  Cancer progression is an evolutionary process that is driven by mutation and
selection in a population of tumor cells. We discuss mathematical models of
cancer progression, starting from traditional multistage theory. Each stage is
associated with the occurrence of genetic alterations and their fixation in the
population. We describe the accumulation of mutations using conjunctive
Bayesian networks, an exponential family of waiting time models in which the
occurrence of mutations is constrained to a partial temporal order. Two
opposing limit cases arise if mutations either follow a linear order or occur
independently. We derive exact analytical expressions for the waiting time
until a specific number of mutations have accumulated in these limit cases as
well as for the general conjunctive Bayesian network. Finally, we analyze a
stochastic population genetics model that explicitly accounts for mutation and
selection. In this model, waves of clonal expansions sweep through the
population at equidistant intervals. We present an approximate analytical
expression for the waiting time in this model and compare it to the results
obtained for the conjunctive Bayesian networks.
