Phase I/II Trial of Pembrolizumab and Androgen-receptor Inhibitor With or Without 225Ac-J591 for Progressive Metastatic Castration Resistant Prostate Cancer

Inclusion Criteria: * Male participants who are at least 18 years of age on the day of signing informed consent with histologically confirmed diagnosis of prostate adenocarcinoma. * A male participant must agree to use a contraception during the treatment period and for at least 4 months after the last dose of study treatment and refrain from donating sperm during this period. * Documented progressive metastatic CRPC based on Prostate Cancer Working Group 3 (PCWG3) criteria, which includes at least one of the following criteria: PSA progression, Objective radiographic progression in soft tissue, New bone lesions * Evaluable for response with at least one of the following: * Measurable disease by RECIST 1.1 * Detectable (\>0) CTC by CellSearch * PSA of at least 5ng/dL * ECOG performance status of 0-1 * Have serum testosterone ≤ 50 ng/dL. Subjects must continue primary androgen deprivation with an LHRH/GnRH analogue (agonist/antagonist) if they have not undergone orchiectomy. * Have previously been treated with at least one of the following in any disease state: Androgen receptor inhibitor (ARI, such as enzalutamide, apalutamide or darolutamide), CYP 17 inhibitor (such as abiraterone acetate). These drugs may have been initiated in the metastatic hormone sensitive (i.e. non-castrate) or non-metastatic (M0) CRPC setting provided they meet criteria for progressive mCRPC at study entry. * Age ≥ 18 years * Patients must have normal organ and marrow function as defined: Absolute neutrophil count \>2,000 cells/mm3, Hemoglobin ≥ 9 g/dL, Platelet count \>150 x 10\^3/mcL, Serum creatinine \<1.5 x upper limit of normal (ULN) or calculated creatinine clearance ≥ 50 mL/min/1.73 m2 by Cockcroft-Gault, Serum total bilirubin \<1.5 x ULN (unless due to Gilbert's syndrome in which case direct bilirubin must be normal), Serum AST and ALT \<3 x ULN in absence of liver metastases; \< 5x ULN if due to liver metastases (in both circumstances bilirubin must meet entry criteria), Serum internalized normalized ratio (INR) OR prothrombin time (PT) AND activated partial thromboplastin time (aPTT) must be ≤1.5 x ULN unless participant is receiving anticoagulant therapy as long as PT or aPTT is within therapeutic range of intended use of anticoagulants. * Ability to understand and the willingness to sign a written informed consent document. Exclusion Criteria: * Prior receipt of chemotherapy for castration-resistant prostate cancer. Prior receipt of docetaxel chemotherapy in the hormone sensitive setting or for localized disease is acceptable provided at least 6 months has passed since the last dose. * Has received prior therapy with an anti-PD-1, anti-PD-L1, or anti PD L2 agent or with an agent directed to another stimulatory or co-inhibitory T-cell receptor (eg, CTLA-4, OX 40, CD137). * 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. * Prior bone-seeking beta-emitting radioisotopes (e.g. Sm-153, Sr-89) or PSMA-targeted radionuclide therapy for metastatic disease; prior radium-223 is allowed provided last dose administered \>12 weeks prior to C1D1 on this study * Has a history of a second malignancy, unless treatment with curative intent has been completed with no evidence of malignancy for 2 years. Patients with treated localized non-melanoma skin care, non-muscle invasive urothelial carcinoma, or carcinoma in-situ of other site are not excluded. * Known history of myelodysplastic syndrome * Has a known history of Human Immunodeficiency Virus (HIV) infection * Has a known history of Hepatitis B (defined as Hepatitis B surface antigen \[HBsAg\] reactive) or known active Hepatitis C virus (defined as HCV RNA \[qualitative\] is detected) infection. * Has a diagnosis of immunodeficiency or is receiving chronic systemic steroid therapy (in dosing exceeding 10 mg daily of prednisone equivalent) or any other form of immunosuppressive therapy within 7 days prior to the first dose of study drug. * Is currently participating in or has participated in a study of an investigational agent or has used an investigational device within 4 weeks prior to the first dose of study intervention. * Has received prior systemic anti-cancer therapy including investigational agents within 4 weeks or \<5 half-lives prior to enrollment. * Has known active CNS metastases and/or carcinomatous meningitis. Participants with previously treated brain metastases may participate provided they are radiologically stable, i.e. without evidence of progression for at least 4 weeks by repeat imaging (note that the repeat imaging should be performed during study screening), clinically stable and without requirement of steroid treatment for at least 14 days prior to first dose of study intervention. * Diagnosis of deep vein thrombosis and/or pulmonary embolus within 1 month of C1D1 * Has a history or current evidence of any condition, therapy, or laboratory abnormality that might confound the results of the study, interfere with the subject's participation for the full duration of the study, or is not in the best interest of the subject to participate, in the opinion of the treating investigator * Has received radiotherapy within 4 weeks of start of study treatment. Participants must have recovered from all radiation-related toxicities, not require corticosteroids, and not have had radiation pneumonitis. A 1-week washout is permitted for palliative radiation to non-CNS disease that is not a measurable target lesion. * Patients on stable dose of bisphosphonates or denosumab, which have been started no less than 4 weeks prior to treatment start, may continue on this medication, however patients are not allowed to initiate bisphosphonate/denosumab therapy during the DLT-assessment period of the study. These drugs may be added after week 12 in phase 1 or phase 2. * Unless azoospermia is present (whether due to surgery or underlying medical condition), having partners of childbearing potential and not willing to use a method of birth control deemed acceptable by the principal investigator and chairperson during the study and for 4 months after last study drug administration. * Is expecting to conceive or father children within the projected duration of the study, starting with the screening visit through 120 days after the last dose of trial treatment * Has received a live vaccine or live-attenuated vaccine within 30 days prior to the first dose of study drug. Administration of killed vaccines are allowed. * Has severe hypersensitivity (≥Grade 3) to pembrolizumab and/or any of its excipients. * Has a history of (non-infectious) pneumonitis/interstitial lung disease that required steroids or has current pneumonitis/interstitial lung disease. * Has an active infection requiring systemic therapy at the time of treatment initiation * Has a known history of active TB (Bacillus Tuberculosis) * Has known psychiatric or substance abuse disorders that would interfere with cooperation with the requirements of the trial * Has had an allogenic tissue/solid organ transplant