Page 35 - 20dynamics of cancer

20 CHAPTER 2

plotted on logarithmic scales. For many types of cancer, incidence tends
to increase approximately logarithmically with age (Armitage and Doll
1954), which can be represented as I = ct n−1 , where I is incidence, t
is age, n − 1 is the rate of increase, and c is a constant. If we take the
logarithm of this expression, we have log(I) = log(c) + (n − 1) log(t).
Thus, a log-log plot of log(I) versus log(t) is a straight line with a slope
of n − 1.
The plots of actual cancer data rarely give perfectly straight lines on
log-log scales. The ways in which cancer incidence departs from log-
log linearity provide interesting information (Armitage and Doll 1954;
Cook et al. 1969; Moolgavkar 2004). For example, Figure 2.1a shows
the number of new cases among males per year. This is a rate, just
as the number of meters traveled per hour is a rate of motion. If we
take the slope of a rate, we get a measure of acceleration. Figure 2.1c
plots the slope taken at each point of Figure 2.1a, giving the age-specific
acceleration of cancer (Frank 2004b). If cancer accelerated at the same
pace with age, causing Figure 2.1a to be a straight line with slope n −
1, then acceleration would be constant over all ages, and the plot in
Figure 2.1c would be a flat line with zero slope and a value of n − 1 for
all ages.
Figure 2.1e takes the age-specific acceleration in Figure 2.1c and re-
scales the age axis to be linear instead of logarithmic. I do this to spread
the ages more evenly, which makes it easier to look at patterns in the
data.
The age-specific acceleration for males in Figure 2.1e shows that can-
cer incidence accelerates at an increasing rate up to about age 50; af-
ter 50, when most cancers occur, the acceleration declines nearly lin-
early. The acceleration plot for females in Figure 2.1f also shows a lin-
ear decline, starting at an earlier age and declining more slowly than
for males. The acceleration plots provide very useful complements to
the incidence plots, because changes in acceleration suggest how cancer
may be progressing within individuals at different ages (Frank 2004b).

2.2 Different Cancers

There is a vast literature on descriptive epidemiology (Adami et al.
2002; Parkin et al. 2002). Those studies examine cancer incidence at