Biomedical Engineering Reference
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a time homogeneous 2-stage model, (c) a time homogeneous 3-stage model,
(d) a time homogeneous 4-stage model, and (e) a time homogeneous 5-
stage model. In models (b)-(e), the mutation rates, the birth rates and the
death rates are assumed to be independent of time. In model (a), while the
mutation rates are independent of time, we assume the birth rate and the
death rate on initiated cells are 2-step piece-wise non-homogeneous with
t
1
= 60 years old as the cut-o time point.
To assess eects of dose level, we let x
i
= log(1 + u
i
), where u
i
is
the mean dose of the ith dose level. Then, based on some preliminary
analysis
63
we assume a Cox regression model for the mutation rates and
assume linear regression models for the birth rates and the death rates.
Thus, we let
0
(i; t) = a
00
(t)e
a
01
(t)x
i
;
j
(i) = a
j0
e
a
j1
x
i
for all models; but
letfb
j
(i) = b
j0
+ b
j1
x
i
; d
j
(i) = d
j0
+ d
j1
(i)x
i
gfor the time homogeneous
models and letfb
1
(i; s) = b
10
(s) + b
11
(s)x
i
; d
1
(i) = d
10
(s) + d
11
(s)x
i
gwith
s = 1 for tt
1
and s = 2 for t > t
1
for the two-stage time nonhomogeneous
model.
Applying the procedures in Section 5, we have estimated the parameters
and tted the data in Table 1. The AIC and BIC values of all models as well
as the p-values for testing goodness of t of the models are given in Table 2.
The p-values are computed using the approximate probability distribution
results
P
m
i=1
P
j=1
(y
ij
^
ij
)
2
=
^
ij
2
(mnk), for large mn, where k is
the number of parameters estimated under the model. The estimates for the
unknown parameters given the 4-stage model are given in Table 3. From the
p-values of the models, apparently that all models except the 5-stage model
t the data well, but the values of AIC and BIC suggested that the 4-stage
model is more appropriate for the data. This 4-stage model seems to t
the following molecular biological model for squamous cell lung carcinoma
proposed recently by Wistuba et al.
67
: Normal epithelium!Hyperplasia
(3p/9p LOH, Genomic Instability)!Dysplasia (Telomerase dysregulation)
!In situ Carcinoma (8p LOH, FHIT gene inactivation, gene methylation)
!Invasive Carcinoma (p53 gene inactivation, k-ras mutation).
From results given in Tables 3, we observe the following interesting
results:
n
(1) The estimates of
0
(i) increases as the dose level increases. This
indicates the tobacco nicotine is an initiator. From molecular bio-
logical studies, this initiation process may either be associated with
the LOH (loss of heterozygosity) of some suppressor genes from
chromosomes 3p or silencing of these genes by epigenetic actions
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