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Angiogenesis is the growth of new capillaries from pre-existing ones. This complex biological phenomenon plays a critical role in the development of cancer, as tumors gain the ability to promote angiogenesis. The new capillaries provide the tumor the necessary nourishment for its fast growth as well as they are used by the cancer cells to metastasize. For this reason, impeding angiogenesis has became a promising cancer therapy. However, many aspects of the physics and biology of angiogenesis are still unknown and researches from multiple disciplines study this phenomenon under different perspectives. We study tumor angiogenesis by way of a hybrid mathematical model. The model includes mobile, agent-based components and high-order partial differential equations. In this talk, I will present the model, and a effective computational model based on isogeometric analysis, a recent generalization of finite element analysis that permits straightforward discretization of higher-order partial-differential operators. I will present full-scale numerical simulations in two and three dimensions.