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Mode-Wise Spectral Criteria for Coupled Mass Transport in Hybrid PDE--ODE Tumor Microenvironments

Jiguang Yu, Louis Shuo Wang, Zonghao Liu, Jingfeng Liu

TL;DR

This paper develops a hybrid PDE–ODE framework for tumor microenvironments with two motile populations $S$ and $R$, non-diffusive microenvironmental switches $P$ and $A$, and a decaying inhibitory field $D$ that obeys a damped diffusion equation. A diffusive chemoattractant $c$ is introduced to realize chemotaxis, with a critical directionality dichotomy: one-way damped coupling yields a block-triangular linearization that cannot destabilize the population spectrum, while two-way feedback produces effective cross-diffusion and enables mode-wise Turing-type instabilities; the authors provide explicit trace/determinant criteria for instability across Neumann Laplacian modes. They establish well-posedness, positivity, and a long-time reduction to the limiting $(S,R)$ kinetics, proving a unique globally attracting coexistence state. The work connects reaction–diffusion dynamics with chemotaxis in a structured hybrid framework, revealing how the direction of signaling controls pattern formation and offering quantitative thresholds for instability that depend on diffusion, chemotactic sensitivities, and coupling terms. These results have implications for understanding spatial heterogeneity in tumor microenvironments and for designing interventions that modulate microenvironmental signaling to influence tissue patterning and diffusion-driven processes.

Abstract

We study coupled mass transport in a tumor--microenvironment setting with two motile densities $(S,R)$ and non-motile state switching $(P,A)$. The populations diffuse and undergo chemotactic drift; $(P,A)$ follow pointwise ODE switching. A decoupled inhibitory field $D$ satisfies a damped Neumann heat equation, giving maximum-principle bounds and exponential decay. Together with the pointwise invariant $P+A$, these identities yield global existence, positivity, and long-time reduction to limiting $(S,R)$ kinetics with a unique globally attracting coexistence state. Neumann eigenmode reduction gives closed dispersion relations. The base $(S,R)$ reaction--diffusion block remains stable for all nonconstant modes for any $d_S,d_R>0$, excluding classical Turing destabilization. Chemotaxis is posed via a diffusive cue $c$, since $\nabla A$ is undefined for non-diffusive $A$. In one-way damped coupling, the linearized mode matrix is block triangular and leaves the $(S,R)$ spectrum unchanged. Two-way coupling adds a feedback rank-one mobility correction, induces effective cross-diffusion, and admits mode growth. We give explicit trace/determinant criteria for unstable Laplacian modes and the resulting instability thresholds.

Mode-Wise Spectral Criteria for Coupled Mass Transport in Hybrid PDE--ODE Tumor Microenvironments

TL;DR

This paper develops a hybrid PDE–ODE framework for tumor microenvironments with two motile populations and , non-diffusive microenvironmental switches and , and a decaying inhibitory field that obeys a damped diffusion equation. A diffusive chemoattractant is introduced to realize chemotaxis, with a critical directionality dichotomy: one-way damped coupling yields a block-triangular linearization that cannot destabilize the population spectrum, while two-way feedback produces effective cross-diffusion and enables mode-wise Turing-type instabilities; the authors provide explicit trace/determinant criteria for instability across Neumann Laplacian modes. They establish well-posedness, positivity, and a long-time reduction to the limiting kinetics, proving a unique globally attracting coexistence state. The work connects reaction–diffusion dynamics with chemotaxis in a structured hybrid framework, revealing how the direction of signaling controls pattern formation and offering quantitative thresholds for instability that depend on diffusion, chemotactic sensitivities, and coupling terms. These results have implications for understanding spatial heterogeneity in tumor microenvironments and for designing interventions that modulate microenvironmental signaling to influence tissue patterning and diffusion-driven processes.

