Androgen Deprivation Therapy Prior to Prostatectomy for Patients With Intermediate and High Risk Prostate Cancer

Degarelix is an approved drug that is used to treat prostate cancer by lowering testosterone levels in the body. Degarelix is commonly given with radiation for prostate cancer, but less frequently with surgery since there has been no proven benefit with this approach. The investigators do not expect the patient to benefit directly from treatment with degarelix since their prostate will be removed shortly after the drug is given. Instead, the investigators hope to learn about how degarelix and other treatment that lowers your testosterone effects prostate cancer cells and use this information to develop better treatments in the future.

Inclusion Criteria: * Histologic confirmation of prostatic adenocarcinoma by MSKCC inclusive of the following: * 3 or more positive biopsy cores or equivalent tumor specimen as confirmed by pathologist * At least 2 cores containing ≥3 mm of tissue with carcinoma or equivalent tumor specimen as confirmed by pathologist * A primary tumor Gleason score ≥ 7 * Adequate primary biopsy tissue or equivalent tumor specimen as confirmed by pathologist available for protocol required analysis (i.e. bladder or TURP specimen) * Planning to have or have had a radical prostatectomy (RP) at MSKCC * Candidates may have a history of deep vein thrombosis, pulmonary embolism, and/or cerebrovascular accident, or require concomitant systemic anticoagulation, if otherwise deemed to be suitable for RP * Karnofsky performance status \>70% (Appendix A) * Sexually active fertile subjects, and their partners, must agree to use medically accepted methods of contraception (eg, barrier methods, including male condom, female condom, or diaphragm with spermicidal gel) during the course of the study and for 3 months after the dose of study drug(s) for Cohorts 1 , 2 and 4, and for 3 months after the surgery for Cohort 3 * For cohorts 1,2 and 4 only:, non-castrate testosterone level (\>100 ng/dL) * For cohort 3 only:, 1-6 months of androgen deprivation therapy (gonadotropin hormone releasing analogs with or without an anti-androgen) prior to prostatectomy with a castrate testosterone level of \