LuCarbo - a Study of 177Lu-PSMA-617 Plus Carboplatin in Metastatic Castrate-resistant Prostate Cancer

Inclusion Criteria: * Participants must have histologically or cytologically confirmed prostate adenocarcinoma without histologic variants comprising \>50% of the sample as determined by pathology review at an academic medical center; men without histologic or cytologic confirmation are eligible provided there is unequivocal evidence of prostate cancer (eg. very high PSA) in the view of the treating physician. * Age ≥ 18years. Children under age 18 are excluded as prostate cancer is a disease of adults. * Progressive disease at study entry, as defined by either one of the following: * Sequence of at least 2 rising PSA values at a minimum of 1-week intervals with the last result being ≥1.0 ng/mL if confirmed PSA rise is the only indication of progression. Patients who received an anti-androgen (flutamide, bicalutamide or nilutamide) must have PSA progression ≥4 weeks after the last dose. * Radiographic progression per RECIST 1.1 for soft tissue and/or per PCWG3 for bone (i.e. appearance of ≥2 new bone lesions), with or without PSA progression. * Presence of ≥1 metastatic lesion metastatic lesion present on baseline CT, MRI, or bone scan imaging obtained ≤28 days prior to beginning study therapy. * Prior receipt of at least one taxane chemotherapy (docetaxel or cabazitaxel) and at least one ARPI (abiraterone, enzalutamide, apalutamide or darolutamide) in the localized, recurrent or metastatic setting. Prior treatment with a PARP inhibitor(s) is permitted. Prior treatment with Ra-223 is permitted, providing that the last dose of Ra-223 was ≥90 days prior to study entry. * Presence of ≥1 PSMA-avid lesion (with uptake \> liver) on baseline/screening 68GaPSMA-11 PSMA-PET. * Serum testosterone level must be ≤50ng/dL (1.73 nmol/L) at the screening visit. Participants who have not undergone bilateral orchiectomy are required to continue LHRH/GnRH agonists/antagonists) throughout the study. Use of relugolix is permitted. * ECOG performance status ≤2 (Karnofsky ≥60%, see Appendix A). * Adequate organ and marrow function as per the below table: --System Laboratory Value * Hematologic * ANC ≥1.5×109/L * Platelets ≥100×109/L * Hemoglobin ≥9g/dL (≥90g/L), independent of transfusions * Hepatic * Total bilirubin ≤1.5 × ULN OR \<2 × ULN if known or suspected Gilbert's syndrome * ALT and AST ≤3 × ULN OR ≤5 × ULN if liver metastases present * Renal --eGFR ≥30 mL/min/1.73 m2 (based on Cockcroft-Gault formula OR 24 hour urine collection * Presence of a recurrent/metastatic lesion (bone or soft tissue) amenable to image-guided percutaneous biopsy adequate for next generation sequencing (NGS), and planned to undergo core biopsy after trial registration but prior to cycle 1 day 1 of therapy. Confirmation of adequacy of this biopsy material for NGS is NOT required for initiation of therapy. * Willingness to undergo core biopsy of a recurrent/metastatic lesion adequate for NGS after approximately 12 weeks of study treatment. * Participants with a prior or concurrent malignancy whose natural history or treatment does not have the potential to interfere with the safety or efficacy assessment of the investigational regimen are eligible for this trial. * The effects of 177Lu-PSMA-617 and carboplatin on the developing human fetus are unknown. For this reason and because chemotherapies and radioligand therapies are known to be teratogenic, men must agree to use adequate contraception (hormonal or barrier method of birth control; abstinence) prior to study entry and for the duration of study participation. Men treated or enrolled on this protocol must also agree to use adequate contraception for the duration of study participation, and at least 14 weeks following the last dose of 177Lu-PSMA-617.Should a woman become pregnant or suspect she is pregnant while her partner is participating in this study, she should inform her treating physician immediately. Female partners of child-bearing potential should also use highly effective birth control methods throughout the male participant's study treatment and for at least 14 weeks following the last dose of 177Lu-PSMA-617. * Ability to understand and the willingness to sign a written informed consent document. (Providing consents in as many languages as possible is encouraged) Exclusion Criteria * Participants who have had chemotherapy or radiotherapy within 4 weeks prior to planned cycle 1 day 1 of study treatment. * Participants who have received anti-neoplastic intervention or experimental antineoplastic therapy within 14 days of planned cycle 1 day 1 of study therapy. * Participants who are receiving any other investigational agents. * Participants who have previously received 177Lu-PSMA-617. * Participants who have not recovered from adverse events due to prior anti-cancer therapy (i.e., have residual toxicities \> Grade 1) with the exception of alopecia. * History of allergic reactions attributed to compounds of similar chemical or biologic composition to 177Lu-PSMA-617 and carboplatin. * Participants with untreated brain metastases. Participants with treated brain metastases are eligible if follow-up brain imaging at least 4 weeks after central nervous system (CNS)-directed therapy shows no evidence of progression and ongoing corticosteroids are not required. Participants with new or progressive brain metastases (active brain metastases) or leptomeningeal disease are eligible if the treating physician determines that immediate CNS specific treatment is not required and is unlikely to be required during the first cycle of therapy. * Symptomatic cord compression, or clinical or radiologic findings indicative of impending cord compression. * Concurrent active malignancy whose natural history or treatment has the potential to interfere with safety or efficacy assessment of the investigational regimen. Patients with non-melanomatous skin cancer, superficial bladder cancer, cancer not needing active therapy for at least 2 years, cancer for which the treating investigator deems the subject to be in remission, or any prior malignancy that was treated with curative intent (no evidence of disease for at least 3 years) are permitted to enroll. * The participant has serious and/or uncontrolled preexisting medical condition(s) that, in the judgment of the investigator, would preclude participation in this study (for example, interstitial lung disease, severe dyspnea at rest or requiring oxygen therapy, severe renal impairment \[e.g. estimated creatinine clearance \<30ml/min\], history of major surgical resection involving the stomach or small bowel, or preexisting Crohn's disease or ulcerative colitis or a preexisting chronic condition resulting in baseline Grade 2 or higher diarrhea).