Most cancer treatment strategies target cell proliferation, angiogenesis, migration, and intravasation of tumor cells in an attempt to limit tumor growth and metastasis. An in vitro platform to assess tumor progression and drug sensitivity could provide avenues to enhance our understanding of tumor metastasis as well as precision medicine. We present a microfluidic platform that mimics biological mass transport near the arterial end of a capillary in the tumor microenvironment. A central feature is a quiescent perfused 3D microvascular network created prior to loading tumor cells or patient-derived tumor organoids in an adjacent compartment. The physiological delivery of nutrients and/or drugs to the tumor then occurs through the vascular network. We demonstrate the culture, growth, and treatment of tumor cell lines and patient-derived breast cancer organoids. The platform provides the opportunity to simultaneously and dynamically observe hallmark features of tumor progression including cell proliferation, angiogenesis, cell migration, and tumor cell intravasation. Additionally, primary breast tumor organoids are viable in the device for several weeks and induce robust sprouting angiogenesis. Finally, we demonstrate the feasibility of our platform for drug discovery and personalized medicine by analyzing the response to chemo- and anti-angiogenic therapy. Precision medicine-based cancer treatments can only be realized if individual tumors can be rapidly assessed for therapeutic sensitivity in a clinically relevant timeframe (≲14 days). Our platform indicates that this goal can be achieved and provides compelling opportunities to advance precision medicine for cancer.