Chemotactic Feedback Controls Patterning in Hybrid Tumor--Stroma Model

TL;DR

This work develops a rigorous hybrid PDE-ODE framework to study spatial patterning and resistance niches in tumor–stroma systems under a single-dose open-loop drug. A key finding is that after drug washout the base damped dynamics exhibit no diffusion-driven instability, but introducing chemotaxis with bidirectional feedback can generate finite-band patterns or, under strong focusing without regularization, aggregation. The authors formalize a directionality–damping principle and provide a clear three-regime classification (stable, finite-band, ill-posed) tied to the effective mobility, supported by reproducible numerical simulations. The results elucidate when spatial heterogeneity can emerge from homogeneous states and highlight the importance of feedback structure and regularization in modeling tumor–stroma interactions with transient therapies, offering mechanistic guidance for therapeutic strategies and model refinement.

Abstract

Motivated by an ongoing collaboration with clinical oncologists and pathologists, we develop a hybrid partial differential equation--ordinary differential equation (PDE--ODE) framework that captures (i) competition between susceptible and resistant phenotypes, (ii) stromal state switching, and (iii) a clinically realistic open-loop, single-dose therapeutic agent $I$ with diffusion and clearance. Clinical management of solid tumors is increasingly limited by spatial heterogeneity and therapy-induced resistance niches that are difficult to predict from well-mixed models. We establish a rigorous mathematical backbone with forward invariance of the nonnegative cone and global-in-time well-posedness. Exploiting the decoupled drug equation $\partial_t I=d_IΔI-γ_I I$, we prove a long-time reduction during washout and show that the damped base dynamics admit no diffusion-driven (Turing-type) instability. We then formulate a directionality--damping principle: unidirectional (open-loop) sensing yields at most transient focusing, whereas bidirectional (closed-loop) feedback reshapes the effective mobility and produces explicit thresholds separating stable homogeneity, finite-band patterning (resistance niche formation), and aggregation when strong parabolicity is violated. Reproducible simulations corroborate this classification and highlight when flux regularization is required for physical realism.

Figures (9)

• Figure 1: Unified mechanism diagram. Top: hybrid PDE--ODE coupling topology for tumor phenotypes $(S,R)$, drug $I$, and stromal switching $(P,F_a)$ under Neumann boundary conditions. Bottom: the directionality--damping principle: (i) drug washout and logistic damping enforce baseline homogenization (no Turing); (ii) unidirectional open-loop sensing yields at most transient focusing; (iii) bidirectional feedback reshapes the effective mobility $M$ and yields stable, finite-band (Turing-type) niche formation, or aggregation/ill-posedness depending on strong parabolicity.
• Figure 2: Comparison of spatial instability mechanisms.(A) Diffusion-driven instability (Turing mechanism): interaction between local reaction kinetics and differential diffusion generates stable, self-organized spatial patterns (e.g., stripes) from a homogeneous state. (B) Aggregation-driven instability (chemotaxis mechanism): directed transport leads to mass concentration and blow-up phenomena (spots) rather than smooth spatial structures. This paper investigates the transition between these regimes.
• Figure 3: Topology of the hybrid PDE--ODE interaction network. The inhibitor $I$ acts as a central signaling hub connecting the tumor layer $(S,R)$ and the microenvironmental layer $(P,F_a)$. Key features include $I$ providing negative feedback to $S$ ($-\delta(I)S$) while activating the microenvironmental stromal cells via the switch $\phi(I)$. The switch promotes resistance via the $\eta\phi(I)F_aR$ term, creating a drug-gated feedback channel from stroma to the resistant phenotype.
• Figure 4: Phase portrait of the reduced $(S,R)$ subsystem. Red and green dashed lines depict the $S$- and $R$-nullclines, respectively. The black dashed line shows the one-dimensional invariant manifold $S+R=K$, where the $S$ and $R$ populations reach the carrying capacity $K$. The solid arrows demonstrate the direction and relative magnitude of the vector field, indicating the attractivity of the unique steady state $\mathbb K^*$ that is represented by the red pentagram. Representative trajectories starting from the block dots approach $\mathbb K^*$. Parameters: $K=10$, $\lambda_S=0.8$, $\lambda_R=0.4$, $\alpha=0.1$, $\xi=0.3$.
• Figure 5: Classification of stability regimes (\ref{['thm:twoway_classification']}). Plot of growth rate $\Re(\lambda)$ vs. wavenumber $k$. Case I (green): unconditional stability. Case II (blue): well-posed finite-band (Turing-type) instability. Case III (red): loss of parabolicity (growth rate diverges), indicating aggregation/ill-posedness.

Theorems & Definitions (25)

• Theorem 3.1: Nonnegativity
• proof
• Proposition 3.2: Bounds for $(I,P,F_a)$
• proof
• Remark 3.3
• Theorem 3.4: Global existence of mild solutions and classical regularity
• proof
• Remark 3.5: Mechanism behind global continuation
• Theorem 3.6: Global attractivity of the coexistence equilibrium
• proof