  Tumor cell migration within the microenvironment is a crucial aspect for
cancer progression and, in this context, hypoxia has a significant role. An
inadequate oxygen supply acts as an environmental stressor inducing migratory
bias and phenotypic changes. In this paper, we propose a novel multi-scale
mathematical model to analyze the pivotal role of Snail protein expression in
the cellular responses to hypoxia. Starting from the description of single-cell
dynamics driven by the Snail protein, we construct the corresponding kinetic
transport equation that describes the evolution of the cell distribution.
Subsequently, we employ proper scaling arguments to formally derive the
equations for the statistical moments of the cell distribution, which govern
the macroscopic tumor dynamics. Numerical simulations of the model are
performed in various scenarios with biological relevance to provide insights
into the role of the multiple tactic terms, the impact of Snail expression on
cell proliferation, and the emergence of hypoxia-induced migration patterns.
Moreover, quantitative comparison with experimental data shows the model's
reliability in measuring the impact of Snail transcription on cell migratory
potential. Through our findings, we shed light on the potential of our
mathematical framework in advancing the understanding of the biological
mechanisms driving tumor progression.
