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I-1

Aggregate Risk of Nonthreshold, Quantal, Toxic End Points Caused by

Exposure to Multiple Agents (Assuming Independent Actions)

The aggregate increased probability *P* of occurrence of any of *n* (presumed) nonthreshold end points caused by exposure to an environmental mixture of *m* toxic agents may be conveniently expressed under a few general assumptions. First, assume that the *m* agents are present in an environmental mixture at corresponding concentrations *C*i, where *i* = 1,2,…,*m*, each of which produce, in exposed people, corresponding lifetime, time-weighted average biologically effective dose rates * D*ij, each causing one or more of *n* quantal (all or none) toxic end points *T*j, where *j* = 1,2,…,*n* (see Figure I-1). Let *O*ij denote the occurrence of a particular *j*th end point * T*j induced by effective dose rate *D*ij, and assume that * T*j has a background occurrence probability of *p*j = Prob(*O*ij ¦ * D*=0) for total effective dose *D* due to all relevant agents and that *O*ij may arise only by events independent of those giving rise to either the background incidence rate of *T*j or to events *O*gh for any *g* and *h* such that *g*≠i, 1≤*g*≤*m*, * h*≠*j*, and 1≤*h*≤*n*. Finally, for very small values of *D*ij, assume that the corresponding increased probability of occurrence of the *T*j is defined by an independent ''one-hit" (nonthreshold, low-dose linear) function of *D*ij. In the following, ∩, ∪, and the overbar denote the logical union, intersection, and negation operations, respectively.

It follows from the stated assumptions and definitions that a *D*ij-induced increased probability *P*ij of * T*j occurrence, conditional on its independent background rate *p*j, is:

(1)