A mathematical model for cancer risk and accumulation of mutations caused by replication errors and external factors

Replication errors influence mutations, and thus, lifetime cancer risk can be explained by the number of stem-cell divisions. Additionally, mutagens also affect cancer risk, for instance, high-dose radiation exposure increases lifetime cancer risk. However, the influence of low-dose radiation exposure is still unclear because this influence, if any, is very slight. We can assess the minimal influence of the mutagen by virtually comparing the states with and without mutagen using a mathematical model. Here, we constructed a mathematical model to assess the influence of replication errors and mutagens on cancer risk. In our model, replication errors occur with a certain probability during cell division. Mutagens cause mutations at a constant rate. Cell division is arrested when the number of cells reaches the capacity of the cell pool. When the number of cells decreases because of cell death or other reasons, cells resume division. It was assumed that the mutations of cancer driver genes occur stochastically with each mutation and that cancer occurs when the number of cancer driver gene mutations exceeds a certain threshold. We approximated the number of mutations caused by errors and mutagens. Then, we examined whether cancer registry data on cancer risk can be explained only through replication errors. Although the risk of leukemia was not fitted to the model, the risks of esophageal, liver, thyroid, pancreatic, colon, breast, and prostate cancers were explained only by replication errors. Even if the risk was explained by replication errors, the estimated parameters did not always agree with previously reported values. For example, the estimated number of cancer driver genes in lung cancer was larger than the previously reported values. This discrepancy can be partly resolved by assuming the influence of mutagen. First, the influence of mutagens was analyzed using various parameters. The model predicted that the influence of mutagens will appear earlier, when the turnover rate of the tissue is higher and fewer mutations of cancer driver genes were necessary for carcinogenesis. Next, the parameters of lung cancer were re-estimated assuming the influence of mutagens. The estimated parameters were closer to the previously reported values. than when considering only replication errors. Although it may be useful to explain cancer risk by replication errors, it would be biologically more plausible to consider mutagens in cancers in which the effects of mutagens are apparent.