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Does indirectness of signal production reduce the explosion-supporting potential in chemotaxis-haptotaxis systems? Global classical solvability in a class of models for cancer invasion (and more)

Christina Surulescu, Michael Winkler

TL;DR

The paper addresses whether indirect production of chemoattractants can curb explosion-like invasion dynamics in chemotaxis-haptotaxis systems. By analyzing a general parabolic-ODE-ODE framework in 2D and introducing a Lyapunov-like quasi-energy structure, the authors prove global classical solvability under mild assumptions, exploiting the indirect production to suppress blow-up. A pivotal step is establishing uniform $L^ty$ bounds for the motile population via a weighted Moser iteration, together with a careful energy-dissipation balance. Simulations corroborate the theory, showing bounded invasion when signal production is indirect and highlighting potential blow-up when production is direct, thereby underscoring the biological relevance of indirect signaling in cancer invasion models.

Abstract

We propose and study a class of parabolic-ODE models involving chemotaxis and haptotaxis of a species following signals indirectly produced by another, non-motile one. The setting is motivated by cancer invasion mediated by interactions with the tumor microenvironment, but has much wider applicability, being able to comprise descriptions of biologically quite different problems. As a main mathematical feature consituting a core difference to both classical Keller-Segel chemotaxis systems and Chaplain-Lolas type chemotaxis-haptotaxis systems, the considered model accounts for certain types of indirect signal production mechanisms. The main results assert unique global classical solvability under suitably mild assumptions on the system parameter functions in associated spatially two-dimensional initial-boundary value problems. In particular, this rigorously confirms that at least in two-dimensional settings, the considered indirectness in signal production induces a significant blow-up suppressing tendency also in taxis systems substantially more general than some particular examples for which corresponding effects have recently been observed.

Does indirectness of signal production reduce the explosion-supporting potential in chemotaxis-haptotaxis systems? Global classical solvability in a class of models for cancer invasion (and more)

TL;DR

The paper addresses whether indirect production of chemoattractants can curb explosion-like invasion dynamics in chemotaxis-haptotaxis systems. By analyzing a general parabolic-ODE-ODE framework in 2D and introducing a Lyapunov-like quasi-energy structure, the authors prove global classical solvability under mild assumptions, exploiting the indirect production to suppress blow-up. A pivotal step is establishing uniform bounds for the motile population via a weighted Moser iteration, together with a careful energy-dissipation balance. Simulations corroborate the theory, showing bounded invasion when signal production is indirect and highlighting potential blow-up when production is direct, thereby underscoring the biological relevance of indirect signaling in cancer invasion models.

Abstract

We propose and study a class of parabolic-ODE models involving chemotaxis and haptotaxis of a species following signals indirectly produced by another, non-motile one. The setting is motivated by cancer invasion mediated by interactions with the tumor microenvironment, but has much wider applicability, being able to comprise descriptions of biologically quite different problems. As a main mathematical feature consituting a core difference to both classical Keller-Segel chemotaxis systems and Chaplain-Lolas type chemotaxis-haptotaxis systems, the considered model accounts for certain types of indirect signal production mechanisms. The main results assert unique global classical solvability under suitably mild assumptions on the system parameter functions in associated spatially two-dimensional initial-boundary value problems. In particular, this rigorously confirms that at least in two-dimensional settings, the considered indirectness in signal production induces a significant blow-up suppressing tendency also in taxis systems substantially more general than some particular examples for which corresponding effects have recently been observed.

Paper Structure

This paper contains 6 sections, 21 theorems, 139 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Local existence and basic estimates
  3. A quasi-energy inequality
  4. $L^\infty$ bounds for $u$
  5. Global extensibility. Proof of Theorem \ref{['theo22']}
  6. Simulations and discussion

Key Result

Theorem 1.1

Let $\Omega\subset \mathbb{R}^2$ be a bounded domain with smooth boundary, assume that $\chi, \xi, \alpha, \beta, D_u$ and $D_h$ are positive, and suppose that $f$, $g$, $\phi$, $\Phi$ and $\psi$ satisfy (fgphipsi_reg), (Hf), (Hg), (Hphi), (HPhi) and (Hpsi). Then for any choice of $(u_0,h_0,v_0,w_0)

Figures (2)

  • Figure 1: Evolution of tumor (first column), tissue (2nd column), MDEs (3rd column), and CAFs (last column) for model \ref{['eq:cafs']} with $\mu =0$. Succesive times from top to bottom, top row: initial conditions.
  • Figure 2: Evolution of tumor (first column), tissue (2nd column), and MDEs (last column) for model \ref{['eq:no-cafs']} with $\mu =0$. Successive times from top to bottom, top row: initial conditions.

Theorems & Definitions (21)

  • Theorem 1.1
  • Lemma 2.1
  • Lemma 2.2
  • Lemma 2.3
  • Lemma 3.1
  • Lemma 3.2
  • Lemma 3.3
  • Lemma 3.4
  • Lemma 3.5
  • Lemma 3.6
  • ...and 11 more