On a cross-diffusion hybrid model: Cancer Invasion Tissue with Normal Cell Involved

TL;DR

This paper introduces a new mathematical model for cancer invasion that includes interactions with normal cells, addressing complex cross-diffusion dynamics and proving well-posedness in low-dimensional spaces.

Contribution

The study develops a novel cross-diffusion hybrid model for cancer invasion involving normal cells and proves its well-posedness in dimensions up to three.

Findings

Established existence and uniqueness of solutions in dimensions ≤ 2.

Proved global existence of solutions in dimension 3.

Developed a decoupling strategy for complex coupled systems.

Abstract

In this paper, we study a well-posedness problem on a new mathematical model for cancer invasion within the plasminogen activation system, which explicitly incorporates cooperation with host normal cells. Key biological mechanisms--including chemotaxis, haptotaxis, recruitment, logistic growth, and natural degradation of normal cells--along with other primary components (cancer cells, vitronectin, uPA, uPAI-1 and plasmin) are modeled via a continuum framework of cancer cell invasion of the extracellular matrix. The resulting model constitutes a strongly coupled, cross-diffusion hybrid system of differential equations. The primary mathematical challenges arise from the strongly coupled cross-diffusion terms, the parabolic operators of divergence form, and the interaction between the cross-diffusion fluxes and the ODE components. We address these by deriving several a priori estimates for dimensions d less or equal to 3. Subsequently, we employ a decoupling strategy to split the system into proper sub-problems, establishing the existence (and uniqueness) for each subsystem. Finally, we demonstrate the global existence and uniqueness of the solution for dimensions d less or equal to 2 and the global existence of a solution for dimension d = 3.