The high-grade malignant brain tumors, glioblastoma multiforme (GBM) and anaplastic astrocytoma (AA), comprise the majority of all primary brain tumors in adults. This group of tumors also exhibits the most aggressive behavior, resulting in median overall survival durations of only 9-12 months for GBM, and 3-4 years for AA. Initial therapy consists of either surgical resection, external beam radiation or both. All patients experience a recurrence after first-line therapy, so improvements in both first-line and salvage therapy are critical to enhancing quality-of-life and prolonging survival. It is unknown if currently used intravenous (IV) therapies even cross the blood brain barrier (BBB). The investigators have shown in a previous phase I trial that a single Super-selective Intraarterial Cerebral Infusion (SIACI) of Bevacizumab (up to 15mg/kg) is safe and effective in the treatment of recurrent GBM. Therefore, this phase I/II clinical research trial is an extension of that trial in that the investigators seek to test the hypothesis that repeated dosing of intraarterial Bevacizumab is safe and effective in the treatment of recurrent malignant glioma. By achieving the aims of this study the investigators will also determine if IV therapy with Bevacizumab should be combined with repeated selected intraarterial Bevacizumab to improve progression free and overall survival. The investigators expect that this project will provide important information regarding the utility of repeated SIACI Bevacizumab therapy for malignant glioma, and may alter the way these drugs are delivered to the patients in the near future.