Health Assessment Document for Diesel Emissions - NSCEP | US ...

1 and
2
m.(t) = (y2i - YI/exp[Ap)a/Yzi - yli)]
J ((1 - Y 1 i) + (y 2 i - l)exp[Ai(t)a/y 2 i - y 1 )])2'
3 where y lj < Yzj are two real roots of ajl- (aj + Pj + 1-Lzjqj)Y + Pj = 0; aj + pj + 1-Lzj = 1; qj = 1 -
4 Pm/amj,j = 1, 2, ... , k.
· 5 When exponential distribution (i.e., AjCt) = Yjt and qj = 1 are as.sumed, the model is
6 equivalent to the MVK model with piece-wise constant parameters. A special case that may be
7 more appropriate than the exponential distribution is when the Gompertz distribution is assumed
8 (i.e., when AjCt) = { 1 - exp[- Yjt]}/yj).
9 For the diesel alternative model, the total time is divided into five (i.e., k = 5)
1 0 subintervals. It is shown in Tan and Chen (1992) that, under the assumption of exponential cell
11 lifetime distribution, the tumor free distribution function, SxCt), can be written as
12
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where S· = t· ifj < k and S· = t.ifj = k, and
J J . J .
where,
211/98
Wr
Zr
Ll.ii(s,t)
k j-1
S(t) = exp{-L [Aitj-l' s) + L Aij(ti-l' s)J}
j =I i=l
=[(a+ p + IJ.zq)2- 4ap]ll2'
= a - p - !Jzq, and
= y i( t- s) if both s and t are in the same closed subinterval [ti-l, tj] and
B-24 DRAFT--DO NOT CITE OR QUOTE