Study Evaluating Dosimetry, Randomized Dose Optimization, Dose Escalation and Efficacy of Ac-225 Rosopatamab Tetraxetan in Participants With PSMA PET-Positive Castration-Resistant Prostate Cancer (CRPC)

Inclusion Criteria: * Progressive CRPC defined as castrate levels of testosterone and progressing by at least one of the following criteria: 1. Serum PSA progression as defined by PCWG3 (rising PSA consisting of two consecutive increases measured at least 3 weeks apart, of a rise of \>25%, and with an absolute rise of 2.0 ng/mL. 2. Soft tissue progression defined as a ≥20% increase in the sum of the diameter (short axis for nodal lesions and long axis for non-nodal lesions) of all target lesions based on the smallest sum of the diameter since the previous treatment was started or the appearance of one or more new lesions by CT/magnetic resonance imaging (MRI) 3. Progression of bone disease defined by PCWG3 as evaluable disease or new bone lesions by bone scan 4. Identification of new soft tissue or bone lesions on PSMA PET imaging * Metastatic disease defined as either or both of the following: 1. Parts 1, 2, 3, and 4: Documented M1 disease on conventional imaging (CT/MRI of the chest/abdomen/pelvis and/or Technetium 99m \[99mTc\] whole-body bone scan) 2. Parts 1, 2, and 4 only: Identification of bone lesion(s), extra-pelvic soft tissue lesion(s), or visceral metastases on PSMA PET imaging with an FDA-approved imaging agent * PSMA PET-positive disease, defined as at least one PSMA-positive metastatic lesion and no PSMA-negative lesions * Progression following treatment with ADT and at least one ARSI (e.g., enzalutamide, apalutamide, darolutamide, and/or abiraterone acetate) * The standard of care use (in the setting of metastatic CRPC with significant burden of active bone metastases) of antiresorptive bone-targeted agents (e.g., zoledronic acid, denosumab) is required for all participants without a contraindication, for at least 4 weeks prior to administration of Ac-225 rosopatamab tetraxetan. * Participants with HIV are eligible if they are well-controlled (i.e, an undetectable HIV viral load (\<50 copies/mL) within 6 months of enrollment and a stable ART regimen for at least 6 months prior to enrollment) and at low risk for HIV-related illness Part 3 Only: * Prior treatment with Lu-177-PSMA-radioligand therapy * Prior treatment with up to only one taxane-based chemotherapy regimen is allowed Exclusion Criteria: * Superscans by nuclear medicine/99mTc bone scan * A known malignancy that is progressing or has required active treatment within the past 3 years other than CRPC, which is expected to alter life expectancy or may interfere with CRPC disease assessment * Prior platinum-based chemotherapy * Prior PARP inhibitors (e.g., olaparib or rucaparib) * Prior treatment with Radium-223, Actinium-225, Strontium-89, Samarium-153, Rheunium-186, or Rhenium-188 * Participants receiving anti-coagulants or anti-platelet drugs (e.g., aspirin or nonsteroidal anti-inflammatory drugs \[NSAIDs\]) who cannot discontinue use if platelet count decreases to \<50,000 Part 2 and 4 Only: * Prior chemotherapy for CRPC. Prior taxane chemotherapy for HSPC is allowed if discontinued ≥1 year prior to randomization * Prior radiopharmaceutical therapy (e.g., Ra-223, Lu-177-PSMA-617, or Lu-177-PSMA-I\&T) * Prior PSMA-targeted therapy, except for bispecific or CAR-T therapies Part 3 Only: * Prior PSMA-targeted therapy (e.g., antibody-drug conjugates or CAR-T therapy), except for Lu-177-PSMA-radioligand therapy and bispecific or CAR-T therapies