A free boundary problem for necrotic tumor growth with angiogenesis

Abstract

This paper deals with a mathematical model for tumor growth with angiogenesis and a necrotic core. The problem is a nonlinear problem which has two free boundaries: the external boundary of the tumor and the external boundary of the internal necrotic core. Using the implicit function theorem, the relationship between the two free boundaries is studied. The asymptotic stability of the unique steady-state solution is proved. The effect of nutrient concentration on tumor growth is studied. The relationship between the stage with a necrotic core and the stage without necrotic core is also studied. The results show that whether the steady-state of tumor growth has a necrotic core or not is determined by the concentration of external nutrients. The stage of tumor with necrotic core and the stage without necrotic core could transform into each other by adjusting the concentration of external nutrients.

Keywords:

• Tumor growth with angiogenesis
• necrotic core
• free boundary problem
• stability

2010 Mathematics Subject Classifications:

• 35K57
• 35Q92
• 35R35

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

This work was partially supported by National Natural Science Foundation of China (11301474), Natural Science Foundation of Guangdong Province (2018A030313536).