A Review on the Qualitative Behavior of Solutions in Some Chemotaxis–Haptotaxis Models of Cancer Invasion

Yifu Wang

doi:10.3390/math8091464

Abstract

Chemotaxis is an oriented movement of cells and organisms in response to chemical signals, and plays an important role in the life of many cells and microorganisms, such as the transport of embryonic cells to developing tissues and immune cells to infection sites. Since the pioneering works of Keller and Segel, there has been a great deal of literature on the qualitative analysis of chemotaxis systems. As an important extension of the Keller–Segel system, a variety of chemotaxis–haptotaxis models have been proposed in order to gain a comprehensive understanding of the invasion–metastasis cascade. From a mathematical point of view, the rigorous analysis thereof is a nontrivial issue due to the fact that partial differential equations (PDEs) for the quantities on the macroscale are strongly coupled with ordinary differential equations (ODEs) modeling the subcellular events. It is the goal of this paper to describe recent results of some chemotaxis–haptotaxis models, inter alia macro cancer invasion models proposed by Chaplain et al., and multiscale cancer invasion models by Stinner et al., and also to introduce some open problems.

Highlights

Chemotaxis is an oriented movement of individual cells in response to some signaling chemical, and is regarded as a universal migration mechanism in a wide range of biological processes such as the migration of embryonic cells to developing tissues, and immune cells to infection sites
In the past several decades, chemotaxis models have received a great deal of attention in the academic literature due to their potential to generate aggregation patterns in several relevant situations [1,2,3,4]
Without the pretension of exhaustiveness, this paper provides a short review of the global bounded results on some cancer invasion models and sketches necessary proofs thereof

Summary

Introduction

Chemotaxis is an oriented movement of individual cells in response to some signaling chemical (chemoattractant), and is regarded as a universal migration mechanism in a wide range of biological processes such as the migration of embryonic cells to developing tissues, and immune cells to infection sites. In the past several decades, chemotaxis models have received a great deal of attention in the academic literature due to their potential to generate aggregation patterns in several relevant situations [1,2,3,4]. Taking into account the microscopic dynamics of intracellular protons and their exchange with extracellular counterparts, a population-based micro–macro model for acid-mediated tumor invasion was proposed by Meral et al in [18]. These continuum micro–macro models explicitly involving subcellular events are rather novel, especially in the context of cancer cell migration [22,23].

Macroscopic Cancer Invasion Models

Multiscale Cancer Invasion Models

Conclusions

Methods