177Lu-PSMA-617 and Pembrolizumab in Treating Patients With Metastatic Castration-Resistant Prostate Cancer

PRIMARY OBJECTIVES: 1. To determine the recommended phase 2 dose and schedule of lutetium Lu 177-PSMA-617 (177Lu-PSMA-617) in combination with pembrolizumab in participants with metastatic castration-resistant prostate carcinoma (mCRPC). (Part A) 2. To determine the objective response rate by Response Evaluation Criteria in Solid Tumors (RECIST) 1.1 criteria. (Part B (Dose Expansion)) SECONDARY OBJECTIVES: 1. To characterize the safety profile of the combination. 2. To determine the median duration of response by RECIST 1.1 criteria. 3. To determine the proportion of participants who experience \>= 50% decline from baseline in serum prostate-specific antigen (PSA). 4. To determine the median PSA progression-free survival. 5. To determine the median time to symptomatic skeletal related event. 6. To determine the 6 month radiographic progression-free survival rate and median radiographic progression-free survival. 7. To determine the median overall survival. CORRELATIVE OBJECTIVES: 1. To assess the lesion-specific response rate by baseline PSMA avidity on gallium Ga 68-labeled PSMA-11 (68Ga-PSMA-11) positron emission tomography (PET). 2. To quantify the change from baseline in T cell repertoire, circulating T cell subsets, tumor infiltrating lymphocytes, and tumor programmed death-ligand 1 (PD-L1) expression by immunohistochemistry after one priming dose of Lu-PSMA radioligand therapy (RLT). 3. To explore the relationship between timing of the 177Lu-PSMA-617 priming dose with initiation of pembrolizumab with respect to immunologic, safety, and efficacy outcomes. 4. To descriptively characterize the patterns of uptake on 68Ga-PSMA-11 PET at the time of disease progression. 5. To explore relationship between tumor genomic profile with clinical outcomes including response rate and progression-free survival. 6. To explore the relationship between tumor dosimetry with objective response. OUTLINE: Participants are assigned sequentially to 1 of 3 treatment schedules. DOSING SCHEDULE 1: Participants receive lutetium Lu 177-PSMA-617 intravenously (IV) over 20-30 minutes on day 1 before beginning pembrolizumab IV over 30 minutes on day 1 in cycle 2 DOSING SCHEDULE 2: Participants receive lutetium Lu 177-PSMA-617 IV over 20-30 minutes concurrently with pembrolizumab IV over 30 minutes on day 1. DOSING SCHEDULE 3: Starting day -21, participants receive pembrolizumab IV over 30 minutes. Participants then receive lutetium Lu 177-PSMA-617 IV over 20-30 minutes on day 1. In all dosing schedules, treatment with pembrolizumab repeats every 21 days for up to 35 cycles (2 years) in the absence of disease progression or unacceptable toxicity. Participants who achieve stable disease (SD) or better may receive 17 additional cycles (approximately 1 year) of pembrolizumab. After completion of study treatment, participants are followed up at 30 days after treatment has been discontinued.