Computational Oncology of Chemotaxis-Driven Tumour–Immune Spatial Patterning and Stability

Jul 16, 2026

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Spatial tumour–immune heterogeneity drives solid-tumour progression through immune infiltration and immune exclusion. A computational oncology reaction–diffusion–chemotaxis model is developed on a bounded tissue domain with no-flux boundaries, coupling tumour-cell density, immune effector cells, and a chemokine signal produced by tumour cells and contact-dependent recruitment. Nondimensional analysis establishes positivity, tumour-density bounds, and immune/chemokine mass estimates, defines an immune-control threshold, and reduces coexistence to a scalar equation. Mode-wise linear stability yields a chemotaxis-driven finite-wavelength instability above a critical sensitivity, validated by a conservative finite-volume scheme with upwind chemotactic flux.

LanguageEnglish

CategoryResearch & Report

FormatPDF

Pages36

Reading Time1 h 31 min

Introduction

Motivation: computational tumour modelling

Biological setting: tumour–immune–chemokine interactions

Computational Oncology of Chemotaxis-Driven Tumour–Immune Spatial Patterning and Stability Zonghao Liua,1 , Jiguang Yub,1 , Lei Suc,d,1 , Louis Shuo Wange,∗,1 , Yang Duc and Jingfeng Liua,f,g,∗ a Department of Hepatopancreatobiliary Surgery, Clinical Oncology School of Fujian Medical University, Fuzhou, 350014, Fujian, China b College of Engineering, Boston University, Boston, 02215, MA, United States c CAS Key Laboratory of Molecular Imaging, Institute of Automation, Chinese Academy of Sciences, Beijing, 100190, China d University of Chinese Academy of Sciences, Beijing, 100190, China e Department of Mathematics, Northeastern University, Boston, 02115, MA, United States f NHC Key Laboratory of Cancer Metabolism, Fuzhou, 350014, Fujian, China g Fujian Key Laboratory of Advanced Technology for Cancer Screening and Early Diagnosis, Fuzhou, 350014, Fujian, China | ARTICLE INFO | AB STRACT | | --- | --- | | Keywords: | Spatial tumour–immune heterogeneity is a key feature of solid-tumour progression, immune | | tumour–immune dynamics; chemo- | infiltration, and immune exclusion. We develop a computational oncology model in which | | taxis; reaction–diffusion; linear sta- | tumour cells, immune effector cells, and a chemokine signal interact through a reaction– | | bility; dispersion relation; pattern | diffusion–chemotaxis system on a bounded tissue domain with no-flux boundaries. Chemokine | | formation; finite-volume method; pos- | is produced by tumour cells and tumour–immune contact, recruits immune cells, and guides | | itivity preservation. | chemotactic migration. After nondimensionalization, we establish positivity, a tumour-density | | 2020 MSC: 35K57; 92C17; 65M08; | bound, and immune/chemokine mass estimates. We identify the tumour-free equilibrium, derive | | 35B36; 92C50 | the immune-control threshold 􀀛0 > 􀀎, and reduce coexistence to a scalar equation. Linear stability analysis about coexistence yields a mode-wise dispersion relation in which chemotaxis appears as a wavenumber amplified coupling, producing finite-wavelength instability above a critical sensitivity. A conservative finite-volume scheme with upwind chemotactic flux verifies the thresholds, dominant unstable modes, sensitivity maps, positivity, convergence, and residual consistency. | 1. Introduction 1.1. Motivation: computational tumour modelling Cancer progression is governed by interacting processes across molecular, cellular, and tissue scales [1, 2] . At the molecular scale, cytokines, chemokines, growth factors, and metabolic signals regulate proliferation, death, migration, and immune activation [3] . At the cellular scale, tumour cells, immune effector cells, stromal cells, and endothelial cells interact through direct contact and through diffusible mediators [4, 5, 6] . At the tissue scale, these local interactions produce spatially heterogeneous patterns of tumour expansion, immune infiltration, immune exclusion, necrosis, invasion fronts, and treatment response [7, 8] . Therefore, tumour progression is not only a temporal growth process; it is also a spatially structured dynamical process. Spatial mathematical models are particularly useful for describing these mechanisms because they connect local cell–signal interactions with macroscopic tumour behaviour. Continuum models based on reaction–diffusion equations describe cell proliferation, death, signal production, degradation, and passive spreading [9] . Chemotaxis terms describe directed cell migration along chemical gradients, which is especially relevant when immune cells respond to chemokine fields generated inside the tumour microenvironment [10] . In this way, reaction–diffusion–chemotaxis systems provide a mechanistic framework for studying how microscopic signalling and cell movement generate macroscopic invasion, immune infiltration, immune exclusion, or coexistence patterns [11, 12, 13, 14, 15, 16, 17] . From a computational viewpoint, such models also provide a

FAQs

What biological interactions does the model represent?

The model represents tumour–immune interactions mediated by tumour-produced chemokines, where chemokine formation and tumour–immune contact recruit immune cells and guide their chemotactic migration.

How is chemotaxis incorporated and what role does it play in stability?

Chemotaxis is included through a reaction–diffusion–chemotaxis coupling that amplifies coupling across wavenumbers. Linear stability shows chemotaxis can trigger a finite-wavelength instability once sensitivity exceeds a critical threshold.

What mathematical and numerical properties are emphasized?

The study proves positivity, derives bounds and mass estimates, establishes an immune-control threshold, and uses a conservative finite-volume method with upwind chemotactic flux to verify stability thresholds, dominant unstable modes, and convergence while preserving no-flux boundaries.

Jul 16, 2026

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