Alpha-Emitting Radionuclide or Beta-Emitting Radionuclide With Metastasis-Directed Stereotactic Body Radiotherapy for the Treatment of Recurrent, Oligometastatic Prostate Adenocarcinoma

Inclusion Criteria: * Oligorecurrent prostate cancer as determined by the presence of 1-5 asymptomatic lesions outside the prostate or prostate bed identified on PSMA PET/CT by local readers * Serum testosterone \> 150 ng/dL * Age ≥ 18 years * Eastern Cooperative Oncology Group (ECOG) performance status ≤ 2 * No indication for urgent or emergent radiation * Histological confirmation of prostate adenocarcinoma (histology from original treatment acceptable) * White blood cell count ≥ 2.5 × 109/L * Platelets ≥ 100 × 109/L * Hemoglobin ≥ 9 g/dL * Total bilirubin ≤ 1.5 × institutional upper limits of normal (ULN) or up to 3 × ULN if known history of Gilbert's syndrome * Alanine aminotransferase or aspartate aminotransferase ≤ 3.0 × ULN or ≤ 5.0 × ULN for patients with liver metastases * Glomerular filtration rate creatinine-cystatin C (GFRcr-cys) ≥ 60 mL/min 1.73m2 using the Chronic Kidney Disease-Epidemiology Collaboration (CKD-EPI) 2021 equation * Serum albumin \> 3.0 g/dL * Partner and patient must use a method of birth control with adequate barrier protection, deemed acceptable by the principal investigator during the study and for 3 months after last study drug administration * Ability to understand, and willingness to sign, the written informed consent Exclusion Criteria: * Patients with neuroendocrine or small cell carcinoma of the prostate * Patients with castrate-resistant disease (i.e., prostate specific antigen \[PSA\] \> 0.5 ng/mL with serum testosterone \<150 ng/dL) * Patients who received androgen deprivation therapy or cytotoxic chemotherapy within 6 months of trial enrolment * Concurrent systemic therapy for a solid organ malignancy * Spinal cord compression * Inability to lie flat * Known hypersensitivity to components of 177-Lu-PSMA-617 or 225-Ac-Lu-PSMA-617 * Inadequate renal function of GFRcr-cys \< 60 mL/min 1.73m2 using the CKD-EPI 2021equation * Total bilirubin \> 1.5 × ULN or \> 3.0 × ULN if known history of Gilbert's syndrome * Alanine aminotransferase or aspartate aminotransferase \> 3 × ULN (or 5 × ULN for patients with known liver metastases) * De novo oligometastatic disease