Mathematical model of colorectal cancer initiation

Abstract

Quantifying evolutionary dynamics of cancer initiation and progression can provide insights into more effective strategies of early detection and treatment. Here we develop a mathematical model of colorectal cancer initiation through inactivation of two tumor suppressor genes and activation of one oncogene, accounting for the well-known path to colorectal cancer through loss of tumor suppressors APC and TP53 and gain of the KRAS oncogene. In the model, we allow mutations to occur in any order, leading to a complex network of premalignant mutational genotypes on the way to colorectal cancer. We parameterize the model using experimentally measured parameter values, many of them only recently available, and compare its predictions to epidemiological data on colorectal cancer incidence. We find that the reported lifetime risk of colorectal cancer can be recovered using a mathematical model of colorectal cancer initiation together with experimentally measured mutation rates in colorectal tissues and proliferation rates of premalignant lesions. We demonstrate that the order of driver events in colorectal cancer is determined primarily by the fitness effects that they provide, rather than their mutation rates. Our results imply that there may not be significant immune suppression of untreated benign and malignant colorectal lesions.

Keywords: cancer evolution; driver mutations; stochastic model.

Fig. 1.

(A) Schematic of CRC initiation. A healthy (wild-type) crypt is in the lower left corner, and a fully malignant crypt at the top right. (B) (Top) Transition rates from APC wild-type genotype (0,*,*) to fully inactivated APC through LOH and mutation (or vice versa) (3,*,*) or double mutation (4,*,*). (Bottom) Transition rate from wild-type KRAS (*,*,0) to activated KRAS (*,*,1).

Fig. 2.

Comparison of Eqs. 1–3, computer simulations of the process of CRC initiation, and lifetime risk of CRC containing mutations in APC, KRAS, and TP53; 8*10⁶ runs of the computer simulation were performed for each setting: when APC provides advantage and KRAS is neutral, compared to Eq. 2, and when both APC and KRAS provide advantage, compared with Eq. 3.

Fig. 3.

Expected number of (A) single- and (B) double-mutant colorectal crypts as a function of time. Formulas for expected numbers of mutated crypts used to produce the plots are given in SI Appendix.

Fig. 4.

Order of acquisition of driver mutations in the mathematical model of CRC initiation. (A) Three example five-step paths (out of 270 possible paths) that a crypt can take in order to become malignant. We classify each path that leads to malignancy within 80 y in our computer simulation into one of the six possible orderings of the three driver mutations. (B) Likelihood of driver mutation order on the path to CRC, obtained from computer simulation of our mathematical model. Note that APC loss as the first event is followed by KRAS gain in the large majority of cases, and TP53 loss in the small minority of situations. KRAS gain as the first event is virtually always followed by APC loss, with TP53 loss as the last event.