Figure 2From: A simple algebraic cancer equation: calculating how cancers may arise with normal mutation rates Colorectal cancer. A: Epidemiology documents increased colorectal cancer incidence with age (data from Ref [9]). B: Equation [8] can approximate epidemiology (gray line) with 5 k rate-limiting mutations and 40 niche stem cells (green dotted line), or 6 k rate-limiting pathway mutations and 8 niche stem cells (red dotted line). See Table 1 for other parameters. C: Increased cancer risks with height in women can be modeled by changing colon lengths or crypts per colon (m). A 1.8 relative risk between the shortest and tallest quintiles can be modeled by changing lengths 28.6% (blue dotted lines) relative to the average colon (red dotted line). Crypts in the shortest quintile are 80% of the tallest quintile. D: Metastatic cancer has a lower incidence and arises later than localized or regional cancer, which can be modeled (dotted lines) by increasing k rate-limiting pathway mutations from 6 to 6.5. E: Lower cancer subtype incidence can also be modeled (dotted lines) with k = 6 and smaller mutational target sizes (u). For all cancers, u = 3 × 10^{-6} but is 2.55 × 10^{-6} for localized or regional cancers, and 2.2 × 10^{-6} for metastatic cancers (also see Fig 4). F: Earlier FAP cancer incidence can be modeled by decreasing k from six in sporadic cancers to five in FAP. This incidence shifts to even younger ages by also increasing crypt niche stem cells (N) from 8 to 16. G: A cancer relative risk of 0.8 with chronic aspirin use can be modeled by decreasing crypt niche stem cells from 8 to 6. H: A similar increase in cancer occurs with a 10% increase in stem cell division rate (green dotted line), or a 10% mutation rate increase (black dotted line). If division increases later in life, the cancer incidence increase is lower (blue dotted line, 10% increase in stem cell division rate starts at 40 years of age).Back to article page