Biomarker Analysis of Castration-resistant Prostate Cancer Undergoing Bipolar Androgen Therapy

Inclusion Criteria: * Willing and able to provide signed informed consent. * Males aged 18 years of age and above. * Histological or cytologic proof of adenocarcinoma of the prostate. * Known castration-resistant disease, defined according to PCWG3 criteria as: castrate serum testosterone level ≤ 50 ng/dL (≤ 1.7 nmol/L). Subjects who have failed initial hormonal therapy, either by orchiectomy or by using a GnRH agonist in combination with an anti-androgen, must first progress through anti-androgen withdrawal prior to being eligible. The minimum timeframe to document failure of anti-androgen withdrawal will be 4 weeks. * Disease progression: serum PSA progression defined as 2 consecutive increases in PSA over a previous reference value within 6 months, each measurement at least 1 week apart, or documented bone lesions by the appearance of ≥ 2 new lesions by bone scintigraphy or dimensionally measurable soft tissue metastatic lesion assessed by CT or MRI. * Absolute PSA ≥ 1.0 ng/mL at screening. * Must have PSA and/or radiographic progression on AT LEAST ONE novel AR- targeted therapy (abiraterone acetate, enzalutamide, apalutamide or darolutamide). One prior chemotherapy agent for mCRPC will be allowed but is not required for inclusion. * Prior treatment with abiraterone, enzalutamide, apalutamide, darolutamide, bicalutamide, and/or ketoconazole is allowed. There is no limit on the maximum number or types of prior hormonal therapies received. * Must be maintained on a GnRH analogue or have undergone orchiectomy. * Radiographic evidence of metastatic disease by CT scan and/or bone scan, performed within the prior 6 months * Karnofsky Performance Status (KPS): ≥ 80% within 14 days before start of study treatment (ECOG \< 2) * Asymptomatic or minimally symptomatic mCRPC according to Brief Pain Inventory - Short Form (BPI-SF) performed during screening: asymptomatic is defined as BPI-SF item #3 score of 0 to 1; minimally symptomatic is defined as BPI-SF item #3 score of 2 to 4. * Archived tumor tissue obtained prior to enrollment from a metastatic tumor lesion or from a primary tumor lesion (formalin fixed paraffin-embedded \[FFPE\] block or unstained tumor tissue sections). Tumor sample may be from core biopsy, punch biopsy, excisional biopsy, or surgical specimen). * Participants must have adequate organ and bone marrow function measured within 28 days prior to administration of study treatment as defined below: * Hemoglobin ≥ 9.0 g/dL with no blood transfusion in the past 28 days * Absolute neutrophil count (ANC) ≥ 1.0 x 109/L * Platelet count ≥ 100 x 109/L * Total bilirubin within institutional upper limit of normal (ULN) (In patients with Gilbert's syndrome, total bilirubin \< 1.5x institutional ULN will be acceptable) * Aspartate aminotransferase (AST) (Serum Glutamic Oxaloacetic Transaminase (SGOT)) / Alanine aminotransferase (ALT) (Serum Glutamic Pyruvate Transaminase (SGPT)) within institutional upper limit of normal * Participants must have Creatinine Clearance estimated using the Modified Cockcroft-Gault equation of ≥ 40 mL/min: Estimated Creatinine Clearance = (140-age \[years\]) x weight (kg) serum creatinine (mg/dL) x 72 * Participants must have a life expectancy ≥ 6 months. * Male participants and their partners, who are sexually active and of childbearing potential, must agree to the use of two highly effective forms of contraception in combination, throughout the period of taking study treatment and for 6 months after the last dose of BAT, to prevent pregnancy in a partner. * No evidence (within 5 years) of prior malignancies (except successfully treated basal cell or squamous cell carcinoma of the skin). Exclusion Criteria: * External-beam radiotherapy within the last 4 weeks prior to start of study treatment. * Prior oral anti-androgen (e.g. bicalutamide, nilutamide, enzalutamide, apalutamide, darolutamide), or androgen synthesis inhibitor (e.g. abiraterone, orteronel) within the past 4 weeks is not permitted. 