Issues in Risk Assessment (1993)
Chapter: 3.4 Model Dependency
To explore the extent to which this result is influenced by sample size, similar calculations were performed for a series of experiments with sample sizes ranging from n=50 to n=1000 animals per group (Table 1). These results indicate that the correlation between the log10(TD 50) and the log10(MTD) remains high, even at the larger sample sizes for which the allowable range of potency values becomes much wider. Even in the limiting case of n = ∞, we have ρ = 0.944 (see annex D).
3.4Model Dependency
Bernstein et al. (1985), Crouch et al. (1987), and Reith and Starr (1989ab) all used a one-stage model to characterize the carcinogenic potency. The one-stage model does not accommodate the majority of dose-response curves which exhibit curvilinearity. Kodell et al. (1990) argue that this limits the range of estimates of potency, and so artificially increases the correlation between the estimates of potency and the
TABLE 1 Correlation Between Carcinogenic Potency and the Maximum Tolerated Dose as a Function of Sample Sizea
Sample Size n | Range of Experimental Outcomes, x/n | Range of Potency Estimates (upper limit ÷ lower limit) | Correlationc | |
Minimum | Maximum | |||
50 | 0.200 | 0.98 | 32 | 0.965 |
100 | 0.150 | 0.99 | 79 | 0.957 |
500 | 0.122 | 0.998 | 247 | 0.950 |
1000 | 0.116 | 0.999 | 349 | 0.949 |
∞b | 0.1 | 1.0 | ∞ | 0.944 |
aBased on a one-stage model and the assumptions in annex D. bLimiting case as n → ∞. cρ=Corr(logTD50,log10MTD) | ||||
MTD. Under the one-stage model, the potency ß is related to the dose D = MTD and the added risk R(D) by
For a population of chemicals, this relationship provides a linear regression of logeß versus loge(1/D) with a slope of unity. The error term, loge[-loge(1 - R(D))], expresses the variation in R(D). Since the extra risk at the MTD is likely to fall in the range of 0.10 to 0.98, the variation about 1/MTD is limited to a range of approximately loge[-loge(1 - 0.10)] = -2.25 to loge[-loge(1 - 0.98)] = 1.36. This corresponds to the approximate 30-fold range noted by Bernstein et al. (1985). Using (2.3), the relationship in (3.3) may be re-expressed as
so that the TD50 is restricted to this same range.
Kodell et al. (1990) suggest relaxing the linear restraints of the one-stage model and using the Weibull model in (2.5) to accommodate curvature. The Weibull model includes the one-stage model as a special case when k = 1. The TD50 and MTD are related by
where k varies from chemical to chemical to accommodate either convexity (upward curvature, k > 1) or concavity (downward curvature, k < 1) in dose-response. This permits additional variation and reduces the correlation between loge(TD50) and loge (MTD) obtained with k fixed at unity. Bailar et al. (1988) demonstrate that an appreciable portion of the National Cancer Institute/National Toxicology Program bioassays exhibit downward curvature (k < 1) (cf. Williams & Portier, 1992). Clearly, the one-stage model limits the values of potency estimates to a rather narrow range determined by the MTD, which contributes to the observed correlation between loge(TD50) and loge (MTD). We expect that the MTD will still be highly correlated with the TD50 derived from a Weibull model, although the degree of correlation will be somewhat less than is observed with the one-stage model.
This expectation is confirmed by the correlation coefficients between
