Pilot Study of ¹⁷⁷Lu-PSMA-617 in Combination With Sipuleucel-T in Patients With Metastatic Castration-Resistant Prostate Cancer

Inclusion Criteria: * Documented informed consent of the participant and/or legally authorized representative * Assent, when appropriate, will be obtained per institutional guidelines * Agreement to allow the use of archival tissue from diagnostic tumor biopsies * If unavailable, exceptions may be granted with study principal investigator (PI) approval * Age: ≥ 18 years * Eastern Cooperative Oncology Group (ECOG) ≤ 1 * Male * Progressive castration-resistant metastatic prostate cancer with pathologically confirmed adenocarcinoma of the prostate without small cell features * Patients must have either: * Measurable disease * For extranodal (visceral) lesions (e.g. lung, liver, etc.) to be considered measurable, they must be ≥ 10 mm in one dimension, using spiral CT * For lymph nodes to be considered measurable (i.e., target or evaluable lesions), they must be ≥ 20 mm in at least one dimension, using spiral CT * OR non-measurable disease * All other lesions, including small lesions (longest diameter \< 20 mm with conventional techniques or \< 10 mm with spiral CT scan) and truly non-measurable lesions * Lesions that are considered non-measurable include bone lesions (only). Progression on first generation ADT * Patients must have been on androgen deprivation therapy with a gonadotrophin releasing hormone (GnRH) analogue, antagonist, or bilateral orchiectomy (i.e., surgical or medical castration) for at least 3 months prior to study entry and maintain castrate levels of serum testosterone \< 50 ng/dL throughout study participation unless intolerant * Fully recovered from the acute toxic effects (except alopecia) to ≤ grade 1 to prior anti-cancer therapy * Absolute neutrophil count (ANC) ≥ 1,500/mm\^3 (within 10 days prior to day 1 of protocol therapy) * White blood cell (WBC) counts \> 2500/uL (within 10 days prior to day 1 of protocol therapy) * Lymphocyte count ≥ 300/uL (within 10 days prior to day 1 of protocol therapy) * Platelets ≥ 100,000/mm\^3 (within 10 days prior to day 1 of protocol therapy) * Hemoglobin ≥ 9g/dL (within 10 days prior to day 1 of protocol therapy) * NOTE: Red blood cell transfusions are not permitted within 14 days of hemoglobin assessment unless cytopenia is secondary to disease involvement * Total bilirubin ≤ 1.5 x upper limit of normal (ULN) (within 10 days prior to day 1 of protocol therapy) (unless has Gilbert's disease, serum bilirubin level ≤ 3 x ULN) * Aspartate aminotransferase (AST) ≤ 2.5 x ULN (within 10 days prior to day 1 of protocol therapy) * Alanine aminotransferase (ALT) ≤ 3.0 x ULN (within 10 days prior to day 1 of protocol therapy) * Alkaline phosphatase ≤ 3 x ULN (within 10 days prior to day 1 of protocol therapy) (Patients with documented bone metastases, alkaline phosphatase \[ALP\] ≤ 5 x ULN) * Serum creatinine ≤ 1.5 x ULN or creatinine clearance of ≥ 50 mL/min per Cockcroft-Gault formula (within 10 days prior to day 1 of protocol therapy) * If not receiving anticoagulants: International normalized ratio (INR) OR prothrombin (PT) ≤ 1.5 x ULN (within 10 days prior to day 1 of protocol therapy) * If on anticoagulant therapy: PT must be within therapeutic range of intended use of anticoagulants (within 10 days prior to day 1 of protocol therapy) * If not receiving anticoagulants: Activated partial thromboplastin time (aPTT) ≤ 1.3 x ULN (within 10 days prior to day 1 of protocol therapy) * If on anticoagulant therapy: aPTT must be within therapeutic range of intended use of anticoagulants (within 10 days prior to day 1 of protocol therapy) * Seronegative for HIV antigen (Ag)/antibody (Ab) combo, hepatitis C virus (HCV), active hepatitis B virus (HBV) (surface antigen negative), and syphilis (rapid plasma reagin \[RPR\]) (within 10 days prior to day 1 of protocol therapy) * If positive, hepatitis C ribonucleic acid (RNA) quantitation must be performed. OR * If seropositive for HIV, HCV or HBV, nucleic acid quantitation must be performed. Viral load must be undetectable * Meets other institutional and federal requirements for infectious disease titer requirements * Note Infectious disease testing to be performed within 28 days prior to day 1 of protocol therapy * For male patients with partners of childbearing potential, agreement (by patient and/or partner) to use highly effective form(s) of contraception or abstain from heterosexual activity for the course of the study through at least 4 months after the last dose of protocol therapy * Childbearing potential defined as not being surgically sterilized (men and women) or have not been free from menses for \> 1 year (women only) Exclusion Criteria: * Any approved or investigational anticancer therapy, including chemotherapy, hormonal therapy (e.g., androgen receptor \[AR\] antagonists, 5 alpha reductase inhibitor, estrogen), or radiotherapy, within 4 weeks prior to initiation of study treatment * Treatment with any of the following medications or interventions within 28 days of registration: * External beam radiation therapy or surgery * Chrysanthemum morifolium/Ganoderma lucidum/Glycyrrhiza glabra/Isatis indigotica/Panax pseudoginseng/Rabdosia rubescens/Scutellaria baicalensis/Serona repens supplement (PC-SPES) (or PC-SPEC) or saw palmetto * Systemic corticosteroids. Use of inhaled, intranasal, and topical steroids is acceptable * Megestrol acetate (Megace®), diethyl stilbestrol (DES), or cyproterone acetate * Ketoconazole * 5-alpha-reductase inhibitors (e.g., finasteride \[Proscar®\], dutasteride \[Avodart®\]) * High dose calcitriol (1,25\[OH\]2 vitamin \[Vit\]D) (i.e., \> 7.0 ug/week) * Prior treatment with 177\^Lu-PSMA-617 and/or sipuleucel-T * Radiation therapy for bone metastasis within 2 weeks or any other radiation therapy within 4 weeks before first dose of study treatment * Known clinically significant liver disease, including active viral, alcoholic, or other hepatitis; cirrhosis; fatty liver; and inherited liver disease * Treatment with any investigational vaccine within 2 years of registration or treatment with any other investigational product within 28 days of registration * Patients with acute leukemias, accelerated/blast-phase chronic myelogenous leukemia, chronic lymphocytic leukemia, Burkitt lymphoma, plasma cell leukemia, or non-secretory myeloma * Known primary central nervous system (CNS) malignancy or symptomatic CNS metastases * Inability to comply with study and follow-up procedures * Any other active malignancy at time of first dose of study treatment or diagnosis of another malignancy within 3 years prior to first dose of study treatment that requires active treatment, except for locally curable cancers that have been apparently cured, such as basal or squamous cell skin cancer, superficial bladder cancer, or carcinoma in situ of the prostate, cervix, or breast * Any other condition that would, in the investigator's judgment, contraindicate the patient's participation in the clinical study due to safety concerns with clinical study procedures * Prospective participants who, in the opinion of the investigator, may not be able to comply with all study procedures (including compliance issues related to feasibility/logistics)