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Source: Crump et al. (1977); original source Walder et al., 1972, Food Cosmet. Toxicol. Vol. 11, pp. 415-432.

Source: Crump et al. (1977); original source Walder et al., 1972, Food Cosmet. Toxicol. Vol. 11, pp. 415-432.

carcinogen is to keep lifetime risk below 10~6 (0.0001%). However, for practical and economic reasons, the dosages in laboratory experiments are selected to produce effects with probability no lower than 10_1. The biological uncertainties with such an extrapolation are discussed below; here we discuss the mathematical aspects of the problem.

Table 19.5 and Figure 19.3 show a typical situation for the carcinogen dieldrin. The data are shown in the first and second columns of the table. The excess risk (the risk due to the toxin) is found by subtracting out the spontaneous risk (0.109 in this case). Cancer bioassays are commonly run at only two or three dosages, plus a control. In the example, the dosages at the three levels produced tumors in 18%, 43%, and 73% of the test animals. How, then, should one extrapolate to estimate the dosage that would produce tumors in 0.0001%? One of the simplest ways would be a linear extrapolation of the two lowest data points, (P1, d1) and (P2, d2). Thus, to compute the dose, d, resulting in risk P:

Or, simple linear regression could be used on all the data. Both of these approaches, however, often produce a threshold below which the probability of harm would be

Linear Threshold

Figure 19.3 Dose-response curve for a carcinogen and extrapolation to low dosages by linear and multistage models.

Linear Threshold

Figure 19.3 Dose-response curve for a carcinogen and extrapolation to low dosages by linear and multistage models.

predicted to be zero. For biological reasons described below, this model is ruled out for carcinogens. The next simplest model is linear extrapolation of the lowest data point to the origin:

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