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One has to be careful when comparing these calculations with data. It is possible that dysplastic crypts can be lost. The model presented here gives the number of dysplastic crypts that are being produced over time, which could be larger than the actual number of dysplastic crypts that patients have at a particular time point. Exact measurements of the incidence of dysplastic crypts will provide important information about the crucial parameters of colorectal cancer initiation.

4.3 Sporadic colorectal cancers, CIN and MSI

Let us now consider the possibility of developing genetic instabilities during cancer initiation. Starting from a population of normal cells, three different events can occur: (i) inactivation of the first copy of the APC gene, (ii) inactivation of the first copy of one of nm MSI genes, and (iii) mutation of one copy of one of nc CIN genes.

We use the notation Xi, Yi and Zi, respectively, for the probability that a crypt consists of normal cells, CIN cells or MSI cells with i copies of the APC gene inactivated, see Table 4.3. Figure 4.3 shows the mutation-selection network of colorectal cancer initiation including CIN and MSI. All the transition rates are calculated as the relevant mutation rate times the probability that the mutant will take over the crypt.

Let us denote the rate of LOH in CIN cells as p. We assume that the crucial effect of CIN is to increase the rate of LOH [Bardelli et al. (2001); Lengauer et al. (1997)], which implies p > pq. Intuitively, the advantage of CIN for the cancer cell is to accelerate the loss of the second copy of a tumor suppressor gene. Similarly, the advantage of MSI is to increase the

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