This paper focuses on the analytic modelling of responses of cells in the body to ionizing radiation. The related mechanisms are consecutively taken into account and discussed. A model of the dose- and time-dependent adaptive response is considered, for two exposure categories: acute and protracted. In case of the latter exposure, we demonstrate that the response plateaus are expected under the modelling assumptions made. The expected total number of cancer cells as a function of time turns out to be perfectly described by the Gompertz function. The transition from a collection of cancer cells into a tumour is discussed at length. Special emphasis is put on the fact that characterizing the growth of a tumour (i.e., the increasing mass and volume) the use of differential equations cannot properly capture the key dynamics - formation of the tumour must exhibit properties of the phase transition, including self-organization and even self-organized criticality. As an example, a manageable percolation-type phase transition approach is used to address this problem. Nevertheless, general theory of tumour emergence is difficult to work out mathematically because experimental observations are limited to the relatively large tumours. Hence, determination of the conditions around the critical point is uncertain.
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