Spatio-Temporal Dynamics of M₁ and M₂ Macrophages in a Multiphase Model of Tumor Growth

This study investigates the complex dynamics of vascular tumors and their interplay with macrophages, key agents of the innate immune response. We model the tumor microenvironment as a multiphase fluid, with each cellular population treated as a distinct, non-mixing phase. The framework also incorporates diffusible species that are critical for processes such as nutrient transport, angiogenesis, chemotaxis, and macrophage activation. A central contribution of this work is the explicit modeling of macrophage infiltration and polarization within the tumor microenvironment. The model captures the divergent roles of ${\text{M}}_1$ (anti-tumor) and ${\text{M}}_2$ (pro-tumor) macrophages and their influence on tumor aggressiveness and progression. Through numerical simulations, we demonstrate the emergence of both spatial and phenotypic heterogeneity in the macrophage population, including their peripheral localization and limited core infiltration -patterns consistent with experimental observations. Furthermore, this is the first multiphase model to incorporate the effects of TGF- $\beta$ -targeting immunotherapy using vactosertib. Our simulations demonstrate that treatment initially enhances the presence of anti-tumor macrophages, followed by a relapse period where tumor dynamics returns to pre-treatment trends. Model parameters are grounded in experimental data and clinically relevant dosage protocols.