5-alpha reductase inhibitor therapies are not allowed as well. * Prior treatment with chemotherapy for the treatment of metastatic hormone- sensitive prostate cancer is allowed if the last dose of chemotherapy was ≥ 6 months prior to enrollment. In addition, one prior chemotherapy agent for mCRPC will be allowed after a minimum wash-out period of 4 weeks prior to enrollment. * Patients who have received prior treatment with bipolar androgen therapy (e.g. testosterone, BAT). * Pain due to metastatic prostate cancer requiring opioid therapy. * Patients with an intact prostate AND urinary obstructive symptoms are excluded (which includes patients with urinary symptoms from benign prostatic hyperplasia (BPH). * Patients receiving anticoagulation therapy are not eligible for study. * Patients with prior history of an arteriovenous thromboembolic event that occurred within the last 12 months are excluded. * Participation in another clinical study with an investigational product during the last 4 weeks. * Evidence of disease in sites or extent that, in the opinion of the investigator, would put the patient at risk from therapy with testosterone (e.g. femoral metastases with concern over fracture risk, severe and extensive spinal metastases with concern over spinal cord compression, extensive liver metastases). * Concurrent use of other anticancer agents or treatments, with the following exceptions: * Ongoing treatment with LHRH agonists or antagonists, denosumab or bisphosphonate (e.g. zoledronic acid) is allowed. Ongoing treatment should be kept at a stable schedule; however, if medically required, a change of dose, compound, or both is allowed. * Any treatment modalities involving major surgery within 4 weeks prior to the start of study treatment. * Symptomatic nodal disease, i.e. scrotal, penile or leg edema (CTCAE ≥ Grade 3). * Patients are excluded if they have active, known brain metastases or leptomeningeal metastases. * Patients should be excluded if they have an active, known or suspected autoimmune disease (e.g. inflammatory bowel disease, rheumatoid arthritis, autoimmune hepatitis, lupus, celiac disease). Subjects are permitted to enroll if they have vitiligo, type I diabetes mellitus, residual hypothyroidism due to autoimmune condition only requiring hormone replacement, psoriasis not requiring systemic treatment, or conditions not expected to recur in the absence of an external trigger. * Patients should be excluded if they have a condition requiring systemic treatment with either corticosteroids (\> 10 mg/day prednisone equivalents) or other immunosuppressive medications within 14 days of study drug administration. Inhaled or topical steroids and adrenal replacement doses \> 10 mg/day prednisone equivalents are permitted in the absence of active autoimmune disease. * Permitted therapies include topical, ocular, intra-articular, intranasal, and inhalational corticosteroids (with minimal systemic absorption). Physiologic replacement doses of systemic corticosteroids are permitted, even if \> 10 mg/day prednisone equivalents. A brief course of corticosteroids for prophylaxis (e.g. contrast dye allergy) or for treatment of non-autoimmune conditions (e.g. delayed- type hypersensitivity reaction caused by contact allergen) is permitted. * History of allergy to study drug components. * Other primary tumor (other than CRPC) including hematological malignancy present within the last 5 years (except non-melanoma skin cancer, low-grade superficial bladder cancer, or cancers that can be cured with local treatment alone). * Has imminent or established spinal cord compression based on clinical findings and/or MRI. * Any other serious illness or medical condition that would, in the opinion of the investigator, make this protocol unreasonably hazardous, including, but not limited to: * Any uncontrolled major infection. * Cardiac failure NYHA (New York Heart Association) III or IV. * Persistent toxicities (CTCAE \> Grade 2) caused by previous cancer therapy, excluding alopecia. * Poor medical risk due to a serious, uncontrolled medical disorder, non-malignant systemic disease, or active, uncontrolled infection. Examples include, but are not limited to, uncontrolled ventricular arrhythmia, recent (within 12 months) myocardial infarction, uncontrolled major seizure disorder, extensive interstitial bilateral lung disease, or any psychiatric disorder that prohibits obtaining informed consent.