The linear–quadratic exponential model often underestimates the risks of high radiation doses, presumably because it describes only the processes of cell initiation and inactivation. A biologically plausible explanation for the persistence of large risks at high fractionated doses delivered during cancer radiotherapy is that cell proliferation during the time intervals between dose fractions and shortly after treatment stops negates much of the effect of cell inactivation. The recent models by Schneider and Kaser-Hotz59 and Schneider and Walsh60 address this issue by providing a semi-empirical or phenomenological treatment of cell proliferation (and possibly other relevant factors that alter the dose response at high doses).

In contrast to phenomenological models, the so-called initiation, inactivation, and proliferation models55–57,64,65 involve a relatively detailed mechanistic treatment of the three short-term processes that give the models their name throughout the course of a typical high-dose (tens of Gy) fractionated radiotherapy regimen and during a subsequent tissue recovery period of a number of weeks. For example, the model developed by Wheldon et al64 and Lindsay65 is based on familiar two-stage concepts: Normal stem cells can be mutated to a premalignant (initiated) state, either spontaneously or by radiation. These once-mutated cells can, with a certain probability, acquire the second mutation, which makes them fully malignant. In addition to elevating the initiation rate, radiation can inactivate (kill, ie, reproductively sterilize) both normal and initiated cells. A key concept in the model involves compensatory proliferation of both normal and initiated cells after some of these cells have been inactivated by radiation. The model is mainly deterministic, but an approximation for stochastic extinction of all initiated cells after high doses of radiation is made. If all initiated cells in the organ are killed, then only normal cells participate in repopulation.

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The initiation, inactivation, and proliferation modeling approach for solid tumors was further refined by Sachs and Brenner.55 Here initiation, inactivation, and proliferation processes were tracked in detail throughout the radiotherapy and subsequent tissue recovery periods, either deterministically or stochastically.56 In particular, cell proliferation of both normal and premalignant cells was calculated in the time gaps between dose fractions, as well as during a recovery period after radiotherapy ends. Eventual risk of cancer was assumed to be proportional to the number of premalignant (initiated) cells present after radiotherapy and recovery, by the time when homeostatic organ repopulation has been completed.

By contrast with the short-term models described so far, another class of biologically motivated quantitative models can be characterized as long-term, in the sense that they track carcinogenesis mechanisms throughout the entire human or animal life span, eg, the Armitage–Doll model,39,66 the Moolgavkar–Venzon–Knudson two-stage clonal expansion (TSCE) model,67–69 the two-stage logistic model,70 and many others.71–76 The main advantages of long-term models are 1) including the modulation of the radiation dose response during the long latency period between radiation exposure and diagnosis of cancer and 2), the fact that radiation carcinogenesis is usually treated as just a perturbation of background carcinogenesis, so that extensive data on spontaneous cancers can be used to help determine the adjustable parameters needed to estimate cancer risks. The main disadvantage is that the early radiation response is typically treated in a less mechanistic, more phenomenological, manner than in the short-term models, and similarly for the dose-rate response. This disadvantage can be partially alleviated by suitable semi-mechanistic assumptions about the short-term aspects of the dose response, eg, the effects of cell repopulation, which partially compensate for cell killing by radiation,77 and cell–cell interactions, which can accelerate the proliferation of premalignant cells causing radiogenic promotion.

This class includes the earliest of the commonly used mathematical models of carcinogenesis – the pioneering models of Nordling38 and Armitage and Doll.39 They are based on the concept that cancer originates from an ancestral target somatic cell, whose lineage has accumulated several relevant alterations (ie, changes that are passed on to daughter cells). In current applications of these models, the target cells are usually thought of as organ-specific stem cells; the relevant alterations are thought to be mutations occurring in oncogenes and tumor suppressor genes, though chromosome rearrangements such as balanced translocations or inversions, copy number changes, or epigenetic changes are also sometimes discussed. Once a cell has accumulated all the necessary mutations, it becomes fully malignant and can subsequently (after some lag period) develop into a clinical cancer. Variants of the Armitage–Doll model intended for prediction of radiation-induced cancer risk have been applied to data sets such as the Japanese atomic bomb survivors.73,74

Two-stage models with clonal expansion are based on the broad paradigm of initiation, promotion, transformation, and progression in carcinogenesis, which has been applied to numerous studies of chemically induced and radiationinduced tumors in experimental animals. They are also sometimes motivated by the concept of “two-hit” recessive oncogenesis, developed by Knudson78 to describe the data on sporadic and inherited forms of human retinoblastoma. The most widely used representative of this class is the TSCE model.67,68 It assumes that a stem cell which has acquired a single relevant mutation has a slight growth or survival advantage relative to normal cells (eg, can proliferate at a faster rate and/or is more resistant to apoptotic signals). Over time, the growth advantage leads to clonal expansion of the mutated premalignant cell. When any cell within the clone acquires a second oncogenic mutation, it becomes a fully malignant cell, which can grow into a clinical cancer.

The TSCE model has been fitted to numerous data sets on spontaneous and carcinogen-induced tumors in animals and humans.79–82 Spontaneous cancer incidence in humans over the age range of 20–70 years is typically described very well, marginally better than by the previously discussed multistage models without clonal expansion, although the small differences in quality of fit are typically not statistically significant. The mechanistic implications are, however, different. In the models without clonal expansion, the “slope” of the spontaneous age-dependent cancer incidence curve is determined by the number of stages (mutations) on the pathway toward cancer. In the TSCE model, it is determined mainly by the net clonal expansion rate of initiated cells (ie, by the difference between proliferation and death/differentiation rates for these cells).

It is well known that cells in many tumors are genomically unstable. Many recent models of spontaneous carcinogenesis focus on genomic instability.73,74 It has also been included in models of radiation-induced cancers.71,72,75 Typically, in such models genomic instability is added to a multistage framework that essentially represents an extension of the concepts used in both the stochastic TSCE model and the Armitage–Doll model.

The incidence of typical adult-onset solid tumors rises quite steeply with age in the age range of 20–70 years. At older ages, however, the increase in incidence slows down and, for some cancers (eg, breast, lung), is apparently reversed – incidence decreases for ages >80 (eg, Surveillance, Epidemiology, and End Results database, Similar trends are seen in some animal cancer data sets.83

The reasons for these old age phenomena are not fully understood. Variations in spontaneous carcinogenesis rates between individuals can also be important: If some individuals are more susceptible to getting cancer, eg, because of some subtle defects in antineoplastic cell signaling or DNA repair capacity, they will get cancer at an earlier age. The oldest age groups will be depleted of these individuals and enriched for those with lower cancer susceptibility. There may also be true physiological reasons for a turnover in cancer incidence at old age. A likely mechanism is senescence of stem cells and/or deterioration of stem cell function, or niche function, with age.84–86