ACTinium in Castrate-RESistant Prostate Cancer After LUTEtium

Phase I 177Lu-PSMA requirement: The first 3 participants treated at each dose level may be 177Lu-PSMA treatment naïve or may have previously received 177Lu-PSMA treatment. Additional participants recruited at any dose level must have received prior 177Lu-PSMA treatment and had a response to therapy, as judged by the treating physician. Phase II 177Lu-PSMA requirement: All participants must have received prior 177Lu-PSMA and had a response to therapy, as judged by the treating physician. Inclusion Criteria: 1. Age ≥ 18 years at time of providing informed consent. 2. Histologically- or cytologically-confirmed diagnosis of prostate adenocarcinoma, which may include small cell or neuroendocrine features. 3. Castration-resistant prostate cancer, defined as a rising PSA despite surgical castration or ongoing medical castration, with serum testosterone ≤ 0.5ng/mL or \<1.7 nmol/L. 4. Progressive mCRPC with rising PSA level, as defined by PCWG3 criteria, or by radiological progression, and must demonstrate a sequence of rising values above baseline at a minimum of 1-week intervals and PSA \> 1 ng/mL. 5. PSMA-avid disease on screening PSMA-PET-CT scan 6. Prior treatment with at least one second-generation androgen receptor pathway inhibitor (ARPI) 7. Prior treatment with at least one but no more than two lines of taxane therapy for prostate cancer, or been deemed ineligible or refused taxane therapy on consultation with their treating physician. 8. At least 4 weeks or 5 half-lives (whichever is longer) elapsed between last anti-cancer treatment administration and the initiation of trial treatment. Anti-cancer treatment includes ARPIs and PARP inhibitors but excludes ADT (e.g. luteinising hormone releasing hormone (LHRH) analogue or gonadotropin-releasing hormone treatment), which should be continued. Prednisone up to 10 mg daily (or equivalent) is also permitted. 9. Prior treatment with 177Lu-PSMA-targeted radiopharmaceutical therapy (e.g. 177Lu-PSMA-617, 177Lu PSMA-I\&T) up to a maximum of 6 cycles and with response to therapy as judged by the treating physician. Exception: in Phase I, the first 3 participants treated at each dose level may be 177Lu-PSMA naïve Note: last treatment with 177Lu-PSMA must be more than 10 weeks prior to study enrolment. 10. Eastern Cooperative Oncology Group performance status (ECOG PS) of 0 or 1. 11. Estimated life expectancy \> 12 weeks. 12. Grade ≤ 1 xerostomia symptoms at time of trial enrolment. 13. Adequate bone marrow, renal, and hepatic function 14. Resolution of all previous treatment-related toxicities to CTCAE v5.0 Grade ≤ 1, except for chemotherapy-induced alopecia, Grade 2 peripheral neuropathy, and Grade 2 urinary frequency, which are permitted. 15. Adequate contraception for participants and their partners. 16. Willing and able to provide written informed consent. Exclusion Criteria: 1. Symptomatic cord compression, or clinical or radiologic findings indicative of impending cord compression. 2. Active metastatic central nervous system (CNS) disease, including leptomeningeal disease. 3. Receipt of 177Lu-PSMA treatment within 10 weeks of trial enrolment. 4. Prior radiotherapeutic treatment for metastatic prostate cancer (e.g. Radium-223) with the exception of 177Lu-PSMA. Note: prior radiotherapeutic treatment for other cancers is permitted (e.g. radioactive iodine for thyroid cancer). 5. Receipt of transfused blood products or erythropoietin stimulating agents within 4 weeks of trial enrolment. 6. Major surgery within 12 weeks of trial enrolment. 7. Other current malignancy, or malignancy diagnosed/relapsed within the past 5 years (other than non melanomatous skin cancer, stage 0 melanoma in situ, or non-muscle invasive bladder cancer that has undergone curative intent therapy). 8. Sjogren's disease or any other medical conditions that in the judgement of the investigator puts the participant at increased risk of xerostomia. 9. Single kidney, renal transplant or any nephrotoxic condition or concomitant therapy that in the judgement of the investigator could put the participant at risk of unacceptable renal toxicity during the trial. 10. Severe urinary incontinence or any other conditions that in the judgement of the investigator would preclude safe disposal of radioactive urine. 11. Any structural kidney/renal tract disease that in the judgement of the investigator could affect excretion of the trial agent (e.g. hydronephrosis), unless addressed with intervention (e.g. ureteric stent insertion with normalisation of renal function). 12. Clinically significant abnormalities on a single 12-lead electrocardiogram (ECG) during screening evaluation. 13. Concurrent serious conditions that in the judgement of the investigator would pose a safety risk or impair trial participation. 14. Radiation therapy within 2 weeks before trial enrolment. 15. Hypersensitivity to the investigational product or any of its constituents. 16. Current participation in another trial with ongoing receipt of an investigational agent. 17. Any psychological, familial, sociological, or geographical condition potentially hampering compliance with the trial protocol and follow-up schedule, or that would pose a risk to the participant's safety.