The Study of 177Lu-TLX591 Plus SOC Versus SOC Alone in Patients With mCRPC (ProstACT Global)

Inclusion Criteria: * Be a male, at least 18 years old, with documented adenocarcinoma of the prostate defined by histological / pathological confirmation. * Be of ECOG Performance Status 0, 1, or 2 and have an estimated life expectancy of ≥6 months from Day 1. * Have metastatic disease (defined as ≥1 metastatic lesion present on baseline CT, MRI or bone scintigraphy). * Have castration-resistant PC (defined as disease progressing despite castration by orchiectomy or ongoing use of luteinizing hormone-releasing hormone \[LHRH\] analogues) and must have a castrate level of serum/plasma testosterone (\<50 ng/dL or \<1.7 nmol/L) at Screening * Must have received a minimum of 12 weeks of prior therapy on an ARPI (abiraterone, apalutamide, darolutamide, or enzalutamide), received in either the mCSPC, nmCRPC, or mCRPC treatment settings, with documented evidence of disease progression while receiving this ARPI. Progression must have occurred on the most recent ARPI. A prior ARPI may have been utilized, but no progression on the prior ARPI is allowed (e,g, ARPI was switched due to poor tolerability or due to adverse events). No washout period is required prior to enrollment into this trial. Participants may have received docetaxel in the mCSPC setting as per the CHAARTED or STAMPEDE treatment regimens (up to 6 cycles of docetaxel), provided the last dose of docetaxel was ≥ 6 months prior to screening and ≥ 4 cycles of docetaxel were administered. * Have a disease that is progressing at study entry, despite a castrate testosterone level (\<50 ng/dL or \<1.7 nmol/L), by the demonstration of at least one of the following: * Two consecutive rising PSA values assessed sequentially at least one week apart, with the final measurement required to be a minimum of 2.0 ng/mL for study entry. Only the last measurement must meet or exceed 2.0 ng/mL. * Progressive disease or new lesion(s) in the viscera or lymph nodes as per RECIST1.1 or in bone as per PCWG3. Any ambiguous results are to be confirmed by other imaging modalities (e.g., CT or MRI scan). * Have disease that is PSMA-positive, as demonstrated by a 68Ga-PSMA-11 PET/CT or PET/MRI scan and confirmed as eligible by the Sponsor's appointed BICR. Imaging-based eligibility review will be performed in two stages: 1. Presence of metastases for exclusion: Screening CT and MRI will be assessed to exclude participants with brain metastasis with long-axis\>1cm 2. PSMA PET eligibility: Screening 68Ga-PSMA-11 PET/CT or PET/MRI will be assessed along with CT, MRI, and bone scans utilizing tumor to liver ratio (TLR) for PSMA positivity-based exclusion. TLR is defined as the ratio of tumor lesion SUVmax to liver SUVmean derived from a 3 cm 3D spherical region of interest (ROI). PSMA positivity is defined as : At least 1 lesion with PSMA TLR≥2. PSMA exclusion critieria: The presence of any of the following will result in the patient being ineligible for this trial: i) visceral metastatic lesions that are ≥1 cm that have a PSMA TLR\< 1 ii) Lytic bone metastatic lesions with a soft tissue component of at least 1 cm with a TLF \<1. iii) At least one metastatic lymph node lesion with short axis ≥2.5 cm with a TLF\<1. * Must have recovered to ≤ Grade 2 from all clinically significant toxicities related to prior therapies (i.e., surgery, local radiotherapy, ARPI, chemotherapy, etc.) with the exception of alopecia. Specific conditions may be discussed with the medical monitor as needed. * Have adequate organ function at Screening: Bone marrow: * Platelets ≥150×109/L. * Absolute neutrophil count ≥1.5 x 109/L. * Hemoglobin \>10g/dL (with no red blood cell transfusion in the previous 4 weeks). Liver function: * Total bilirubin ≤ 1.5× the upper limit of normal (ULN). For participants with known Gilbert's Syndrome ≤3× ULN is permitted. * Alanine aminotransferase (ALT) or aspartate aminotransferase (AST) ≤3× ULN. Renal function: * Creatinine clearance ≥45 mL/min determined using the Cockcroft-Gault formula. * Must understand the study and agree to adhere to all protocol requirements. * Participants must comply with the radiation protection rules (including hospital admissions and isolation) that are used by the treating institution to protect their contacts and the public, especially if a female partner of the participant is or could be pregnant. * Must agree to practice adequate precautions to prevent pregnancy in a partner and to avoid potential problems associated with radiation exposure to the unborn child (Recommendations related to contraception and pregnancy testing in clinical trials Version 1.1 \[Clinical Trial Coordination Group {CTCG, 2024}\]). Exclusion Criteria: * Is unable to understand or is unwilling to sign a written informed consent document or to follow investigational procedures in the opinion of the Investigator. * Has PC associated with pathological findings consistent with small cell or any histology other than adenocarcinoma of the prostate. If there are minor (\<20%) elements of neuroendocrine histology, this is acceptable. * Participants with a history of other malignancies that could significantly impact life expectancy or interfere with disease assessment will be excluded. Exceptions apply to participants with: 1. Prior malignancy that has been adequately treated and has remained disease-free for at least 3 years (maybe confirmed by a scan, etc.). 2. Adequately treated non-melanoma skin cancer. 3. Superficial (non-muscle invasive) bladder cancer that is controlled and stable. * Has received prior treatment with monoclonal antibody (mAb) J591 or HuJ591 or any other PSMA targeted therapy. * Have received chemotherapy in the mCRPC or non-metastatic prostate cancer (nmCRPC) settings (note: prior docetaxel use in the mCSPC setting with CHAATERED or STAMPEDE regimens is permitted if the last dose of therapy was ≥6 months prior to screening and ≥4 cycles of docetaxel were administered). * Has known allergies, hypersensitivity, or into