This phase II trial studies how well giving testosterone at levels higher than normally found in the body (supraphysiological) works to enhance chemotherapy treatment, and Lutetium 177Lu-prostate specific-membrane antigen (PSMA)-617 (LuPSMA) in patients with prostate cancer that has progressed despite being previously treated with androgen therapies and has spread from where it first started (prostate) to other places in the body (metastatic castration-resistant prostate cancer). In patients that have developed progressive cancer in spite of standard hormonal treatment, administering supraphysiological testosterone may result in regression of tumors by causing deoxyribonucleic acid (DNA) damage in tumor cells that have adapted to low testosterone conditions. Carboplatin is in a class of medications known as platinum-containing compounds. Carboplatin works by killing, stopping or slowing the growth of tumor cells. Etoposide is in a class of medications known as podophyllotoxin derivatives. It blocks a certain enzyme needed for cell division and DNA repair and may kill tumor cells. Radioactive drugs, such as LuPSMA, may carry radiation directly to tumor cells and not harm normal cells. Giving supraphysiological levels of testosterone and carboplatin or etoposide or LuPSMA together may be an effective treatment for metastatic castration-resistant prostate cancer.

OUTLINE:

Patients are assigned based on personal preference to 1 of 3 cohorts.

COHORT I: Patients are then assigned to 1 of 3 sub-cohorts within cohort I.

COHORT Ia: Patients continue to receive ADT and receive testosterone cypionate intramuscularly (IM) on day 1 of cycle 1. Patients then receive testosterone cypionate IM and carboplatin intravenously (IV) on day 1 of subsequent cycles. Cycles repeat every 28 days in the absence of disease progression or unacceptable toxicity.

COHORT Ib: Patients continue to receive ADT and receive carboplatin IV on day 1 of cycle 1. Patients then receive testosterone cypionate IM and carboplatin IV on day 1 of subsequent cycles. Cycles repeat every 28 days in the absence of disease progression or unacceptable toxicity.

COHORT Ic: Patients continue to receive ADT and receive testosterone cypionate IM and carboplatin IV on day 1 of cycle 1. Patients then receive testosterone cypionate IM and carboplatin IV on day 1 of subsequent cycles. Cycles repeat every 28 days in the absence of disease progression or unacceptable toxicity.

COHORT II: Patients are then assigned to 1 of 3 sub-cohorts within cohort II.

COHORT IIa: Patients continue to receive ADT and receive testosterone cypionate IM on day 1 of cycle 1. Patients then receive testosterone cypionate IM on day 1 and etoposide orally (PO) once daily (QD) on days 1-14 of subsequent cycles. Cycles repeat every 28 days in the absence of disease progression or unacceptable toxicity.

COHORT IIb: Patients continue to receive ADT and receive etoposide PO QD on days 1-14 of cycle 1. Patients then receive testosterone cypionate IM on day 1 and etoposide PO QD on days 1-14 of subsequent cycles. Cycles repeat every 28 days in the absence of disease progression or unacceptable toxicity.

COHORT IIc: Patients continue to receive ADT and receive testosterone cypionate IM on day 1 and etoposide PO QD on days 1-14 of cycle 1. Patients then receive testosterone cypionate IM on day 1 and etoposide PO QD on days 1-14 of subsequent cycles. Cycles repeat every 28 days in the absence of disease progression or unacceptable toxicity.

COHORT III: Patients are assigned to 1 of 3 sub-cohorts within cohort III.

COHORT IIIa: Patients continue to receive ADT and receive testosterone cypionate intramuscularly (IM) on day 1 of cycles 1-6. Patients receive LuPSMA IV on day 1 of cycles 2-6. Cycles repeat every 6 weeks for up to 6 cycles in the absence of disease progression or unacceptable toxicity. Patients then receive testosterone cypionate intramuscularly (IM) on day 1 of subsequent cycles. Cycles repeat every 28 days in the absence of disease progression or unacceptable toxicity. Patients also undergo 68Ga-PSMA positron emission tomography (PET) at screening and single photon emission computed tomography (SPECT)/CT throughout the study.

