Turing pattern induced by cross-diffusion in cancer immunotherapy model

TL;DR

The paper extends a cancer–immune reaction-diffusion model by incorporating cross-diffusion of IL-2 toward tumor regions, and analyzes both untreated and Adoptive Cellular Immunotherapy (ACI)-treated scenarios. It derives and studies steady states, linear stability, and diffusion-driven (Turing) instabilities, complemented by Hopf bifurcation analysis, and confirms findings with numerical simulations (FDM and COMSOL). The results reveal that IL-2 cross-diffusion can trigger stationary spatial patterns (spots and stripes) whose form is reshaped by therapy, suggesting spatial considerations are crucial for treatment efficacy. The work emphasizes cross-diffusion as a key mechanism shaping spatial tumor–immune structure and offers insights that could inform theoretical development and immunotherapy design.

Abstract

In this paper, we investigate a mathematical model describing the interactions between effector cells (E), cancer cells (T), and the IL-2 compound (IL). The model considered here is a generalization, taking into account some cross-diffusion effects, of a spatial cancer immunotherapy model proposed by S. Suddin et al in 2021. These modifications allow us to describe two biologically relevant scenarios: a patient treated with Adoptive Cell Immunotherapy (ACI) and a patient not receiving any treatment/therapy. Cross-diffusion effects are particularly relevant in the interactions between tumor cells and the immune system, in fact they play a key role in immune response dynamics and cannot be neglected. We analyze the equilibrium points of the homogeneous system, along with their stability and bifurcation mechanisms. Furthermore, adopting the Turing approach for reaction-diffusion systems, we investigate the diffusion-driven instability and the emergence of spatial regular structures (stationary in time), i.e. the patterns. Finally, numerical simulations based on the Finite Difference Method (FDM) are presented for the two previously mentioned scenarios.

Figures (10)

• Figure 1: Region of the $p_2\,c$-plane where the Routh conditions are satisfied: (a) untreated case and (b) treated case. Note that a logarithmic scale is applied to the $p_2$-axis. The remaining parameters are listed in Table \ref{['tab:par']}.
• Figure 2: The solid and dotted lines represent the real part $\text{Re}(\lambda)$ and the imaginary part $\text{Im}(\lambda)$ of the eigenvalues $\lambda$, respectively, both with and without therapy. The parameters values are given in Table \ref{['tab:par']}, with $c=0.25$.
• Figure 3: First row: $s_1=s_3=0$ and Second row: $s_1=0.0035$, $s_3=0.2$. The remaining parameters are listed in Table \ref{['tab:par']}, with $c=0.25$. The time evolutions for two different initial conditions are shown in the first and second column, i.e.$u_0=0.1$, $v_0=0.3$, $w_0=1$ and $u_0=1.5$, $v_0=2$, $w_0=0.5$, respectively. The third column shows the limit cycle corresponding to the first set of initial conditions.
• Figure 4: Spatial distributions of $u,\,v,\, w$ with $d_{32}=-0.01$ in case of patient non-treated. The parameters are $c=0.25$, $p_2=0.5$, $d_{11}=0.001$, $d_{22}=1.99\cdot 10^{-5}$, $d_{33}=0.01$ and the remaining ones are provided in Table \ref{['tab:par']}
• Figure 5: Spatial distributions of $u,\,v,\, w$ with $d_{32}=0.01$ in the case of an untreated patient. The parameters are $c=0.25$, $p_2=0.5$, $d_{11}=0.001$, $d_{22}=1.99\cdot 10^{-5}$, $d_{33}=0.01$; the remaining ones are listed in Table \ref{['tab:par']}

Theorems & Definitions (7)

• Theorem 1
• proof
• Theorem 2
• proof
• Remark 3
• Theorem 4
• proof