We have presented a biologically motivated mathematical model of background and radiation-induced cancer risk, and applied it to data on nine solid adult-onset second cancer types. The model takes into account initiation, inactivation, and repopulation of target stem cells (*iir* processes) throughout the comparatively short-term period of radiotherapy and recovery by a stochastic formalism. This formalism is integrated with a deterministic long-term two-stage carcinogenesis model, which follows the kinetics of pre-malignant stem cells throughout the entire human lifetime, before and after radiation exposure.

Such an approach, unifying short- and long-term models, has some advantages over currently existing methods, as discussed in the previous article. Briefly, our model allows mechanistic risk predictions to be made at high radiotherapeutic doses as well as at low doses, can track the age and time dependencies of risk mechanistically, and is qualitatively better at describing background cancer incidence at old ages than the commonly used two-stage clonal expansion (TSCE) model, with the same number of adjustable parameters. Radiation-induced risks are calculated analytically, using plausible assumptions about underlying biology.

The current model is an improvement over our previous models. For example, in our deterministic solid tumor model (Sachs and Brenner

2005), the shape of the ERR dose response was determined by the relative proliferation rate of pre-malignant stem cells compared with normal ones—parameter

*r*. To describe the data,

*r* had to be smaller than 1, e.g. 0.8, implying that pre-malignant cells proliferate more slowly than normal ones, at least over the short term, i.e. during radiotherapy and for some weeks afterwards. Such an interpretation is at odds with the long-term tendency of pre-malignant cells to clonally expand. In our stochastic analysis of the same problem (Sachs et al.

2007), this inconsistency was removed, and a fit to the data was possible even if pre-malignant cells were assumed to proliferate as fast as, or faster than, their normal counterparts, i.e.

*r* ≥ 1. However, the stochastic algorithm involved additional adjustable parameters. In the current model, repopulation of pre-malignant cells during radiotherapy is assumed to occur at the same rate as that of normal ones, i.e.

*r* = 1 implicitly. The reasoning is that any proliferative advantage that pre-malignant cells have manifests itself only on the scale of multiple years in humans. The model describes the data adequately, so there is no need for

*r* to be smaller than 1, and no need for additional parameters.

Our model applied to second cancer data can also provide some insight into the underlying biological mechanisms of carcinogenesis. An important outcome of our analysis is the finding that for many cancer types, radiation-induced risk, especially at ages >20, may be dominated by promotion, rather than initiation. Such an important role of promotion in low-LET radiation-induced risk for many cancers is consistent with the findings of other authors using the TSCE model on Japanese atomic bomb survivor data (e.g. Heidenreich et al.

2007). Our results indicate that promotion is the dominant mechanism of radiogenic risk for lung, colon, rectal, pancreatic, bladder, and CNS tumors; initiation and promotion both contribute to stomach and breast cancer risk; and thyroid cancer risk is dominated by initiation (see best-fit parameter values in Table ). For example, we estimated that a 1 Gy dose leads to almost a doubling in the number of existing pre-malignant cells in the breast. Of course, this conclusion about the importance of radiogenic promotion is dependent on model assumptions, and needs to be tested experimentally.

For most cancers analyzed here, promotion appears to be permanent, i.e. pre-malignant niches are not reduced in size or number over time after irradiation. However, a gradual reduction in niche size after exposure, at a rate of ~1% per year, is suggested for lung cancer, and at a slower rate for some other cancers. These findings indicate that the radiation-induced hyper-proliferation of pre-malignant cells may be either permanent, or transient, depending on the organ.

The

*iir*-based models in general, including the one described here, produce a dose response that has the same basic shape, with a maximum ERR at some intermediate dose, as the shape generated by older linear–quadratic–exponential (LQE) models, which neglected repopulation by cell proliferation (Little

2001; Bennett et al.

2004; Dasu et al.

2005). The main difference is in the dose range at which the maximum occurs. Models without repopulation predict that maximum ERR would occur at relatively low doses (below 5 Gy), and at typical clinically used doses (above 30 Gy) ERR would be essentially zero, due to the inactivation of nearly all target stem cells. In

*iir* models, repopulation of both normal and pre-malignant stem cells between dose fractions compensates for much of the inactivation. Consequently, ERR peaks at much higher doses, e.g. 20–60 Gy. This prediction is much more consistent with the recent epidemiological data, which indicates that ERR can remain substantial even at doses as high as 40–50 Gy (e.g. Travis et al.

1996,

1997,

2002,

2003,

2005).

In the context of the model presented here, the radiation dose at which ERR peaks is determined by the carrying capacity for pre-malignant cells per niche, *Z*, in addition to the cell killing parameters α and β and the repopulation rate λ. Physiologically, *Z* can be interpreted as an estimate for the number of stem cells that contribute to repopulation within a given location of the target organ. For most cancers analyzed here, the best-fit values of *Z* (Table ) suggested that up to several thousand stem cells may cooperate for local repopulation. The exceptions were breast and CNS cancers, where the upper bound of the 95% CI for the best-fit value of *Z* is infinity, suggesting that all the stem cells in the entire organ may participate in repopulation after irradiation.

The ability to reasonably predict cancer ERR at both low and high radiation doses using a biologically based mathematical model, incorporating formalisms for both short-term (

*iir*) and long-term processes, should enable this model to be used for optimization of radiotherapy protocols, by introducing second cancer risk as an additional criterion. This can be done if a dose-volume histogram for the protocol is available, as is usually the case (e.g. Hodgson et al.

2007; Koh et al.

2007).

Almost by definition, all mathematical models are greatly simplified representations of complex and incompletely understood biological processes. Our model has the main inherent drawbacks of other two-stage and *iir* carcinogenesis formalisms; extension to more stages and processes may improve biological realism, but at the cost of extra parameters. For example, the model can be extended by treating long-term clonal expansion stochastically instead of deterministically, by incorporating more detailed cell–cell interactions other than just a slowdown in net cell proliferation after filling of a pre-malignant stem cell niche, by allowing the number of pre-malignant cells at birth to be greater than 0, by accounting for variability in the lag period for the development of clinical cancer after the appearance of the first malignant cell, etc.

In the future, we intend to apply the model to additional second cancer data sets, where only a single estimate of radiation dose is available, but more information can be gained about age/time dependencies of radiation-induced risk. For example, some such studies with long follow-up times and large numbers of patients (e.g. Chaturvedi et al.

2007; Brown et al.

2007) suggest that these age/time dependencies may have different trends from the age/time dependencies suggested by Japanese atomic bomb survivors data, which is used here. Using data on older radiotherapy protocols optimally will be an important step in credibly projecting second cancer risks of current and future protocols decades into the future.