COHORT IIIb: Patients continue to receive ADT and LuPSMA IV on day 1 of cycles 1-6. Patients also receive receive testosterone cypionate IM on day 1 of cycles 2-6 . Cycles repeat every 6 weeks for up to 6 cycles in the absence of disease progression or unacceptable toxicity. Patients receive testosterone cypionate IM on day 1 of subsequent cycles. Cycles repeat every 28 days in the absence of disease progression or unacceptable toxicity. Patients also undergo 68Ga-PSMA PET at screening and single SPECT/CT throughout the study.

COHORT IIIc: Patients continue to receive ADT and receive testosterone cypionate IM on day 1 and LuPSMA IV on day 1 of cycles 1-6. Cycles repeat every 6 weeks for up to 6 cycles in the absence of disease progression or unacceptable toxicity. Patients then receive testosterone cypionate IM on day 1 of subsequent cycles. Cycles repeat every 28 days in the absence of disease progression or unacceptable toxicity. Patients also undergo 68Ga-PSMA PET at screening and single SPECT/CT throughout the study.

All patients undergo a biopsy on study and blood sample collection on study, and bone scans, dual x-ray absorptiometry (DEXA) and computed tomography (CT) scans throughout the trial. Patients may also undergo an optional second biopsy at the end of study treatment.

After completion of study treatment, patients are followed up at 30 days, and then every 6 months for 2 years.

Inclusion Criteria:

• Must be willing to provide informed consent prior to any study specific procedures
• Age >= 18 years
• Documented histologically confirmed adenocarcinoma of the prostate
• Patient must have evidence of castration resistant prostate cancer as evidenced by PSA progression (per Prostate Cancer Working Group 3 [PCWG3] criteria) and a castrate serum testosterone level (i.e., =< 50 mg/dL)
• PSA must be at least 2 ng/ml and rising on two successive measurements at least two weeks apart
• Patients must have progressed on at least one prior next-generation androgen receptor-signalling inhibitor (e.g., abiraterone, enzalutamide, etc.). There must be at least a 2-week washout period after stopping the most recent approved therapy for metastatic castration-resistant prostate cancer (mCRPC) (e.g., abiraterone, enzalutamide, Ra-223, sipuleucel-t) prior to cycle 1, day 1. If applicable, patients should be weaned off steroids at least 1 week prior to starting treatment
• Subjects enrolling to Cohort 3 must demonstrate evidence of PSMA expression on 68Ga-PSMA-11 PET as defined in the VISION trial
• No prior chemotherapy for the treatment of mCRPC. Patients may have received docetaxel for the treatment of hormone-sensitive prostate cancer
• Prior treatment with non-chemotherapy investigational agents is permitted. There must be at least a 2-week washout period after stopping any investigational cancer agent prior to cycle 1, day 1
• Hemoglobin >= 9 g/dL with no blood transfusion in the past 28 days (within 30 days prior to administration of study treatment)
• Absolute neutrophil count (ANC) >= 1.5 x 10^9/L (within 30 days prior to administration of study treatment)
• Platelet count >= 100 x 10^9/L (within 30 days prior to administration of study treatment)
• Total bilirubin =< 1.5 x institutional upper limit of normal (ULN) (within 30 days prior to administration of study treatment)
• Aspartate aminotransferase (AST) (serum glutamic oxaloacetic transaminase [SGOT]) / alanine aminotransferase (ALT) (serum glutamic pyruvate transaminase [SGPT]) =< 2.5 x institutional upper limit of normal unless liver metastases are present in which case they must be =< 5x ULN (within 30 days prior to administration of study treatment)
• Patients must have creatinine clearance estimated using the Cockcroft-Gault equation or based on 24 hour urine test of >= 51 mL/min (within 30 days prior to administration of study treatment)
• Eastern Cooperative Oncology Group (ECOG) performance status =< 2
• Patients must have a life expectancy >= 16 weeks
• Patient is willing and able to comply with the protocol for the duration of the study including undergoing treatment and scheduled visits and examinations
• At least one lesion (measurable and/or non-measurable) that can be accurately assessed at baseline by CT, positron emission tomography (PET), magnetic resonance imaging (MRI) and/or bone scan and is suitable for repeated assessment
• Must be willing to undergo metastatic biopsy and have a lesion amenable for biopsy
• Male patients and their partners, who are sexually active and of childbearing potential, must agree to the use of two highly effective forms of contraception in combination, throughout the period of taking study treatment and for 6 months after last dose of study drug(s) to prevent pregnancy in a partner

