Chemotactic Feedback Controls Patterning in Hybrid Tumor--Stroma Model

TL;DR

This paper develops a hybrid PDE-ODE model to understand how chemotactic feedback influences pattern formation and resistance niches in tumor-stroma interactions, with implications for therapy resistance.

Contribution

It introduces a novel hybrid PDE-ODE framework incorporating chemotactic feedback, stromal switching, and realistic therapy modeling, with rigorous mathematical analysis and pattern classification.

Findings

Unidirectional sensing causes transient focusing.

Bidirectional feedback leads to patterning and resistance niches.

Thresholds distinguish between homogeneity, patterning, and aggregation.

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 $\partial_t I=d_I\Delta I-\gamma_I I$, 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.