Cabazitaxel +/- Carboplatin vs 177Lu-PSMA-617 in Metastatic Castrate-resistant Prostate Cancer

Inclusion Criteria:

* Participants must have histologically or cytologically confirmed adenocarcinoma of prostate
* Evidence of metastatic castrate-resistant prostate cancer that has previously been treated with an androgen receptor pathway inhibitor. Prior docetaxel exposure is recommended but not mandatory. Tissue is not mandatory, but a pathologic report is required at time of enrollment.
* Patients must have a PSMA-positive 18F-rhPSMA-7.3 performed within 12 weeks from C1D1 with ≥1 site with SUVmax ≥10. An alternative PSMA PET tracer is permitted at baseline if performed within 8 weeks prior to randomization.
* Eligible patients have evidence of mCRPC who have progressed on prior novel hormonal agent(s) to include at least one of the following:

  * Baseline PSMA SUVmean \<10 OR
  * ≥1 visceral metastasis OR
  * ≥5 bone metastases OR one of the following (using Next Generation Sequencing on file within 5 years)
  * TP53
  * PTEN
  * mutation.
* Age \> 18 years.
* ECOG performance status of 0 to 2.
* Participants must have adequate organ and marrow function as defined below to be suitable for the randomized treatment outlined in this

  * Absolute neutrophil count \>1000/μL; platelet count \>90 000/μL; hemoglobin \>8.5 g/dL) at screening.
  * Note: Participants must not have received any growth factors within 7 days or blood transfusions within 14 days prior to the hematologic laboratory values obtained at screening).
  * Total bilirubin (TBIL) \<2.5 × the upper limit of normal (ULN) at screening, except participants with documented Gilbert syndrome who must have a TBIL \<3 mg/dL
  * Alanine aminotransferase (ALT) and aspartate aminotransferase (AST) \< 5 ULN at screening
  * Creatinine clearance ≥40 mL/min and/or estimated glomerular filtration rate (eGFR) ≥30
  * Albumin \>30 g/L (3.0 g/dL) at screening
* Participants receiving bisphosphonates or other approved bone-targeting therapy (e.g., denosumab) must be on a stable dose for at least 14 days before the start of study treatment.
* Participants of child-producing potential agree to use highly effective contraceptive methods (i.e., barrier contraception measures such as a male condom with spermicide during intercourse) and avoid sperm donation during the study treatment and for 3 months after the last dose of study treatment. A man is considered to be of child-producing potential, unless he has had a bilateral vasectomy with documented aspermia or a bilateral orchiectomy. Partners of patients must also practice approved forms of birth control
* Participants must have the ability to understand and the willingness to sign a written informed consent form (ICF).
* Members of all races and ethnic groups are eligible for this trial

Exclusion Criteria:

* Evidence of hormone-sensitive prostate cancer (HSPC)
* Evidence of small cell prostate cancer
* Participants receiving any other investigational agents.
* Diagnosis of another clinically significant malignancy within the previous 2 years other than curatively treated non-melanomatous skin cancer or superficial urothelial carcinoma and other in situ or noninvasive malignancies, as determined by the PI or Co-PI.
* Participants with brain metastases/central nervous system (CNS) disease that are treated prior to enrollment will be allowed in this clinical trial.
* Known or suspected significant hypersensitivity to any components of the formulation used for Cabazitaxel, carboplatin or 177Lu-PSMA-617.
* 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 considered by the Investigator to limit compliance with study requirements.
* Prior treatment toxicities not resolved to ≤ Grade 2 according to NCI CTCAE Version 5.0