Tumor Angiogenesis and Vascular Patterning: A Mathematical Model

Understanding tumor induced angiogenesis is a challenging problem with important consequences for diagnosis and treatment of cancer. Recently, strong evidences suggest the dual role of endothelial cells on the migrating tips and on the proliferating body of blood vessels, in consonance with further events behind lumen formation and vascular patterning. In this paper we present a multi-scale phase-field model that combines the benefits of continuum physics description and the capability of tracking individual cells. The model allows us to discuss the role of the endothelial cells' chemotactic response and proliferation rate as key factors that tailor the neovascular network. Importantly, we also test the predictions of our theoretical model against relevant experimental approaches in mice that displayed distinctive vascular patterns. The model reproduces the in vivo patterns of newly formed vascular networks, providing quantitative and qualitative results for branch density and vessel diameter on the order of the ones measured experimentally in mouse retinas. Our results highlight the ability of mathematical models to suggest relevant hypotheses with respect to the role of different parameters in this process, hence underlining the necessary collaboration between mathematical modeling, in vivo imaging and molecular biology techniques to improve current diagnostic and therapeutic tools.

解析肿瘤诱导血管生成（tumor induced angiogenesis）是一项兼具挑战性与重要临床价值的研究课题，其成果对癌症的诊断与治疗均具有关键意义。近年来，大量可靠证据表明，内皮细胞（endothelial cells）在血管迁移尖端与增殖主体中发挥着双重作用，且与管腔形成及血管模式构建背后的一系列后续事件协同发生。本文提出了一种多尺度相场模型（multi-scale phase-field model），该模型兼具连续介质物理描述的优势与单个细胞追踪的能力。借助此模型，我们得以探讨内皮细胞的趋化应答与增殖速率作为塑造新生血管网络的关键因素所发挥的作用。尤为重要的是，我们还将理论模型的预测结果与展现出独特血管模式的小鼠相关实验数据进行了对照验证。该模型能够复现新生血管网络的体内（in vivo）形态，在分支密度与血管直径两项指标上得到了与小鼠视网膜实验测量结果量级相符的定量与定性结果。本研究结果证实了数学模型可针对该过程中不同参数的作用提出合理假说，进而凸显了数学建模、体内成像（in vivo imaging）与分子生物学技术之间的必要协同，以优化现有的癌症诊断与治疗手段。