A mathematical model for the age distribution of cancer in man

Abstract Incidence data that have been collected by cancer registries in different parts of the world provide an opportunity for examining the relationship between cancer incidence and age on a wider scale and more accurately than has been possible before. In the present paper the relationship has been examined in a large group of cancers whose incidence increases progressively from young adult life into old age. Five‐year age‐specific incidence rates between 35 and 74 years of age were recorded for 31 types of cancer (22 in men and 9 in women) in 11 populations, and the median age of each age group was used to characterize the group. The data were used to find the best fitting constants for each type of cancer in each population, in the equation: where I is the incidence at age t, and b and k are constants. With this equation a straight line is obtained, when incidence and age are plotted on a double logarithmic scale. On visual examination, 21% (72 out of 338) of the sets of data appeared to fit the relationship closely. In 54% (181 out of 338) the lines showed downward curvature; that is, the rate of increase with age was less than predicted by the equation; and in 25% (85 out of 338) the lines showed upward curvature. In one‐third of the total the differences were too great to be readily attributable to chance. An alternative hypothesis was examined which postulated that the risk of cancer was determined, not by the age of the subject, but by the prevalence of carcinogenic agents and the length of time the subject had been exposed to them. The equation was then written: where w is a constant and t‐w represents the ‘effective exposure’ between first exposure and the first appearance of cancer as a pathological entity. The data for cancer of the prostate were found to fit the equation adequately in all 11 populations, when w was 32.5 years. No evidence was found to suggest that the shape of the observed relationship could be attributed to attenuation of a limited pool of susceptibles. Changes in prevalence of carcinogenic agents could account for the variation in the pattern of the observed relationship, and the conditions under which this might occur are discussed. Differences in the estimated value of k (the power of the effective exposure time in the equation), were examined between (i) types of cancer and (ii) populations. The results were not wholly consistent but suggested that the value of k might be a biological constant characteristic of the tissue in which the cancer is produced.