Abstract

We study coupled mass transport in a tumor--microenvironment setting with two motile densities and non-motile state switching . The populations diffuse and undergo chemotactic drift; follow pointwise ODE switching. A decoupled inhibitory field satisfies a damped Neumann heat equation, giving maximum-principle bounds and exponential decay. Together with the pointwise invariant , these identities yield global existence, positivity, and long-time reduction to limiting kinetics with a unique globally attracting coexistence state. Neumann eigenmode reduction gives closed dispersion relations. The base reaction--diffusion block remains stable for all nonconstant modes for any , excluding classical Turing destabilization. Chemotaxis is posed via a diffusive cue , since is undefined for non-diffusive . In one-way damped coupling, the linearized mode matrix is block triangular and leaves the spectrum unchanged. Two-way coupling adds a feedback rank-one mobility correction, induces effective cross-diffusion, and admits mode growth. We give explicit trace/determinant criteria for unstable Laplacian modes and the resulting instability thresholds.
Paper Structure (42 sections, 14 theorems, 116 equations, 7 figures, 1 table)

This paper contains 42 sections, 14 theorems, 116 equations, 7 figures, 1 table.

Table of Contents

  1. Introduction
  2. Model formulation and standing assumptions
  3. Hybrid PDE--ODE core model
  4. Structural identities
  5. Functional-analytic setting and solution concepts
  6. Spaces and notation.
  7. Abstract form and analytic semigroup.
  8. Chemotaxis extensions with a diffusive signal $c$
  9. Template I: one-way damped coupling (no feedback from $(S,R)$ to $c$).
  10. Template II: two-way coupling (feedback from $(S,R)$ to $c$).
  11. Fast-relaxation closure used in the mode-wise criteria.
  12. Main results
  13. Well-posedness and positivity
  14. Long-time reduction and global attractivity of the limiting kinetics
  15. Absence of diffusion-driven (Turing) instability in the base reaction--diffusion subsystem
  16. ...and 27 more sections

Key Result

Lemma 2.2

Let $(P,A)$ solve the last two equations of eq:full_system for a given (measurable) signal $D(x,t)$. Then for a.e. $x\in U$ and all $t\ge 0$, In particular, if $P_0,A_0\ge 0$ a.e. in $U$, then $P(x,t),A(x,t)\ge 0$ for all $t\ge 0$, and

Figures (7)

  • Figure 1: Comparison of pattern formation mechanisms and the proposed hybrid framework.(A) Classical diffusion-driven (Turing) instability generates spatial patterns from homogeneity (Eq. \ref{['eq:intro_RD_general']}). (B) Chemotaxis (Keller--Segel) drives directed transport leading to aggregation or blow-up (Eq. \ref{['eq:intro_KS']}). (C) The proposed hybrid TME framework couples motile populations (PDE) to non-motile microenvironmental states ($A$, ODE) via a diffusive signal ($c$) to rigorously define chemotactic drift.
  • Figure 2: Directionality dichotomy in stability results (Main Analytical Contributions).(Left) Under one-way damped coupling, the Jacobian is block-triangular, maintaining the stability of the population spectrum (all eigenvalues $\Re \lambda < 0$). Spatially, the system remains homogeneous. (Right) Under two-way feedback, the interaction generates effective cross-diffusion, causing eigenvalues to cross the imaginary axis ($\Re \lambda > 0$). This leads to the recovery of spatial Turing patterns.
  • Figure 3: The diffusive and non-diffusive components.
  • Figure 4: The model structure and their interactions.
  • Figure 5: Graphs of $\phi$ and $\delta$.
  • ...and 2 more figures

Theorems & Definitions (39)

  • Remark 2.1: Chemotaxis extensions
  • Lemma 2.2: Pointwise conservation in the non-diffusive $(P,A)$ subsystem
  • proof
  • Proposition 2.3: Maximum principle and exponential energy decay for the signal $D$
  • proof
  • Remark 2.4: Spectral representation
  • Definition 2.5: Mild solution
  • Remark 2.6: Unshifted form
  • Lemma 3.1: Local Lipschitz continuity of the reaction map
  • Theorem 3.2: Nonnegativity
  • ...and 29 more