The uneven irradiation of a target cell and its dynamic movement can mathematically explain incubation period for the induction of cancer by internally deposited radionuclides.

Irradiation from internally deposited radionuclides induces malignant tumors. Ingested radionuclides accumulate in specific organs, which are irradiated over a lifelong period. Our aim is to elucidate why the development of malignant tumors requires long-term internal exposure, on the order of decades, despite the fact that irradiation is continuous over this period. Three major factors are considered to be responsible for the long incubation time in carcinogenesis caused by internally deposited alpha-emitters: uneven distribution of radionuclides, limited range of irradiation, and dynamic movement of tumor precursor cells. We hypothesized that target cells susceptible to malignant transformation may undergo one event by alpha particles and may then migrate outside of the range of alpha particles, thereby avoiding immediate induction of successive additional events that would lead to cell death or neoplastic changes. Based on this hypothesis, we further proposed a mathematical model to predict the relationship between dose rate and incubation period of tumors induced by internally deposited alpha-emitters. The function was non-linear and included terms of both direct and indirect radiation effects. It well fitted both human Th-ICC cases and rat Pu-induced lung cancer, suggesting that indirect radiation effects are independent from dose rate. The significance of parameters of the model is discussed.