Exclusion Criteria:

• Involvement in the planning and/or conduct of the study

• Other malignancy unless curatively treated with no evidence of disease for >= 2 years except: adequately treated non-melanoma skin cancer, non-muscle invasive bladder cancer

• Persistent toxicities (Common Terminology Criteria for Adverse Event (CTCAE) grade > 2) caused by previous cancer therapy, excluding alopecia

• Patients with symptomatic uncontrolled brain metastases. A scan to confirm the absence of brain metastases is not required. Patients with spinal cord compression unless considered to have received definitive treatment for this and evidence of clinically stable disease for 28 days

• Use of corticosteroids at a dose equivalent to > 10 mg of prednisone daily

• Planning to receive concurrent treatment with another systemic cancer therapy, aside from a luteinizing hormone releasing hormone (LHRH) analogue

• Use of warfarin is not permitted. Low-molecular weight heparin and direct oral anticoagulants are allowed, but their use should be discussed with the principal investigator (PI) first

• Major surgery within 2 weeks of starting study treatment and patients must have recovered from any effects of any major surgery

• Patients considered a poor medical risk due to a serious, uncontrolled medical disorder, non-malignant systemic disease or active, uncontrolled infection. Examples include, but are not limited to, uncontrolled ventricular arrhythmia, uncontrolled major seizure disorder, uncontrolled hypertension (blood pressure [BP] >= 165/100), history of prior stroke, unstable spinal cord compression, superior vena cava syndrome, extensive interstitial bilateral lung disease or any psychiatric disorder that prohibits obtaining informed consent

• Patients with a known hypersensitivity to the testosterone cypionate, etoposide, carboplatin or any of the excipients of these products

• Patients with known active hepatitis (i.e., hepatitis B or C) due to risk of transmitting the infection through blood or other body fluids

• Evidence of serious and/or unstable pre-existing medical, psychiatric or other condition (including laboratory abnormalities) that could interfere with patient safety or provision of informed consent to participate in this study

• Any psychological, familial, sociological, or geographical condition that could potentially interfere with compliance with the study protocol and follow-up schedule

• Evidence of disease that, in the opinion of the investigator, would put the patient at risk from testosterone therapy (e.g. femoral metastases with concern over fracture risk, spinal metastases with concern over spinal cord compression, lymph node disease with concern for ureteral obstruction)

• Patients with pain attributable to their prostate cancer.

  • Excluded due to concern for pain flare due to testosterone supplementation

• Tumor causing urinary outlet obstruction that requires catheterization for voiding. Patients that require catheterization to void secondary to benign strictures or other non-cancer causes will be permitted to enroll. Patients with percutaneous nephrostomy tubes will also be permitted to enroll

• Prior history of deep venous thrombosis or pulmonary embolism within 5 years prior to enrollment in the study and not currently on systemic anticoagulation.

  • Excluded due to risk of venous thromboembolism from hormone supplementation

• Patients with NYHA (New York Heart Association) class III or IV heart failure or history of a prior myocardial infarction (MI) within 5 years of enrollment to the study.

  • Excluded due to increased risk of cardiovascular events with testosterone supplementation