Mathematical model of tumor-macrophage interactions: Elucidating the tumor-driven macrophage phenotype reprogramming

TL;DR

A mathematical model that incorporates tumor cells, M1 type macrophages, M2 type macrophages and an M3 type macrophage population characterized by dual phenotypic features is developed and proposes two potential clinical prognostic markers: the level of M1 type macrophage infiltration and the peak time of M3 type macrophage activation.

Abstract

The interplay between tumor cells and macrophages plays a central regulatory role in cancer progression. In this study, we developed a mathematical model that incorporates tumor cells, M1 type macrophages, M2 type macrophages and an M3 type macrophage population characterized by dual phenotypic features. First, we analyzed the fundamental mathematical properties of the model and derived the conditions under which the system attains a tumor free stable state or a coexistence state of tumor and immune cells. Second, global sensitivity analysis revealed that key parameters governing macrophage polarization and intercellular communication vary dynamically during tumor development. Bifurcation analysis further identified the polarization rate of M1 type macrophages $κ$ and the baseline level of resting macrophages $M_0$ as critical determinants of the system's dynamical behavior. Notably, using approximate Bayesian computation for parameter inference and dynamic simulations, the model successfully recapitulated the evolutionary trajectories of eight tumor samples. The results demonstrate that lower tumor burden is significantly associated with higher M1 type macrophage infiltration and delayed peak time of M3 type macrophage activation. Moreover, survival analysis indicated that both enhanced M1 type macrophage infiltration and delayed peak time of M3 type macrophage activation are correlated with longer survival time. In summary, this study not only provides a theoretical framework for understanding the dynamic mechanisms underlying tumor macrophage interactions but also proposes two potential clinical prognostic markers: the level of M1 type macrophage infiltration and the peak time of M3 type macrophage activation.

Figures (9)

• Figure 1: The dynamic regulatory network of the interaction between tumors and macrophages.
• Figure 2: Model validation and global sensitivity analysis. (A) Simulated tumor growth trajectories (solid line) closely match experimental data (dots), with $R^2 = 0.988$. (B) Temporal trajectories of M1‑type, M2‑type, and M3‑type macrophage subpopulations predicted by the calibrated model. (C) Temporal trajectories of the relative fractions of the three macrophage phenotypes. (D) - (G) Sobol' global sensitivity analysis quantifies the influence of individual parameters on tumor cell and macrophage phenotype dynamics.
• Figure 3: Bifurcation analysis of tumor-macrophage dynamics with respect to polarization rate of M1-type macrophages $\kappa$ and baseline level of resting macrophages $M_0$. (A) Two‑parameter bifurcation diagram of tumor cell counts $(C)$ in the ($\kappa$, $M_0$) plane. Color‑coded regions correspond to distinct qualitative regimes: tumor-free equilibria (pink), bistability (sky-blue) and sustained tumor presence (wheat). (B)-(C) One‑parameter bifurcation diagrams of tumor cell counts as a function of $\kappa$ with different values of $M_0$ ($3 \times 10^6$ and $5.4 \times 10^6$ denoted $M_{03}$ and $M_{05}$ shown in (A), respectively). Dashed and solid curves represent unstable and stable steady states, respectively. Blue and magenta lines represent tumor-free equilibrium and positive steady states, respectively.
• Figure 4: Bifurcation analysis with respect to polarization rate of M1-type macrophages $\kappa$ and baseline level of resting macrophages $M_0$. (A)-(D) Bifurcation diagrams illustrating the dependence of the positive steady‑state counts of (A) tumor cells, (B) M1-type, (C) M2-type and (D) M3-type macrophages on $\kappa$ for five different values of $M_0$ ($M_{10}$, $M_{20}$, $M_{30}$, $M_{40}$ and $M_{50}$ in Fig. \ref{['Fig_bif1']}A), respectively. (E)-(H) Bifurcation diagrams showing the steady‑state counts of (E) tumor cells, (F) M1‑type, (G) M2‑type and (H) M3‑type macrophages as functions of $M_0$ for five fixed values of $\kappa$ ($\kappa_1$, $\kappa_2$, $\kappa_3$, $\kappa_4$ and $\kappa_5$ in Fig. \ref{['Fig_bif1']}A), respectively. Solid and dashed lines represent stable and unstable steady states, respectively.
• Figure 5: Population-level dynamics of tumor growth and macrophage polarization. (A) Numerical simulation range of tumor cell counts over time, compared with established experimental time-course data Reda.NatCommun.2022. (B) The probability density distributions of the coefficient of determination ($R^2$).

Theorems & Definitions (10)

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