Page 156 - 20dynamics of cancer

THEORY II 141

For carcinogen experiments, Druckrey and others have noted an ex-
cellent linear fit on log-log scales between the median time to tumor, m,
and dosage, d, such that

log (m) = k 1 − (1/s) log (d) ,

which means that, in the form usually given in publications,

s
k 1 = dm .
To use these empirical relations in the incidence formulae above, where
n
patterns depend on t and on m, we can use s = n/r, thus
k 1 = dm n/r

and
r/n
m = (k 1 /d) .
Substituting for m in our previous formulae,

r n
− ln (0.5)d t r n
CI = μ(t) = r = k 2 d t ,
k 1
which suggests that cumulative incidence depends on the rth power of
dose and the nth power of age, with k values fit to the data.
Note that if d = 0, this formula for incidence suggests no cancer
in the absence of carcinogen exposure. If there is a moderate to high
dosage, then almost all cancers will be excess cases induced by carcino-
gens. However, one may wish to correct for background cases, either
r
by interpreting CI as excess incidence or by substituting (d + δ) for d,
where δ> 0 explains the background cases.

CONCLUSIONS
This section provided the technical details to analyze experimental
studies of carcinogens. Those studies measure the relation between
tumor incidence and age at different dosage levels. The analysis then
estimates the effect of dosage on the time to tumor development. Most
studies fit well to a model in which the cumulative incidence up to age
r n
t rises with d t , where d is dose, t is age, the exponent r is the log-
log slope for incidence versus dosage, and n is the log-log slope for
cumulative incidence versus age.