The impact of time delay and angiogenesis in a tumor model

Abstract

<p style='text-indent:20px;'>We consider a free boundary tumor model under the presence of angiogenesis and time delays in the process of proliferation, in which the cell location is incorporated. It is assumed that the tumor attracts blood vessels at a rate proportional to <inline-formula><tex-math id="M1">\begin{document}$ \alpha $\end{document}</tex-math></inline-formula>, and a parameter <inline-formula><tex-math id="M2">\begin{document}$ \mu $\end{document}</tex-math></inline-formula> is proportional to the 'aggressiveness' of the tumor. In this paper, we first prove that there exists a unique radially symmetric stationary solution <inline-formula><tex-math id="M3">\begin{document}$ \left(\sigma_{*}, p_{*}, R_{*}\right) $\end{document}</tex-math></inline-formula> for all positive <inline-formula><tex-math id="M4">\begin{document}$ \alpha $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M5">\begin{document}$ \mu $\end{document}</tex-math></inline-formula>. Then a threshold value <inline-formula><tex-math id="M6">\begin{document}$ \mu_\ast $\end{document}</tex-math></inline-formula> is found such that the radially symmetric stationary solution is linearly stable if <inline-formula><tex-math id="M7">\begin{document}$ \mu<\mu_\ast $\end{document}</tex-math></inline-formula> and linearly unstable if <inline-formula><tex-math id="M8">\begin{document}$ \mu>\mu_\ast $\end{document}</tex-math></inline-formula>. Our results indicate that the increase of the angiogenesis parameter <inline-formula><tex-math id="M9">\begin{document}$ \alpha $\end{document}</tex-math></inline-formula> would result in the reduction of the threshold value <inline-formula><tex-math id="M10">\begin{document}$ \mu_\ast $\end{document}</tex-math></inline-formula>, adding the time delay would not alter the threshold value <inline-formula><tex-math id="M11">\begin{document}$ \mu_\ast $\end{document}</tex-math></inline-formula>, but would result in a larger stationary tumor, and the larger the tumor aggressiveness parameter <inline-formula><tex-math id="M12">\begin{document}$ \mu $\end{document}</tex-math></inline-formula> is, the greater impact of time delay would have on the size of the stationary tumor.</p>