In Response:
Mdzinarishvili, Gleason, and Sherman correctly comment that we did not discuss the uncertainty in variables a and k of our “β formula,” which we have fit to the age-dependant incidence of many cancers. We deliberately only gave confidence intervals for the incidence values, and not for the variables, because our point was to give evidence that US cancer incidence commonly turns over at around age 80 years, and has done so for the past 25 years. Additionally fitting the “β formula” to the data shows that by incorporating even a one variable linear “correction” term to Armitage and Doll's equation, we can account for the peak and decline in cancer rates. But we caution that our modification to Armitage and Doll's simple formula is largely empirical and highly speculative. Although the “β model” has a plausible biological justification in senescence ( 1, 2) and is nicely parsimonious, it is easy to pick out many other equations that also offer a good fit to the data. We find I(t) = atk−1(1−(bt)2) and I(t) = atk−1(1−bt)2 particularly appealing. However, the best values of a and k differ for each equation, although the number of adjustable variables seems to be the same. More complicated formulas would be needed to account for small childhood peaks and the incidences of Hodgkin's lymphoma and testicular cancer.
Mdzinarishvili and colleagues suggest modifying the “β formula” by varying the starting point of susceptibility to cancer and precancerous stages. Yet, one great success of the multistage model was to show that cancer rates can increase dramatically with age even when the probability of proceeding from each precancerous stage to the next is age independent. Although shifting the starting point of susceptibility has been suggested by Doll and colleagues ( 3, 4) as an approximation for smoking-dominated and reproductive cancers, we do not know if Mdzinarishvili and colleagues' suggestion for Pancreatic cancer is biologically plausible. In any case, we are not surprised that reducing the time domain by ∼30% can decrease the best value of the exponent k by ∼40%.
Finally, our analysis relies on US census and Surveillance, Epidemiology, and End Results population data, for which we would like to see more verification work. This is a “cross-sectional” study, with all of its limitations, and the data comes from the United States alone. We would like to see the crucial points emphasized by future studies, particularly from a well-conducted, blind, large cohort study.
Disclosure of Potential Conflicts of Interest
No potential conflicts of interest were disclosed.
- ©2009 American Association for Cancer Research.
Volume 69, Issue 1
