177Lu-PSMA-617 vs. Androgen Receptor-Directed Therapy in the Treatment of Progressive Metastatic Castrate Resistant Prostate Cancer

Inclusion Criteria:

* Signed informed consent must be obtained prior to participation in the study.
* Participants must be adults \>= 18 years of age.
* Participants must have an ECOG performance status of 0 to 1.
* Participants must have histological pathological, and/or cytological confirmation of adenocarcinoma of the prostate.
* Participants must be 68Ga-PSMA-11 PET/CT scan positive, and eligible as determined by the sponsor's central reader.
* Participants must have a castrate level of serum/plasma testosterone (\< 50 ng/dL or \< 1.7 nmol/L).
* Participants must have progressed only once on prior second generation ARDT (abiraterone, enzalutamide, darolutamide, or apalutamide).

  * First generation androgen receptor inhibitor therapy (e.g. bicalutamide) is allowed but not considered as prior ARDT therapy.
  * Second generation ARDT must be the most recent therapy received.
* Participants must have progressive mCRPC. Documented progressive mCRPC will be based on at least 1 of the following criteria:

  * Serum/plasma PSA progression defined as 2 increases in PSA measured at least 1 week apart. The minimal start value is 2.0 ng/mL; 1.0 ng/mL is the minimal starting value if confirmed rise in PSA is the only indication of progression.
  * Soft-tissue progression defined \[PCWG3-modified RECIST v1.1 (Eisenhauer et al 2009, Scher et al 2016)\].
  * Progression of bone disease: two new lesions; only positivity on the bone scan defines metastatic disease to bone (PCWG3 criteria (Scher et al 2016)).
* Participants must have \>= 1 metastatic lesion that is present on baseline CT, MRI, or bone scan imaging obtained prior to randomization.
* Participants must have recovered to =\< Grade 2 from all clinically significant toxicities related to prior therapies (i.e. prior chemotherapy, radiation, etc.) except alopecia.
* 11\. Participants must have adequate organ function:

  * Bone marrow reserve:
  * ANC \>= 1.5 x 109/L
  * Platelets \>= 100 x 109/L
  * Hemoglobin \>= 9 g/dL
  * Hepatic:
  * Total bilirubin \< 2 x the institutional upper limit of normal (ULN). For participants with known Gilbert's Syndrome =\< 3 x ULN is permitted.
  * ALT or AST =\< 3.0 x ULN OR =\< 5.0 x ULN for participants with liver metastases.
  * Renal:
  * eGFR \>= 50 mL/min/1.73m2 using the Modification of Diet in Renal Disease (MDRD) equation.
* Albumin \>= 2.5 g/dL. -. Candidates for change in ARDT as assessed by the treating physician: • Participants cannot have previously progressed nor had intolerable toxicity to both enzalutamide and abiraterone.

Exclusion Criteria:

* Previous treatment with any of the following within 6 months of randomization: Strontium-89, Samarium-153, Rhenium-186, Rhenium-188, Radium-223, hemi-body irradiation.
* Previous PSMA-targeted radioligand therapy.
* Prior treatment with cytotoxic chemotherapy for castration resistant or castrate sensitive prostate cancer (e.g., taxanes, platinum, estramustine, vincristine, methotrexate, etc.), immunotherapy or biological therapy \[including monoclonal antibodies\]. \[Note: Taxane exposure (maximum 6 cycles) in the adjuvant or neoadjuvant setting is allowed if 12 months have elapsed since completion of this adjuvant or neoadjuvant therapy. Prior treatment with sipuleucel-T is allowed\].
* Any investigational agents within 28 days prior to day of randomization.
* Known hypersensitivity to any of the study treatments or its excipients or to drugs of similar classes.

  -. Concurrent cytotoxic chemotherapy, immunotherapy, radioligand therapy, PARP inhibitor, biological therapy, or investigational therapy.
* Transfusion or use of bone marrow stimulating agents for the sole purpose of making a participant eligible for study inclusion.
* Participants with a history of CNS metastases who are neurologically unstable, symptomatic, or receiving corticosteroids for the purpose of maintaining neurologic integrity. Participants with CNS metastases are eligible if received therapy (surgery, radiotherapy, gamma knife), asymptomatic and neurologically stable without corticosteroids. Participants with epidural disease, canal disease and prior cord involvement are eligible if those areas have been treated, are stable, and not neurologically impaired.
* Symptomatic cord compression, or clinical or radiologic findings indicative of impending cord compression.
* History or current diagnosis of the following ECG abnormalities indicating significant risk of safety for study participants:

  * Concomitant clinically significant cardiac arrhythmias, e.g. sustained ventricular tachycardia, complete left bundle branch block, high-grade AV block (e.g., bifascicular block, Mobitz type II and third degree AV block).
  * History of familial long QT syndrome or known family history of Torsades de Pointe.
  * Cardiac or cardiac repolarization abnormality, including any of the following: History of myocardial infarction (MI), angina pectoris, or CABG within 6 months prior to starting study treatment.
* Concurrent serious (as determined by the Principal Investigator) medical conditions, including, but not limited to New York Heart Association class III or IV congestive heart failure, history of congenital prolonged QT syndrome, uncontrolled infection, known active hepatitis B or C or other significant co-morbid conditions that in the opinion of the investigator would impair study participation or cooperation.

  * HIV-infected participants who are at a low risk of AIDS-related outcomes may participate in this trial.
  * Participants with an active documented COVID-19 infection (any grade of disease severity) at time of informed consent may be included only when completely recovered (in accordance with local guidance).
* Diagnosed with other malignancies that are expected to alter life expectancy or may interfere with disease assessment. However, participants with a prior history of malignancy that has been adequately treated and who have been disease free and treatment free for more than 3 years prior to randomization, are eligible, as are participants with adequately treated non-melanoma skin cancer and superficial bladder cancer.
* Sexually active males unwilling to use a condom during intercourse while taking study treatment and for 14 weeks after stopping study treatment. A condom is required for all sexually active male participants to prevent them from fathering a child AND to prevent delivery of study treatment via seminal fluid to their partner.

In addition, male participants must not donate sperm for the time period specified above. If local regulations deviate from the contraception methods listed above to prevent pregnancy, local regulations apply and will be described in the ICF.

* Unmanageable concurrent bladder outflow obstruction or urinary incontinence. Note: Participant with bladder outflow obstruction or urinary incontinence, which is manageable and controlled with best available standard of care (incl. pads, drainage) are allowed.
* History of somatic or psychiatric disease/condition that may interfere with the objectives and assessments of the study.
* Any condition that precludes raised arms position.
* Eligible for treatment(s) other than ARDT based on presence of any mutations or biomarkers that are known as predictors of better response (e.g., AR-V7 or BRCA).
* Not able to understand and to comply with study instructions and requirements.