# Mathematical model of colorectal cancer initiation

## Behind the paper

Written by Ivana Bozic - October 01, 2020

### Mathematical model of colorectal cancer initiation

Chay Paterson, Hans Clevers, Ivana Bozic

The seed of the idea for the paper was planted in 2016, during my visit to the Hubrecht Institute in Utrecht, Netherlands. I met with Hans Clevers, who told me that they recently measured the rate of accumulation of mutations in human tissues using organoids derived from healthy donor stem cells1. We both agreed that it would be very interesting to use that measured in vivo mutation rate in a mathematical model of colorectal tumor evolution. After returning to Boston, I started thinking about the idea discussed with Hans and made some preliminary calculations, but it took a while before the project actually took off. I soon moved to Seattle to start a new position at the University of Washington. Finally, in 2018 Chay Paterson joined my group as a postdoc and started working on the project.

We wanted to understand the detailed mathematical model of how the famous APC-KRAS-TP53 driver gene sequence2 evolves from healthy colorectal tissue, and to parametrize the model using experimentally derived parameter values. Complicating matters was the fact that APC and TP53 are tumor suppressors, and that typically both copies of these genes had to be inactivated independently; furthermore, each copy could be inactivated through either mutations or loss of heterozygosity events (Figure 1). We reasoned that any ordering of these oncogenic events may be possible in principle, so our model included a complex network of 50 premalignant genotypes and 270 distinct paths on the way to colorectal cancer. The model is initiated with $N$ healthy colorectal crypts at age 0 and assumes that crypts collect driver mutations stochastically, with rates that are the product of the background mutation rate and the number of driver positions in each gene.

Figure 1. (A) Schematic of colorectal cancer initiation. Healthy (wild type) crypt is in the lower left corner, and a fully malignant crypt in 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).

We started with a simpler model in which the three driver events accumulate neutrally and showed that the neutral assumption leads to colorectal cancer (CRC) incidence that is many orders of magnitude lower than reported. Then we tackled the full model, in which we allowed colorectal crypts with driver mutations to divide according to a stochastic birth process. We were lucky that two groups measured the rates of expansion of colorectal crypts containing APC and KRAS mutations in human subjects in vivo3,4, and that another group showed that TP53 may not be providing growth advantage on its own5, so we were able to parametrize the model fully. Strikingly, we found that the reported lifetime risk of colorectal cancer can be recovered using our mathematical model of CRC initiation together with experimentally measured mutation rates in colorectal tissues and proliferation rates of premalignant lesions.

One of the most interesting things we discovered is that the most likely order in which driver mutations are accumulated on the way to CRC in our mathematical model is APC-KRAS-TP53, exactly the same ordering that has been reported experimentally2. Not only that, we found that the most likely order of driver mutations is one that maximizes fitness along the carcinogenic path, so the driver mutation that provides maximum fitness advantage over healthy tissue is collected first and so on. We also realized that the question of mutation order is a bit subtle, and that one should distinguish between the order in which driver mutations are accumulated in typical crypts (large majority of which will not end up cancerous) and the most likely order of driver mutations that will result in cancer within a patient’s lifetime. For example, we find that healthy crypts will typically first get an activating mutation in KRAS, but crypts that will become cancerous will typically first inactivate APC.

In sum, our detailed genetic model contributes to the quantitative understanding of the process of colorectal carcinogenesis in patients; in the future it could be extended to include other significant driver genes, and to allow the mutation rate and fitness advantage provided by drivers to vary between patients and throughout the patients’ lifetimes, reflecting differences in genetic background and the tumor microenvironment6.

## References

1. F. Blokzijl et al., Tissue- specific mutation accumulation in human adult stem cells during life. Nature 538, 260-264 (2016).
2. B. Vogelstein et al., Genetic alterations during colorectal-tumor development. N Engl J Med 319, 525-532 (1988).
3. A. M. Baker et al., Quantification of crypt and stem cell evolution in the normal and neoplastic human colon. Cell Rep. 8, 940-947 (2014).
4. A. M. Nicholson et al., Fixation and spread of somatic mutations in adult human colonic epithelium. Cell Stem Cell 22, 909-918 (2018).
5. L. Vermeulen et al., Defining stem cell dynamics in models of intestinal tumor initiation. Science 342, 995-998 (2013).
6. Fane, M. & Weeraratna, A. T. How the ageing microenvironment influences tumour progression. Nat. Rev. Cancer 20, 89–106 (2020).