Probabilistic approach to a proliferation and migration dichotomy in the tumor cell invasion

The proliferation and migration dichotomy of the tumor cell invasion is examined within a two-component continuous time random walk (CTRW) model. The balance equations for the cancer cells of two phenotypes with random switching between cell proliferation and migration are derived. The transport of tumor cells is formulated in terms of the CTRW with an arbitrary waiting time distribution law, while proliferation is modelled by a logistic growth. The overall rate of tumor cell invasion for normal diffusion and subdiffusion is determined.

💡 Research Summary

The paper addresses the well‑observed proliferation–migration dichotomy of invasive tumor cells by constructing a two‑component continuous‑time random walk (CTRW) model that explicitly incorporates stochastic switching between a proliferative phenotype and a migratory phenotype. The authors first define two subpopulations, (C_P(t,\mathbf{x})) (proliferative cells) and (C_M(t,\mathbf{x})) (migratory cells), and introduce random transition rates (\lambda_{1}) (from proliferative to migratory) and (\lambda_{2}) (from migratory to proliferative). These rates are allowed to be time‑dependent, thereby capturing non‑Markovian switching that may be driven by micro‑environmental cues such as growth‑factor gradients or extracellular‑matrix stiffness.

Cell movement is modeled through a CTRW framework. The waiting‑time (or residence‑time) distribution (\psi(\tau)) governs how long a cell remains stationary before making a jump, while the jump‑length distribution (w(\boldsymbol{\xi})) describes the spatial displacement. The authors keep the jump‑length distribution Gaussian with zero mean and variance (\sigma^{2}), ensuring isotropic random motion. Crucially, (\psi(\tau)) is left arbitrary; two cases are examined in detail: (i) an exponential waiting‑time (\psi(\tau)=\gamma e^{-\gamma\tau}) leading to normal diffusion, and (ii) a heavy‑tailed power‑law (\psi(\tau)\propto \tau^{-1-\alpha}) with (0<\alpha<1) that generates subdiffusive dynamics.

Proliferation is described by a logistic growth term (r C_P (1-C/K)) where (r) is the intrinsic proliferation rate, (K) the carrying capacity of the tissue, and the logistic nonlinearity limits uncontrolled expansion. Migratory cells do not proliferate; their dynamics are purely governed by the CTRW. By coupling the two subpopulations through the switching terms, the authors derive a system of integro‑differential balance equations:

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