y = 0.5 Gy⁻¹, δ = 0.01 y⁻¹, Z = 10.0 cells/niche, α = 0.3 Gy⁻¹, β = 0.0 Gy⁻², λ = 0.35 day⁻¹, T_x = 30 y, T_y = 0.0 y, and L = 10 y

Modulation of promotion-driven excess relative cancer risk (ERR) over time after exposure: a low doses (either acute or fractionated produce similar results, since cell inactivation is negligible), b high fractionated doses (fractions are given once daily, with gaps on weekends). It is seen that, contrary to a proportional hazards assumption, not only the amplitude but also the shape of the dose–response curve is altered by long-term processes. The parameters were the same as in previous figures

Schematic representation of proposed pre-malignant cell kinetics. Each square represents a tissue niche. Cyan color normal cells; other colors pre-malignant cells—each color indicates a different clone. Black arrows invasion of an adjacent niche. The order of panels denotes a hypothetical time sequence: a only niches filled with normal cells are present at a young age. b Some niches filled with pre-malignant cells are spontaneously initiated (red and blue). c–e The pre-malignant clones expand over time by taking over adjacent niches. f Radiation exposure (lightning symbols) perturbs the system: some normal and pre-malignant cells are inactivated (niches become smaller), and some niches are completely inactivated (shown by gaps). Some new pre-malignant clones are initiated (green and brown). g After exposure, all niches filled with pre-malignant cells (red, blue, green, brown) expand in cell number (undergo promotion, shown by an increase in the size of the squares). Possible promotion of normal niches is neglected. Surviving normal and pre-malignant niches replace the niches inactivated by radiation (shown by disappearance of the gaps). h Over many years/decades after irradiation, roughly normal tissue architecture is restored. The promoting effect may disappear, as all niches return to the default size. Meanwhile, replication of existing pre-malignant niches continues (shown by arrows), and some new clones appear by spontaneous initiation (shown by the black square). i Subsequent mutations in one of the pre-malignant clones produce the first fully malignant cell, which grows into a malignant tumor (gray mass)

The typical shape for age dependence of background incidence for a specific type of adult-onset solid cancer is reasonably reproduced by our model. The curve was generated using the following parameter values: a = 1.0 × 10⁻⁸ y⁻², b = 0.25 y⁻¹, c = 1.75 × 10⁻³ y⁻², L = 10 y

Combined effects of age at exposure and time after exposure on radiation-induced excess relative cancer risk (ERR). The ERR is calculated at age 70, which is intended to approximate lifetime risk. As discussed in the text, only age at exposure matters if parameter δ = 0, but if δ > 0, time after exposure affects the ERR as well. The parameter values were the same as in Fig. 5, except for T_x and T_y, and a reference radiation dose of 1 Gy