Stochastic and deterministic models for the metastatic emission process: formalisms and crosslinks
Résumé
Although the detection of metastases radically changes prognosis of and treatment decisions for a cancer patient, clinically undetectable micrometastases hamper a consistent classification into localised or metastatic disease. This chapter discusses mathematical modelling efforts that could help to estimate the metastatic risk in such a situation. We focus on two approaches: 1) a stochastic framework describing metastatic emission events at random times, formalised via Poisson processes, and 2) a deterministic framework describing the micrometastatic state through a sizestructured density function in a partial differential equation model. Three aspects are addressed in this chapter. First, a motivation for the Poisson process framework is presented and modelling hypotheses and mechanisms are introduced. Second, we extend the Poisson model to account for secondary metastatic emission. Third, we highlight an inherent crosslink between the stochastic and deterministic frameworks and discuss its implications. For increased accessibility the chapter is split into an informal presentation of the results using a minimum of mathematical formalism and a rigorous mathematical treatment for more theoretically interested readers.