Bifurcation analysis of a free boundary model of vascular tumor growth with a necrotic core and chemotaxis

TL;DR

This paper performs a bifurcation analysis on a vascular tumor growth model with a necrotic core and chemotaxis, revealing how chemotaxis influences tumor boundary instability and morphology.

Contribution

It introduces a bifurcation analysis based on explicit steady-state solutions, extending previous models by incorporating a necrotic core and chemotaxis effects.

Findings

Chemotaxis affects the monotonicity of bifurcation points.

Existence of bifurcation branches proven using Crandall-Rabinowitz theorem.

Parameter of cell proliferation influences tumor boundary stability.

Abstract

A considerable number of research works has been devoted to the study of tumor models. Several biophysical factors, such as cell proliferation, apoptosis, chemotaxis, angiogenesis and necrosis, have been discovered to have an impact on the complicated biological system of tumors. An indicator of the aggressiveness of tumor development is the instability of the shape of the tumor boundary. Complex patterns of tumor morphology have been explored by Lu, Min-Jhe et al. [Nonlinear simulation of vascular tumor growth with chemotaxis and the control of necrosis, Journal of Computational Physics 459 (2022): 111153]. In this paper, we continue to carry out a bifurcation analysis on such a vascular tumor model with a controlled necrotic core and chemotaxis. This bifurcation analysis, to the parameter of cell proliferation, is built on the explicit formulas of radially symmetric steady-state solutions. By perturbing the tumor free boundary and establishing rigorous estimates of the free boundary system, %applying the Hanzawa transformation, we prove the existence of the bifurcation branches with Crandall-Rabinowitz theorem. The parameter of chemotaxis is found to influence the monotonicity of the bifurcation point as the mode $l$ increases both theoretically and numerically.