Bipolar Androgen Therapy (BAT) and Radium-223 (RAD) in Metastatic Castration-resistant Prostate Cancer (mCRPC)

Inclusion Criteria: * Histologically documented adenocarcinoma of the prostate confirmed by pathology report from prostate biopsy or a radical prostatectomy specimen. If prostatic tumor is of mixed histology, \> 50% of the tumor must be adenocarcinoma. * Bone metastases as manifested by one or more lesions on a Technetium 99m bone scan performed within 2 months of screening * Castrate-resistant prostate cancer, in the setting of castrate levels of testosterone (≤ 50 ng/dL), defined as current or historical evidence of disease progression concomitant with surgical castration or androgen deprivation therapy (ADT), as demonstrated by two consecutive rises in PSA OR new lesions on bone scan: * PSA progression will be defined as 2 rising PSA values compared to a reference value, measured at least 7 days apart and the second value is ≥ 2 ng/mL. Appearance of one or more new areas of abnormal uptake on bone scan when compared to imaging studies acquired during castration therapy or against the precastration studies if there was no response. Increased uptake of pre-existing lesions on bone scan does not constitute progression. It must be documented within 8 weeks of screening 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. * Serum PSA ≥ 2.0 ng/mL * Patients must be on bone health agents, either zoledronic acid or denosumab, for at least 4 weeks before enrollment. These treatments must then be continued during the study. * Screening Eastern Cooperative Oncology Group (ECOG) performance status ≤ 1 * Asymptomatic or minimally symptomatic disease (no opioids) * Prior treatment with no more than one novel AR targeted drug (abiraterone, enzalutamide, darolutamide or apalutamide) is permitted, but not required. Prior first-generation AR targeted therapies such as bicalutamide or nilutamide are permitted as previous therapy and does not count as novel AR targeted therapy. * Prior chemotherapy for hormone-sensitive prostate cancer (given ≥ 12 months prior to study entry) is allowed, but not necessary. * Adequate bone marrow, renal and liver function (Absolute Neutrophil count \> 1,000, Platelets \>100,000, Hemoglobin ≥ 9g/dL aspartate aminotransferase/ alanine amino transferase (AST)/(ALT) within normal limits (WNL); Total Bilirubin WNL. * No evidence (within 5 years) of prior malignancies (except successfully treated basal cell or squamous cell carcinoma of the skin). * All patients must have tissue for genomic analysis. A biopsy of a metastatic site may be done during the screening; however, archive tissue will be allowed. Prostate tissue from prostate biopsy will be allowed. Exclusion Criteria: * The presence of known visceral metastasis, including lung, liver and brain metastases. * Spinal cord compression, imminent long bone fracture, or any other condition that, in the opinion of the investigator, is likely to require radiation therapy and/or steroids for pain control during the active phase. * Previous treatment with chemotherapy for mCRPC, or chemotherapy for any reason within 12 months prior to registration. (Chemotherapy in the adjuvant setting or for hormone-sensitive prostate cancer is permitted, as long as it was completed more than 6 months before registration). * History of radiation therapy, either via external beam or brachytherapy within 28 days prior to registration. * Systemic therapy with strontium-89, samarium-153, rhenium-186 or rhenium-188 for the treatment of bony metastases within previous 24 weeks * Use of opioid analgesics for cancer-related pain such as oxycodone, morphine or methadone. Weak opioid analgesics such as codeine or tramadol are permitted. * Use of experimental drug within 4 weeks of treatment. * 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 with warfarin are not eligible for study. Patients on other anticoagulants such as rivaroxaban, dabigatran, apixaban are permitted. * Symptomatic nodal disease, i.e. scrotal, penile or leg edema. * Poor medical risk due to a serious, uncontrolled medical disorder, non-malignant systemic disease, or active, uncontrolled infection or a disease that may compromise safety. Examples include, but are not limited to, diabetes, heart failure, chronic obstructive pulmonary disease (COPD), ulcerative colitis, or Crohn's disease, Paget's disease, ventricular arrhythmia, recent (within 12 months) myocardial infarction, thromboembolic events or any psychiatric disorder that prohibits obtaining informed consent. Any medical intervention, any other condition, or any other circumstance which, in the opinion of the investigator, could compromise adherence with study requirements or otherwise compromise the study's objectives. * 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, etc). Patients with low volume visceral metastasis are permitted at the discretion of the investigator, however bone disease must be predominant.