A fast numerical integration scheme for clonal expansion processes on graphs

Compound birth-death processes are widely used to model the age-incidence curves of many cancers. There are efficient schemes for directly computing the relevant probability distributions in the context of linear multi-stage clonal expansion (MSCE) models. However, these schemes have not been generalised to models on arbitrary graphs, forcing the use of either full stochastic simulations or mean-field approximations, which can become inaccurate at late times or old ages. Here, we present a numerical integration scheme for directly computing survival probabilities of a first-order birth-death process on an arbitrary directed graph, without the use of stochastic simulations. As a concrete application, we show that this new numerical method can be used to infer the parameters of an example graphical model from simulated data.Author summary: In this work, we develop an algorithm that solves a first-order stochastic multi-stage model faster than existing methods. The main application is modelling cancer incidence as a function of age, but many other applications are imaginable. In this class of models, cancer cells are produced from a large number of healthy cells by a series of a few rare and random mutations. Multi-stage models have been very successful, and are currently widely used to explain why cancer is more common in old age. However, these multi-stage models only consider the overall number of stages, and do not consider different types of mutation, which could occur in different orders. The different orders in which different types of mutation can happen form a network or “graph”. This is because there was not an efficient algorithm for working out the age-specific incidence for multi-stage models on graphs. We have now found an algorithm for accurately computing this, at least ten million times faster than other proposals. We also show how this algorithm could be used to fit models to data from a simulated clinical study.