Prostate Specific Membrane Antigen (PSMA) or (FACBC) PET/CT Site-Directed Therapy for Treatment of Prostate Cancer, Flu-BLAST-PC Study

Inclusion Criteria: * Patient must have histologically or cytologically documented evidence of prostate adenocarcinoma * Patient must previously have undergone radical prostatectomy * Patient must previously have undergone either adjuvant or salvage radiation therapy to the prostatic fossa +/- whole pelvis * Patient must have a prostate specific antigen (PSA) \>= 0.2 and \< 10 ng/mL. If there is only one PSA value that has risen to \>= 0.2 with this biochemical recurrence, a second PSA value must be confirmed to be within \>= 0.2 and \< 10 ng/mL at least 2 weeks from the first value and within 28 days of enrollment * PSA doubling time must be calculated utilizing either all PSA measurements \> 0.1 ng/mL from most recent biochemically-recurred (BCR) or the most recent 3 PSA measurements \> 0.1 ng/mL (if the latter, all 3 PSA measurements must be \> 2 weeks apart to be used in the calculation). PSA doubling time must be \> 3 months and \< 18 months. The Memorial Sloan Kettering PSA doubling time calculator should be used * Patient must have no previous evidence of radiographically detectable metastatic prostate cancer by conventional CT and bone scan imaging * Patient must have total testosterone level \> 120 ng/dL demonstrated within 42 days of enrollment * Patient must have Eastern Cooperative Oncology Group (ECOG) performance status of 0 or 1 * Absolute neutrophil count (ANC) \>= 1.0 X 10\^9/L * Platelet count \>= 100 X 10\^9/L * Hemoglobin \>= 9 g/dL * Potassium \>= 3.5 * …