177Lu-PSMA, Niraparib/AA Plus Prednisone for Prostate Cancer

Inclusion Criteria: * Signed informed consent must be obtained prior to participation in the study * Adults ≥ 18 years of age * Eastern Cooperative Oncology Group (ECOG) performance score of 0 to 2 * Life expectancy of greater than six months as determined by the treating physician * Evidence of metastatic castration resistant prostate cancer with histological or cytology confirmed adenocarcinoma. This can be based on prior biopsy of prostate or metastatic disease. 1. Should have previous therapy with at least one androgen receptor pathway inhibitor in castration resistant or sensitive setting, and 2. Should have prior therapy with at least one line of therapy with taxane chemotherapy in castration resistant or sensitive setting * Evidence of disease progression on current therapy which is based on either: 1. PSA progression: based on rising values at a minimum of 1-week intervals. The minimum starting value of PSA being 1 ng/mL 2. Radiologic progression: growth of known metastases or evidence of new metastases. * Evidence of PSMA positive disease based on PSMA positron emission tomography (PET)/computed tomography (CT) scan utilizing radiotracers F-18 piflufolastat PSMA or Ga-68 PSMA-11. PSMA positive is defined as having at least one tumor lesion with radiotracer uptake greater than normal liver. Patients will be excluded if any lesions exceeding size criteria in short axis \[organs ≥ 1 cm, lymph nodes ≥ 2.5 cm, bones (soft tissue component) ≥ 1 cm\] have uptake less than or equal to uptake in normal liver. * Patients must have evidence of metastatic disease with at least one of the following: 1. Metastatic disease in the bones visible on technetium-99m-MDP (methylene diphosphonate) bone scan ("bone scan") and/or 2. Pathologically enlarged lymph nodes of any distribution and/or 3. Visceral metastasis of any size or distribution. * Adequate organ function: 1. Bone marrow reserve: absolute neutrophil count (ANC) ≥ 1500 cells/microliter (uL), platelets ≥ 100,000 cells/uL, hemoglobin ≥ 9 g/dL 2. Hepatic function: total bilirubin ≤ 2 × institutional upper limit of normal (ULN), for patients with known Gilbert syndrome ≤ 3 × institutional ULN is permitted. Alanine transaminase (ALT) or aspartate transaminase (AST) ≤ 3 × institutional ULN or ≤ 5 × institutional ULN if liver metastases 3. Renal function: estimated glomerular filtration rate (eGFR) ≥ 45 mL/min * Able to swallow the study medication tablets whole. * While on study medication and for 4 months following the last dose of study medication, participants must agree to use condom and an adequate contraception method for partner of childbearing potential. Participants must agree not to donate sperm while on study treatment and for a minimum of 4 months following the last dose of study medication. Exclusion Criteria: * Pathological finding consistent with small cell or neuroendocrine carcinoma of the prostate * Prior therapy with poly adenosine diphosphate (ADP)-ribose polymerase (PARP) inhibitor * Prior therapy with 177Lu-PSMA 617 therapy. Exception: prior therapy with radium 223 is allowed if last dose was ≥ 6 months from study therapy. Note that other prior radiation therapy (such as external beam radiation therapy or brachytherapy) to prostate or other metastatic sites is allowed. * History of adrenal dysfunction * Active malignancies (i.e., progressing or requiring treatment change in the last 24 months) other than the disease being treated under study. The only allowed exceptions are: 1. Non-muscle invasive bladder cancer 2. Skin cancer (non-melanoma or melanoma) treated within the last 24 months that is considered completely cured 3. Malignancy that is considered cured with minimal risk of recurrence * History or current diagnosis of myelodysplastic syndrome/acute myeloid leukemia (MDS/AML) * Current evidence within 3 months of study consent of any of the following: severe/unstable angina, myocardial infarction, symptomatic congestive heart failure, clinically significant arterial or venous thromboembolic events (i.e., pulmonary embolism), or clinically significant ventricular arrhythmias. * Presence of sustained uncontrolled hypertension (systolic blood pressure \>160 mm Hg or diastolic blood pressure \>100 mm Hg). Participants with a history of hypertension are allowed, provided that blood pressure is controlled to within these limits by an antihypertensive treatment. * Known allergies, hypersensitivity, or intolerance to the excipients of Nira/AA tablets. * Current evidence of any medical condition that would make prednisone use contraindicated. * Received an investigational intervention (including investigational vaccines) or used an invasive investigational medical device within 30 days before the planned first dose of study medication * Participants who have had the following ≤ 28 days prior to enrollment 1. A transfusion (platelets or red blood cells); 2. Hematopoietic growth factors; 3. Major surgery * Participants with known history of human immunodeficiency virus with 1 or more of the following: 1. Not receiving highly active antiretroviral therapy or on antiretroviral therapy for less than 4 weeks 2. Receiving antiretroviral therapy that may interfere with the study medication 3. CD4 (type of white blood cell) count \<350 at screening. 4. An acquired immunodeficiency syndrome-defining opportunistic infection within 6 months of the start of screening. 5. Human immunodeficiency virus load \>400 copies/mL * Known history of active or symptomatic viral hepatitis or chronic liver disease; encephalopathy, ascites or bleeding disorders secondary to hepatic dysfunction. * Moderate or severe hepatic impairment (Class B and C per Child-Pugh classification system).