Continuous Branching Processes with Settlement in Cancer Metastasis: Stochastic Modelling and the Feller Property

Motivated by models of cancer metastasis, this paper introduces a type of (multi-type) branching process that records the positions of particles, representing tumor cells or clusters. Particles may be absorbed (removed from the state space), move, or settle. The process is rigorously constructed, and the Markov property is established via embedding into a multidimensional process that tracks the labels, positions, and phases (moving or resting) of living particles. The Feller property for the associated semigroup is investigated. It is proved for a simplified model that tracks the number of particles in each class, and an explicit generator is derived, enabling Feynman-Kac-type formulas in this framework.

💡 Research Summary

This paper introduces a novel stochastic modeling framework for cancer metastasis, focusing on the spatial dissemination and colonization of tumor cells. The authors propose a multi-type continuous branching process that explicitly tracks the spatial positions of particles (representing individual cells or clusters), incorporating key biological mechanisms: random motility, settlement (extravasation), reproduction (shedding of new cells), and death.

The core model is rigorously constructed. Particles are labeled within a tree structure. Each particle moves independently in a locally compact metric space E according to a cadlag Markov process Z. After an independent exponential time with rate μ_S, a particle “settles” at its current location, ceasing movement. Once settled, it can produce offspring according to a compound Poisson process with rate μ_B and jump distribution ν, meaning it gives birth to random batches of new particles at random times. Particles may die during movement (at rate δ_M) or after settlement (at rate δ_S), and can also be absorbed upon hitting certain sets. The parent remains alive after reproduction, distinguishing it from classical branching models.

The authors first prove that this complex interactive system retains the Markov property. This is achieved by embedding the process into a higher-dimensional process that tracks the labels, spatial positions, and phases (moving/settled) of all living particles. They then investigate the Feller property of the associated semigroup, a crucial analytical feature that ensures continuity and enables connections to differential equations. For a simplified version of the model that only tracks the number of particles in each class (e.g., moving vs. settled) rather than their individual positions, the Feller property is established.

A significant contribution is the derivation of an explicit formulation for the generator of this simplified process. The generator describes the infinitesimal evolution of the system. Its explicit form is pivotal as it opens the door to Feynman-Kac-type formulas, which link expectations over particle trajectories to solutions of linear partial differential equations. The paper highlights that this framework provides a foundational step towards establishing McKean-type representations, where solutions to nonlinear reaction-diffusion equations (such as the Fisher-KPP equation, widely used in modeling tumor invasion fronts) can be expressed as expectations over the entire branching population.

In summary, this work provides a mathematically rigorous and general stochastic model for metastatic spread that incorporates spatial structure and settlement dynamics. By establishing its Markov and Feller properties and deriving an explicit generator, it creates a robust probabilistic foundation. This framework not only advances the theoretical understanding of cancer metastasis but also bridges stochastic modeling with analytical PDE theory, offering new tools for future analysis and simulation of multi-scale dispersal phenomena.