A Stochastic Model for Cancer Metastasis: Branching Stochastic Process with Settlement

We introduce a new stochastic model for metastatic growth, which takes the form of a branching stochastic process with settlement. The moving particles are interpreted as clusters of cancer cells while stationary particles correspond to micro-tumors and metastases. The analysis of expected particle location, their locational variance, the furthest particle distribution, and the extinction probability leads to a common type of differential equation, namely, a non-local integro-differential equation with distributed delay. We prove global existence and uniqueness results for this type of equation. The solutions asymptotic behavior for long time is characterized by an explicit index, a metastatic reproduction number R0: metastases spread for R0 > 1 and become extinct for R0 < 1. Using metastatic data from mouse experiments, we show the suitability of our framework to model metastatic cancer.