由血管生成驱动的肿瘤侵袭的自由边界分析
A free boundary analysis of tumor invasion driven by angiogenesis
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中文总结 AI 辅助
研究由血管生成驱动的肿瘤侵袭的自由边界模型，通过该模型描述肿瘤细胞扩散等情况，证明肿瘤能长期存活并得出相关表达式，还根据比率κ区分肿瘤发展的两种情况。
中文摘要 AI 辅助
我们讨论了一个肿瘤侵袭的自由边界模型，该模型描述了一群细胞，它们既能扩散，又能沿着趋化方向的向量场漂移。该模型捕捉了实体肿瘤的演变，包括血管生成过程，即形成新血管为肿瘤提供氧气和其他营养物质，从而促进其扩散和生长。我们证明，一旦形成，肿瘤会长期存活，保持严格的正厚度，并根据初始数据得出了一个明确的表达式。此外，我们根据肿瘤细胞扩散与肿瘤质量增长的比率κ区分了两种情况。如果κ足够大，肿瘤会随时间呈指数增长并侵袭整个宿主组织。相反，如果κ足够小，肿瘤要么随时间保持大小受限，要么可能会迅速收缩。
英文摘要
We discuss a free boundary model for tumor invasion that describes a cloud of cells that diffuse and, at the same time, are drifted along the vector field of the chemotactic direction. The model captures the evolution of a solid tumor, including the process of angiogenesis, which consists in the formation of new blood vessels that supply the tumor with oxygen and other nutrients, thereby promoting its spread and growth. We prove that, once formed, the tumor survives through time, maintaining strictly positive thickness. An explicit expression in terms of the initial data is derived. Moreover, we distinguish two regimes depending on the ratio $κ$ between the spreading of tumor cells and the growth of the tumor mass. If $κ$ is sufficiently large, then the tumor grows exponentially in time and invades the entire host tissue. In contrast, if $κ$ is small enough, then either the tumor remains bounded in size over time or may experience a fast contraction.
