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Appendix G DEVELOPMENT OF SOME EQUAT IONS USED FOR QUANTITATIVE RISK ASSESSMENT This appendix develops the equations necessary for arriving at the cohort-specific ad justment to calculate c in equation ( 12) of Chapter 7 from the values of the constant b in Table 7-1. The method of calculating the risk of mesothelioma mortality at age t for an exposure starting at age to is also given. In addition, justification is provided for the calculations in Chapter 7 that are used to determine the contribution to lifetime risk resulting from an exposure beginning at to. To begin, the instantaneous mortality (or hazard) function at age u for an exposure of dose level D incurred at age v is defined as follows: i(u,v,D) - aD(u-v)~~2, u > v, (Gl) where a ~ O and k ~ 2 are specified constants. D is the constant exposure level (concentration) over the period of exposure. By contrast, d in Chapter 7 and later in this appendix is the equivalent average continuous exposure level from time of first exposure until the time that all exposure ceases. It will become apparent from the following development that this equation for the instantaneous hazard from a brief constant exposure was selected to be consistent with equation 7 in Chapter 7. The hazard function i(u,v,D) means that the probability of death in the short time interval (u,u + Au) of length u from an exposure to dose D at prior time v is given by i(u,v,D)(au). If this is the case, then the cumulative mortality (i.e., cumulative hazard) up to time t from the exposure D starting at time v is given by I(t,v,D) = i i(u,v,D)du v = sum of instantaneous hazards from time v to t. 311 l (G2)

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312 Now consider the case of a continuous exposure of length Q starting at to, where exposure occurs during the time interval from to to to + Q. If one now calculates the cumulative mortality (hazard) at age t, it becomes I(t,to, Q,D) = ItO tI(t,v,D)dv o = NATO Ivi (u, v, D) dudv = sum of the instantaneous hazards of all u and v so that v ~ u < t and to < v ' to + Q . (G3) Using i~u,v,D) as given in equation (G1), one can calculate the integrals in equation (G3) as follows: I ~ t, to, Q. D) = bt t-to~k, with where c = a/k~k-l). b = cD{l-[ l- Q/(t-to) Ski, (G4) (G5) Novice that equation (G4) is in the form given by Peto et al. (1982), who estimated k as 3. 2. The corresponding values of b for various worker cohorts are given in Table 7-1. Equation (GS) gives the correction term to obtain c in equation ( 12) with d = (0. 219)D = D/4. 56. The choice of d = (0. 219)D is justified as follows: equations (G1), (G4), and (GS) all assume a continuous daily exposure to dose D. Assuming a worker is employed 240 days per year at ~ hours per day, a rough est imate of a continuous 24-hour exposure to dose d based on an S-hour workday exposure D is given by d = (0. 219)D, since 0 219 = 8 x 240 Equation (12) in Chapter 7 is based on this adjustment to convert workday B-hour exposures to daily 24-hour exposures along with the adjustment shown in equation (G5) for a partial exposure of length Q from to to to ~ Q , as compared to a continuous exposure to to t, i.e., of duration it - to). If Q = t - to, equation (G4) can be simplified to equation (7) with D replac ing d . Equations (G4) and (G5) also provide the framework for the risk assessments in Chapter 7, which are based on partial exposures at levels higher than the assumed environmental level d = 0.002 fibers/cm3.

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313 REFERENCE Peto, J., H. Seidman, and I. J. Selikoff. 1982. Mesothelioma mortality in asbestos workers: Implications for models of carcinogenesis and risk assessment. Br. J. Cancer 45:124-135. ill 1, ',