A kinetic derivation of spatial distributed models for tumor-immune system interactions

TL;DR

This work develops a kinetic framework to model spatial tumor–immune interactions by introducing distribution functions for tumor and immune cell states and two static backgrounds. Through two kinetic setups—one conservative (no immune proliferation) and one proliferative (immune growth and death)—the authors derive macroscopic diffusion-type models via hydrodynamic limits, obtaining linear diffusion, nonlinear cross-diffusion, and nonlinear self-diffusion regimes. They perform qualitative analyses of spatially homogeneous reductions to identify equilibria and bifurcations, revealing robust correspondences across macroscopic models derived from the same kinetic framework. Numerical simulations in 2D validate the analytical predictions and illustrate how diffusion structure shapes spatial patterns and tumor clearance. The framework offers a structured path to explore therapy strategies and can be extended to heterogeneous environments and optimal control of immune-modulating factors like interleukins, with potential applications beyond cancer to other immune-related pathologies.

Abstract

We propose a mathematical kinetic framework to investigate interactions between tumor cells and the immune system, focusing on the spatial dynamics of tumor progression and immune responses. We develop two kinetic models: one describes a conservative scenario where immune cells switch between active and passive states without proliferation, while the other incorporates immune cell proliferation and apoptosis. By considering specific assumptions about the microscopic processes, we derive macroscopic systems featuring linear diffusion, nonlinear cross-diffusion, and nonlinear self-diffusion. Our analysis provides insights into equilibrium configurations and stability, revealing clear correspondences among the macroscopic models derived from the same kinetic framework. Using dynamical systems theory, we examine the stability of equilibrium states and conduct numerical simulations to validate our findings. These results highlight the significance of spatial interactions in tumor-immune dynamics, paving the way for a structured exploration of therapeutic strategies and further investigations into immune responses in various pathological contexts.

Figures (9)

• Figure 1: Qualitative evolution of systems \ref{['Homo_2pop_Lin1']} and \ref{['Homo_Cross']}. Qualitative evolution of the trajectories of model \ref{['Homo_2pop_Lin1']} (blue dashed line: tumor ($Y_1$), red dashed line: active immune system ($Y_2$)) and \ref{['Homo_Cross']} (solid green line: tumor ($Y_1$)) as they approach the equilibrium point $E_1$. The parameter values are set to $A = 5,\,B = 2,\,C = 1,\,I = 0.33$. The trajectories illustrate the dynamics leading to the stable disease-free equilibrium, with differences in transient behaviors observed between the two systems.
• Figure 2: Bifurcation analysis on system \ref{['Homo_2pop_Nonlin']}. Panel A: bifurcation diagram highlighting existence and stability of equilibria $E_2$ and $E_3$ in system \ref{['Homo_2pop_Nonlin']}, setting the parameter values to ${A=G=1.5}$, $P=0.5$, and setting $B=D$. Panel B: Hopf limiting cycle appearing in system \ref{['Homo_2pop_Nonlin']} for the parameter values $A=G=1.5$, $P=0.5$, $B=D=0.25$, $H=4.5$.
• Figure 3: Qualitative evolution of systems \ref{['Homo_3pop_Lin2']} and \ref{['Homo_2pop_Nonlin']}. Panels A-B: qualitative evolution of the trajectories of model \ref{['Homo_3pop_Lin2']} (blue dashed line: tumor ($Y_1$), red dashed line: immune system ($Y_N$)) and \ref{['Homo_2pop_Nonlin']} (green line: tumor ($Y_1$), orange line: immune system ($Y_N$)) as they approach the equilibrium point $E_2$, taking $B=D=0.05$ and $H=1.5$. Panels C-D: qualitative evolution of the trajectories of model \ref{['Homo_3pop_Lin2']} (blue dashed line: tumor ($Y_1$), red dashed line: immune system ($Y_N$)) and \ref{['Homo_2pop_Nonlin']} (solid green line: tumor ($Y_1$), solid orange line: immune system ($Y_N$)) as they approach the equilibrium point $E_3$, taking $B=D=0.47$ and $H=4.5$. In all cases, the remaining parameter values are fixed at $A=1.5$, $C=7$, $E=3.5$, and $P=0.5$.
• Figure 4: Test 1: evolution of system \ref{['Adim_Lin1']}. Numerical simulation of model \ref{['Adim_Lin1']} with parameter values $A=50$, $B=0.5$, and $C=1$ and the diffusion coefficients $\mathcal{D}_1=10^{-6}$, $\mathcal{D}_2=0.05$, and $\mathcal{D}_3=0.0002$. The columns represent the tumor population ($Y_1$), active immune cells ($Y_2$), and passive immune cells ($Y_3$), respectively. The rows correspond to three different time steps: $T=0.004$, $0.008$, and $0.04$, respectively.
• Figure 5: Test 1: evolution of system \ref{['Adim_Cross']}. Numerical simulation of model \ref{['Adim_Cross']} with parameter values $A=50$, $B=0.5$, and $C=1$ and the diffusion coefficients $\mathcal{D}_1=10^{-6}$, $\mathcal{D}_2=0.05$, and $\mathcal{D}_3=0.0002$. The columns represent the tumor population ($Y_1$) and the total immune system ($Y_N$), respectively. The rows correspond to three different time steps: $T=0.02$, $0.1$, and $0.5$, respectively.