If, on the contrary, the background incidence of a given tumor type is heavily influenced by host or environmental promoting factors (*e.g.*, breast cancer, stomach cancer), the effects of these factors can be thought to affect the expansion (increasing α to a value α_{r}, decreasing β to β_{r}, or both) of the clone of initiated or transformed cells and, thus, the expression of tumors. The resulting relative risk would then be of the form:

This formulation is independent of the magnitude of the effect of promoting factors on the cell division and mortality rates α_{r} and β_{r}. Hence the ERR can be exported directly from one population to another.

Expressed in simple terms, low-LET radiation (viewed here as a tumor initiator) will tend to act additively with other tumor initiators and multiplicatively with tumor promoters. Thus, in the case of a radiogenic tumor type with a strong influence of promoters (*e.g.*, stomach cancer), one would favor an RR transportation model, while in the case of a tumor with strong influence of initiators, one would favor an AR transportation model.

The preceding formulations are consistent with more general analyses of the nature of risk relationships involving exposure to two carcinogens (Kodell and others 1991; Zielinski and others 2001).

First consider fractionated acute exposures. If the relative risk due to the sum of *K* acute exposures of equal dose, *D* / *K*, administered at separate times, is the sum of the individual relative risks, and if an LQ dose-response model describes the effects at each fraction, then the total relative risk due to all *K* exposures is

Thus, for a given total dose *D*, the importance of the quadratic term diminishes with increasing number of fractions of exposure. The RR due to a protracted exposure may be thought of, at least approximately, as the limit as *K* approaches infinity. In this way, the total RR due to a protracted exposure is simply α*D*, where α is the linear coefficient in the LQ model. Therefore, if a risk estimate corresponding to a protracted exposure *D* is based on an LSS linear model, it should be adjusted to correspond to the linear component of the estimated LQ model, which is exactly what the DDREF presented in this chapter is designed to do.

Figure 10B-1 shows data from mouse experiments that fitted to the model above (data from Table 6 of Edwards 1992). These data show that the slope in the linear dose-response for chronic exposure approximates the linear component of the LQ model for acute exposure.

An LQ model for ERR or EAR, with curvature constrained to be θ, may be written as α_{LQ}[*Dose* + θ*Dose*^{2}]. A Bayesian statistical analysis is used to update information about dose-response curvature from animal carcinogenesis studies with the information concerning curvature from the LSS cohort of Japanese A-bomb survivors (over the dose range 0–1.5 Sv). A posterior distribution for LSS DDREF follows directly from this, via its definition as a function of θ. The LSS DDREF is essentially 1 + θ for the 0–1.5 Sv dose range and for values of θ of interest here. Pierce and Vaeth (1991) provide a more detailed discussion of this relationship over different dose ranges.

Two forms of animal experiment data were used to estimate curvature: estimated cancer risks and mean survival times (referred to as life-shortening data). These are two different summarizing results from the same experiments, so they are not independent but address the curvature in different ways. LQ models for risk as a function of dose can be estimated for each separate cancer and combined to form a single estimate of curvature, θ. On the other hand, the life-shortening studies ignore cause of death and therefore represent a cumulative effect of all radiation-induced deaths, the majority of which are cancer related. By using the relationship between survival rate and risk, the curvature of interest can be estimated from these, as detailed below.

The estimated risks of relevant cancers, plotted versus radiation dose in Figure 10B-2, were extracted from the summary tables of Edwards (1992), but exclude (1) the results in Tables 1 and 2 because those risk estimates were not adjusted for competing causes of death; (2) results for doses greater than 2 Gy; and (3) results on lymphomas, ovarian cancer, reticulum cell carcinoma, and nonmyeloid leukemias, because these are thought to arise via atypical biological mechanisms, as discussed in Chapter 3, or to reflect an ill-defined combination of cancer types. The risks presented here are based on acute exposures only.

There is substantial evidence that the curvature, θ, is not the same in all 11 situations (*p*-value < .0001, from a likelihood ratio test). Despite this evidence, the model with common curvature explains 97% of the variability and the model with different curvatures explains 98% of the variability in estimated risks, so the practical significance of the different curvatures may not be too important. Note in Figure 10B-2 that although the LQ curves seem to be highly divergent, the