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A non-linear mathematical model of cell turnover, differentiation and tumorigenesis in the intestinal crypt

We present a development of a model of the relationship between cells in three compartments of the intestinal crypt: stem cells, semi-differentiated cells and fully differentiated cells. Stem and semi-differentiated cells may divide to self-renew, undergo programmed death or progress to semi-differentiated and fully differentiated cells respectively. The probabilities of each of these events provide the most important parameters of the model. Fully differentiated cells do not divide, but a proportion undergoes programmed death in each generation. Our previous models showed that failure of programmed death - for example, in tumorigenesis - could lead either to exponential growth in cell numbers or to growth to some plateau. Our new models incorporate plausible fluctuation in the parameters of the model and introduce non-linearity by assuming that the parameters depend on the numbers of cells in each state of differentiation. We find that the model is characterized by bifurcation between increase in cell numbers to stable equilibrium or 'explosive' exponential growth; in a restricted number of cases, there may be multiple stable equilibria. Fluctuation in cell numbers undergoing programmed death, for example caused by tissue damage, generally makes exponential growth more likely, as long as the size of the fluctuation exceeds a certain critical value for a sufficiently long period of time. In most cases, once exponential growth has started, this process is irreversible. In some circumstances, exponential growth is preceded by a long plateau phase, of variable duration, mimicking equilibrium: thus apparently self-limiting lesions may not be so in practice.