Nivolumab + Docetaxel + ADT in mHSPC Patients With DDRD or Inflamed Tumors

Inclusion Criteria: * Newly diagnosed histologically confirmed prostate adenocarcinoma within 6 months prior to study registration with evidence of high-volume distant metastasis on conventional imaging * Distant metastasis is defined by non-regional lymph node(s) metastasis (M1a), bone metastasis (M1b), and/or other site(s) of metastatic disease (M1c). * Conventional imaging consists of CT, MRI or radionuclide bone scan * High volume of disease is defined by presence of four or more bone lesions with at least one beyond the vertebral bodies or pelvis or any site of visceral metastasis. * Age ≥18 years * ECOG performance status ≤2 (Karnofsky ≥60%, see Appendix A) * Subjects with ECOG performance status of 2 are only eligible if the performance status decline is attributed to metastatic prostate cancer * Serum PSA \> 4.0 ng/mL before initiation of ADT * Serum testosterone \> 100 ng/dL before initiation of ADT * Subjects whose testosterone level is unknown before initiation of ADT may be allowed after discussion with Sponsor-Investigator. * Grade ≤ 1 peripheral neuropathy, defined as asymptomatic or paresthesia and/or decreased deep tendon reflexes is allowed. * Subjects must have adequate organ and marrow function as defined below: * Absolute neutrophil count ≥1,500 /mcL * Platelets ≥100,000 /mcL * Total bilirubin ≤1.5 × institutional upper limit of normal. Exception: Subjects with confirmed Gilbert's syndrome (persistent or recurrent hyperbilirubinemia that is predominantly unconjugated in the absence of hemolysis or hepatic pathology) may be allowed after consultation with treating physician * AST(SGOT) and ALT(SGPT) ≤2.5 × institutional upper limit of normal. Exception: ≤5 x institutional upper limit of normal in subjects with liver metastasis * Creatinine (Cr) and creatinine clearance (CrCl) Cr \<1.6 mg/dL or CrCl ≥30 mL/min; CrCl should be calculated using the Cockcroft-Gault formula: CrCl (mL/min) = (140-Age) x Body weight (Kg)/72 x Serum creatinine (mg/dL) * PT, INR and PTT ≤ 1.5 x institutional upper limit of normal. Exception: Subjects who are on a stable regimen of therapeutic anticoagulation for an appropriate clinical indication may be enrolled * Availability of adequate baseline prostate biopsy tissue for integral biomarker analysis and correlative studies: * Sources of tumor tissue allowed are (1) prostate biopsy, (2) transurethral resection of the prostate tissue (TURP) , (3) trans urethral resection of bladder tumor tissue (TURBT) with contiguous spread of prostate cancer to the bladder, and (4) metastatic biopsy tissue excluding bone and lymph node metastases (e.g. lung or liver biopsies are acceptable). * For OncoPanel, submit at least one (1) H\&E slide and ten (10) 5-micron thick serially sectioned unstained formalin-fixed paraffin-embedded (FFPE) slides. Biopsy should contain at least 20% tumor involvement with the highest Gleason score(s). If requested tissue is unavailable, a lower number of 4-micron or 5- micron slides and/or slides containing lower tumor involvement may be accepted after discussion with the Sponsor-Investigator. * For ImmunoProfile, submit at least one (1) H\&E slide and one (1) 5-micron thick serially sectioned unstained, freshly cut, FFPE slide. Biopsy should contain at least 50% tumor involvement with the highest Gleason score(s). If requested tissue is unavailable, a 5-micron slide containing lower tumor involvement may be accepted after discussion with the study Sponsor-Investigator. * Submission of one (1) H\&E slide and at least one (1) FFPE tissue core block with at least 3mm2 tumor area with the highest Gleason score is an acceptable alternative to unstained FFPE slides. * Subjects who have insufficient baseline prostate biopsy tissue for OncoPanel analysis but have baseline metastatic biopsy tissue available may have OncoPanel analysis performed using metastatic biopsy tissue. Successful OncoPanel testing (but not ImmunoProfile) of metastatic biopsy tissue is acceptable from any source including lymph node or bone, after discussion with the study Sponsor- Investigator. * For Exploratory Correlative Studies, at least 1 tissue core block (preferred) or one (1) H\&E slide and twelve (12) 5-micron thick