Predicted stem-cell loss: division ratios required to confer latent periods of more than 80 years on clones initiated by mutations conferring “exponential phenotype.” Example solutions of the identity E(t) = (logφ-log2-logμ)/(log2(λ-μ)) (see “Latent period equations” under “Results” for derivation) are shown in the central graph for E(t) = 80 years. Two lines are shown, the upper corresponding to 20 malignant cells being required for diagnosis and the lower, 100 cells. A range of expected stem-cell division rates shown on the horizontal axis to a maximum of once per year with stem-cell loss (differentiation or apoptosis) rates, expressed as a proportion of expected stem-cell division rates, shown on the vertical axis. The panels surrounding the main graph illustrate predicted log-log age-specific incidence curves from a simulation program described in the main text. Each panel shows the predicted curves corresponding to the stem-cell division and loss rates indicated by the lines connecting to circles on the central graph. Predicted curves corresponding to 20 cells per year contributing to erythropoiesis are joined to the connecting line on the simulation at a circle, while those corresponding to 100 cells are joined at a square. Observed data are indicated by diamonds unconnected by lines. Combinations of stem-cell division and loss parameter values below the illustrated lines give latencies of less than 80 years, corresponding to simulations that result in flat age-specific incidence curves at ages earlier than that observed (panels F and G, both 20 and 100 cells to present; panel D, 20 cells to present). Combinations of stem- cell division and loss parameter values above the illustrated lines give latencies of more than 80 years, corresponding to simulations that result in age-specific incidence curves that mainly fail to flatten at higher ages. However, 2 other effects are also apparent. First, clones may present after chance expansion and this process results in relatively flat age-specific presentations. This effect is illustrated in panels B and C, resulting from stem-cell loss/division ratios of 1, and is particularly noticeable when numbers of stem cells required to present and stem-cell division rates are low. Indeed, in this simple model, malignant stem-cell division rates of over once per year are incompatible with exponential phenotypes causing observed log-log behavior. In addition, if stem-cell division rates are low, then high stem-cell loss/division ratios fail to cause a sufficiently high number of presenting cases (A, especially lack of simulated line corresponding to 100 cells/year contributing to erythropoiesis at loss/division ratio of 0.9). Approximate log-log behavior in this simple model for 20 malignant stem cells contributing to erythropoiesis thus results from combinations of stem-cell division and loss parameter values in the shaded area (panels A and E).