Lu Min-Jhe, Hao Wenrui, Hu Bei, Li Shuwang
Department of Mathematics, University of California at Irvine, Irvine, CA, 92617, USA.
Department of Mathematics, Pennsylvania State University, University Park, PA, 16802, USA.
J Math Biol. 2023 Jan 7;86(1):19. doi: 10.1007/s00285-022-01862-9.
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 in Lu et al. (J Comput Phys 459:111153, 2022). 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, 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.
大量的研究工作致力于肿瘤模型的研究。已经发现一些生物物理因素，如细胞增殖、凋亡、趋化性、血管生成和坏死，会对肿瘤复杂的生物系统产生影响。肿瘤边界形状的不稳定性是肿瘤发展侵袭性的一个指标。Lu等人（《计算物理杂志》459:111153，2022年）探索了肿瘤形态的复杂模式。在本文中，我们继续对这样一个具有可控坏死核心和趋化性的血管肿瘤模型进行分岔分析。这种对细胞增殖参数的分岔分析是基于径向对称稳态解的显式公式构建的。通过扰动肿瘤自由边界并建立自由边界系统的严格估计，我们用克兰德尔 - 拉宾诺维茨定理证明了分岔分支的存在。从理论和数值上都发现，随着模型的增加，趋化性参数会影响分岔点的单调性。
