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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.
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