  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.
