Study Evaluating the Addition of Pembrolizumab to Radium-223 in mCRPC

Inclusion Criteria: * Histologically confirmed adenocarcinoma of the prostate * Castration-resistant prostate cancer requires the following 3 criteria: * Progression after surgical castration or on GnRH agonist or antagonist * A castrate level of testosterone (\<50ng/dL) * Prostate cancer progression documented by PSA rise or bone progression according to PCWG2 * There is no limit to number of prior therapies * Metastatic disease by bone scan * Age ≥18 years * Eastern Cooperative Oncology Group (ECOG) performance status of 0 or 1 * Be willing to undergo a core or excisional biopsy of a bone metastasis prior to study drug initiation for tumor tissue. Newly-obtained is defined as a specimen obtained up to 12 weeks (84 days) prior to initiation of treatment on Day 1. Bone biopsy can have been done prior to screening; archival specimens of bone metastasis are permitted if done for other purpose and available. --If biopsy is non-diagnostic, patient must undergo repeat biopsy as proof of tumor tissue by pathology review. Proof of tumor specimen is required for eligibility. * Be willing to undergo a second core or excisional biopsy of a bone metastasis on therapy (approximately after 8 weeks of study therapy or after 2 doses of radium-223 if delays have occurred). * Demonstrate adequate organ function as defined below, all screening labs for eligibility should be performed within 30 days prior to treatment initiation. * Hematological * Absolute neutrophil count (ANC) ≥ 1,500 /mcL * Platelets ≥ 100,000 / mcL * Hemoglobin ≥ 9 g/dL or ≥ 5.6 mmol/L; transfusions permitted * Renal --Serum creatinine OR Measured or calculated creatinine clearance (GFR can also be used in place of creatinine or CrCl) ≤ 1.5 X institutional upper limit of normal (ULN) OR ≥30 mL/min for subject with creatinine levels \> 1.5 X institutional ULN * Hepatic * Serum total bilirubin ≤ 1.5 X institutional ULN OR Direct bilirubin ≤ institutional ULN for subjects with total bilirubin levels \> 1.5 institutional ULN * AST (SGOT) and ALT (SGPT) ≤ 2.5 X institutional ULN * Albumin \> 2.5 mg/dL * Coagulation --International Normalized Ratio (INR) or Prothrombin Time (PT) ≤1.5 X institutional ULN unless subject is receiving anticoagulant therapy as long as PT or PTT is within therapeutic range of intended use of anticoagulants * Activated Partial Thromboplastin Time (aPTT) ≤1.5 X institutional ULN unless subject is receiving anticoagulant therapy as long as PT or PTT is within therapeutic range of intended use of anticoagulants * Creatinine clearance should be calculated per institutional standard. * The effects of radium-223 and pembrolizumab on the developing human fetus are unknown. For this reason, men must agree to use adequate contraception. Specifically, they must agree to use a condom (even men with vasectomies) and another effective method of birth control if he is having sex with a woman of childbearing potential or agree to use a condom if he is having sex with a woman who is pregnant while on study drug and for 120 days (4 months) following the last dose of study drug. They must also agree not to donate sperm during the study and for 4 months after receiving the last dose of study drug. * Ability to understand and the willingness to sign a written informed consent document. Exclusion Criteria: * Pathology consistent with majority of specimen having small cell carcinoma of the prostate (prostate cancer with other neuroendocrine features is acceptable). * Prior treatment with radium-223 * Prior treatment with a PD-1, PD-L1, or PD-L2 blocking therapy * Evidence of nodal disease greater than or equal to 15 mm in short axis as these findings are concerning for metastases that would not be targeted with radium-223 alone (Arm B). However, lymph nodes with short axis measurements between 1.5-3cm that have not enlarged more than 5mm (to account for reader variability) over the last 6 months and which are not inducing symptoms, causing obstruction, or in the opinion of the investigator pose a risk of impending obstruction of any structures, will be allowed. * Pulmonary nodules \>10 mm --Pulmonary nodules \>10mm that have been stable for \>6 months and are not clearly metastatic disease per the treating investigator are permitted * Soft tissue components of bone metastases ≥ 1.0 cm in longest axis --Soft tissue components of bone metastases \< 1.0 cm that have been stable for \>6 months (must not have enlarged \> 5mm) are permitted * Soft tissue lesions ≥ 1.0 cm in longest axis --Soft tissue lesions \< 1.0 cm that have been stable for \> 6 months (must not have enlarged \> 5mm) are permitted. * If present, primary disease in the prostate must be stable for \> 6 months (defined as no growth \> 5mm) * Evidence of local recurrence in the prostate bed * Evidence of liver metastases or visceral disease * Has a diagnosis of immunodeficiency * Receiving systemic steroid therapy or any other form of immunosuppressive therapy within 7 days prior to the first dose of trial treatment. Patients who have received acute, low-dose, systemic immunosuppressant medications (e.g., limited low-dose dexamethasone for nausea, multiple doses for contrast allergy) may be enrolled in the study and would not require a 7 day washout. * Has a known history of active TB (Bacillus Tuberculosis) * Has had a prior anti-cancer monoclonal antibody (mAb) within 4 weeks prior to study Day 1 or who has not recovered (i.e., ≤ Grade 1 or at baseline) from adverse events due to agents administered more than 4 weeks earlier. * Has had prior systemic therapy (exception: GnRH agonist or antagonist) or radiation therapy for prostate cancer within 2 weeks prior to study Day 1. There must be at least a 2 week washout period from last dose of any prior systemic or radiation therapy for prostate cancer prior to Day 1 of study treatment (including nonsteroidal antiandrogens). Screening may commence during this washout window. * Any patient who has not recovered (i.e., ≤ Grade 1 or at baseline) from adverse events due to a previously administered systemic agent or radiation therapy. * Exceptions: Subjects with ≤ Grade 2 neuropathy, hot flashes, or hypertension may qualify for the study if all other eligibility criteria met. * Other toxicity or complications that are deemed by the treating investigator as not clinically significant (e.g., urinary incontinence from past prostatectomy) * Diethylstilbestrol, estrogens, saw palmetto, or other preparations that are known to have possible endocrine effects on prostate cancer that have been started within the past 8 weeks, as they may affect PSA levels or response. These are allowed if the patient has been on a stable dose for at least 8 weeks prior to C1D1. * Receiving other investigational agents * History of allergic reactions to compounds with similar biologic or chemical composition to pembrolizumab or Radium-223 * Has a known additional malignancy that is progressing or requires active treatment. Exceptions include basal cell carcinoma of the skin or squamous cell carcinoma of the skin that has is planned for curative therapy. * Has known active central nervous system (CNS) metastases and/or carcinomatous meningitis. Subjects with previously treated brain metastases may participate provided they are stable (without evidence of progression by imaging for at least four weeks prior to the first dose of trial treatment and any neurologic symptoms have returned to baseline), have no evidence of new or enlarging brain metastases, and are not using steroids for at least 7 days prior to trial treatment. This exception does not include carcinomatous meningitis, which is excluded regardless of clinical stability. * Has active autoimmune disease that has required systemic treatment in the past 2 years (i.e. with use of disease modifying agents, corticosteroids or immunosuppressive drugs). Replacement therapy (eg., thyroxine, insulin, or physiologic corticosteroid replacement therapy for adrenal or pituitary insufficiency, etc.) is not considered a form of systemic treatment and is allowed. * Has known history of, or any evidence of active, non-infectious pneumonitis. A history of radiation pneumonitis which is asymptomatic with no signs of active process is allowed. * Has a history or current evidence of any condition, therapy, or laboratory abnormality that might confound the results of the trial, interfere with the subject's participation for the full duration of the trial, or is not in the best interest of the subject to participate, in the opinion of the treating investigator. * Has known psychiatric or substance abuse disorders that would interfere with cooperation with the requirements of the trial. * Has a known history of Human Immunodeficiency Virus (HIV) (HIV 1/2 antibodies). HIV-positive participants on combination antiretroviral therapy are ineligible because of the potential for pharmacokinetic interactions with pembrolizumab. In addition, these participants are at increased risk of lethal infections when treated with marrow-suppressive therapy. Appropriate studies will be undertaken in participants receiving combination antiretroviral therapy when indicated. * Has known active Hepatitis B (e.g., HBsAg reactive) or Hepatitis C (e.g., HCV RNA is detected). * Has received a live vaccine within 30 days of planned start of study therapy. * Note: Seasonal influenza vaccines for injection are generally inactivated flu vaccines and are allowed; however intranasal influenza vaccines (e.g., Flu-Mist®) are live attenuated vaccines, and are not allowed. * Uncontrolled intercurrent illness including, but not limited to, ongoing or active infection, severe or unstable angina, myocardial infarction, symptomatic congestive heart failure (defined as New York Heart Association Grade II or greater), arterial or venous thromboembolic events (e.g., pulmonary embolism, cerebrovascular accident including transient ischemic attacks) or clinically significant ventricular arrhythmias within 6 months prior to randomization; or significant vascular disease (e.g., aortic aneurysm, aortic dissection), symptomatic peripheral vascular disease * Evidence of QTc prolongation as defined as QTcF \> 470 ms * Inability to comply with study and/or follow-up procedures