Evaluation of Weekly Ixabepilone With or Without Biweekly Bevacizumab

The primary objective of this study is as follows: * To assess the activity of ixabepilone with bevacizumab compared to ixabepilone alone in patients with recurrent or persistent platinum-resistant/refractory epithelial (non-mucinous) ovarian, fallopian tube, or primary peritoneal cancer. We will assess this by comparing the ixabepilone +bevacizumab (experimental) arm to the ixabepilone-alone (control) arm for an improvement in median progression free survival (PFS). The secondary objectives of this study are as follows: * To compare the experimental arm to the control arm for increases in objective response rate (ORR) and durable disease control rate (DDCR). * To compare the experimental arm to the control arm for an increase in overall survival (OS). * To assess the safety profile of ixabepilone in combination with bevacizumab in ovarian, fallopian tube, or primary peritoneal cancer patients. * To assess whether prior treatment with bevacizumab impacts future response to bevacizumab in combination with ixabepilone. In addition to the primary and secondary objectives of this study, there are additional exploratory/correlative objectives. The exploratory/correlative objectives of this study are as follows: * To characterize number, length and composition (e.g., class III β-tubulin expression) of microtentacles (McTNs) isolated from circulating tumor cells isolated from whole blood of patients undergoing treatment with ixabepilone with or without bevacizumab, and correlate with best response, PFS, and OS. * To observe McTNs on circulating tumor cells in blood using a novel polyelectrolyte multi-layer (PEM) tethering technology. * To correlate ex vivo response of McTNs to drug treatment with clinical response in order to develop a real-time assay to predict response to therapy. * To explore use of circulating tumor (ct) DNA as a biomarker for disease response and compare its performance to CA-125. * To examine whether clinical response to ixabepilone with or without bevacizumab differs between high and low expressors of class III β-tubulin.