EVOLUTION: 177Lu-PSMA Therapy Versus 177Lu-PSMA in Combination With Ipilimumab and Nivolumab for Men With mCRPC

Inclusion Criteria: 1. Aged 18 years or older, with histologically confirmed adenocarcinoma of the prostate. 2. Castration-resistant metastatic prostate cancer (defined as disease progressing despite castration by orchidectomy or ongoing luteinising hormone-releasing hormone agonist or antagonist). 3. Patients must have progressed on prior novel AR targeted agents for treatment of prostate cancer. Progressive disease defined by at least one of the following: * PSA progression, minimum of two rising PSA values from a baseline measurement with an interval of ≥ 1 week between each measurement. The PSA value at screening should be ≥ 5ng/ml * Soft tissue or visceral disease progression as per RECIST 1.1 * Bone progression: ≥ 2 new lesions on bone scan as per PCWG3 4. Target or non-target lesions according to RECIST 1.1 and PCWG3 5. Significant PSMA avidity on PET/CT using 68GaPSMA, defined as SUVmax ≥15 at a site of disease, and SUVmax ≥ 10 at other sites of disease ≥10mm (where there is no impact from partial voluming. 6. Eastern Cooperative Oncology Group (ECOG) performance status of 0-1. 7. Adequate bone marrow, hepatic and renal function documented within 28 days of registration, defined as: * Haemoglobin ≥90 g/L independent of transfusions (no red blood cell transfusion in last 4 weeks) * Absolute neutrophil count ≥1.5x109/L * Platelets ≥100 x109/L * Total bilirubin ≤1.5 x upper limit of normal (ULN) except for patients with known Gilbert's syndrome * Aspartate transaminase (AST/SGOT) and alanine transaminase (ALT/SGPT) ≤2.5 × ULN or ≤5 × ULN for participants with liver metastases * Serum creatinine ≤1.5 x ULN or a calculated creatinine clearance \> 50mL/min (Cockcroft-Gault equation) 8. Patients must have a life expectancy ≥ 24 weeks. 9. Willing and able to comply with all study requirements, including treatment, timing and/or nature of required assessments 10. Signed, written informed consent. Exclusion Criteria: 1. Prostate cancer with known significant sarcomatoid, spindle cell or neuroendocrine cell components, or metastasis of other cancers to the prostate. 2. 18F-FDG-PET/CT SUVmax ≥10 at a site of measurable disease with no concurrent PSMA expression \> 10mm 3. Prior treatment with anti-PD1, anti-PD-L1/L2, anti-CTLA-4 antibody, or any other antibody or drug specifically targeting T cell co-stimulation or checkpoint pathways. 4. Patients must not have had more than one line of chemotherapy. If a patient has had docetaxel chemotherapy for hormone sensitive or castrate resistant setting, this will be considered one line. 5. Prior treatment with 177Lu-PSMA. 6. Patients with active, known, or suspected autoimmune disease. Sjogren's syndrome is considered an autoimmune disease. Exceptions: Patients with vitiligo, type I diabetes mellitus, residual hypothyroidism due to an autoimmune condition only requiring hormone replacement, or conditions not expected to recur in the absence of an external trigger, may be eligible. 7. Patients with a condition requiring systemic treatment with either corticosteroids (\>10mg daily prednisone equivalents) or other immunosuppressive medications within 14 days of registration. Inhaled or topical steroids, and adrenal replacement doses ≤ 10 mg daily prednisone equivalents are permitted in the absence of active autoimmune disease. 8. Participants must have recovered from all AE due to previous therapies to ≤Grade 1 or baseline. Participants with ≤ Grade 2 neuropathy may be eligible. 9. Active malignancies within the previous 2-years with \>30% probability of recurrence within 1 year. Melanoma in situ, basal cell or squamous cell carcinomas of skin, are permitted. 10. Significant infection, including chronic active hepatitis B, hepatitis C, or HIV. Testing for these is not mandatory unless clinically indicated. 11. Radiation or surgery within 2 weeks of randomisation. 12. Previous history of interstitial lung disease or non-infectious pneumonitis. 13. Administration of a live vaccine within 30 days prior to the first dose of study drug. 14. Uncontrolled intercurrent illness including, but not limited to, ongoing or active infection, symptomatic congestive heart failure, unstable angina pectoris, cardiac arrhythmia, or psychiatric illness/social situations that would limit compliance with study requirements. 15. Inadequate contraception. Men must have been surgically sterilised or use a (double if required) barrier method of contraception.