FFPE slides with unstained, freshly cut, serial sections from biopsy cores containing at least 20% tumor involvement with the highest Gleason score(s) will be requested, if available. * Tissue should be submitted with redacted pathology report. * Successful OncoPanel and ImmunoProfile biomarker analysis for allocation into a study cohort during pre-screening * Subjects whose tumors harbor somatic or germline homozygous deletions and/or deleterious mutations in a DDR gene using OncoPanel will be assigned to Cohort 1, regardless of ImmunoProfile results * DDR genes include and are not limited to BRCA2, ATM, CHEK2, BRCA1, PALB2, RAD51D, ATR, NBN, PMS2, GEN1, MLH1, MSH2, MSH6, RAD51C, MRE11A, BRIP1, FAM175A, and CDK12 * Deleterious mutations are defined as loss of function, splice site, nonsense, or frameshift mutations, and determination will be made between DFCI molecular pathology and study Sponsor-Investigator * Tumors identified as mismatch repair deficient (MMR-d) or microsatellite instability high (MSI-H) will also be included in Cohort 1 * Patients with germline DDRD or MMR-d/MSI-H (Lynch Syndrome) or tumors with DDRD or MMR-d/MSI-H identified in another CLIAcertified laboratory (e.g., Foundation Medicine) using prostate or metastatic tissue may be assigned to Cohort 1 after discussion with the Sponsor-Investigator. If archival tissue is available, it will be requested for OncoPanel testing; however, results will not influence eligibility. * Subjects whose tumors are PD-L1 positive and/or CD8+ T cell inflamed using ImmunoProfile without the presence of DDRD will be assigned to Cohort 2 * PD-L1 positivity will be defined as Combined Positive Score (CPS) ≥ 1, which is the number of PD-L1 staining cells (e.g., tumor cells, immune cells) divided by the total number of tumor cells, multiplied by 100 * CD8+ T cell inflammation will be defined as CD8+ T cell density ≥ 200, which is the number of CD8+ cells divided by the surface area of a region of interest (mm2) * Subjects whose tumors do not harbor DDRD and are PD-L1 negative with low CD8+ T cell infiltration will be assigned to Cohort 3 * Subjects whose prescreening is unsuccessful for cohort allocation or whose biomarker status matches that of a filled cohort will not be eligible * Subjects who underwent successful ImmunoProfile pre-screening but failed OncoPanel pre-screening may be allocated to Cohort 2 or Cohort 3 based on ImmunoProfile results and assuming DDRD negativity, at the discretion of the Sponsor-Investigator. * Before one of the study cohorts enrolls 15 of 20 subjects (Cohort 3 is anticipated to complete accrual first), subjects may undergo main study screening when ImmunoProfile and OncoPanel analyses are ongoing, and may proceed to study treatment if they meet all eligibility criteria with the exception that OncoPanel analysis is ongoing. These patients will be allocated into their respective cohort after OncoPanel results return. * Willingness to provide leftover metastatic biopsy tissue for correlative studies, if obtained for clinical purposes * Based on its mechanism of action and data from animal studies, nivolumab can cause fetal harm. For this reason non-sterilized men who are sexually active with a female partner of childbearing potential treated or enrolled on this protocol must agree to use adequate contraception prior to the study, for the duration of study participation, and for 7 months after last dose of nivolumab administration * Adequate contraception includes male condom plus spermicide * Not engaging in sexual activity is an acceptable practice; however, occasional abstinence, the rhythm method, and the withdrawal method are not acceptable methods of contraception * Subjects in this study should refrain from sperm donation * Ability to understand and the willingness to sign a written informed consent document, or have a legally authorized representative sign on the subject's behalf Exclusion Criteria: * Subjects must not have received prior ADT (LHRH analogue +/- antiandrogen), chemotherapy, or immunotherapy for prostate cancer. The following exception is allowed: * Subjects who have initiated ADT prior to study registration and are able to complete biomarker pre-screening, cohort allocation, and start C1D1 study chemoimmunotherapy ≤120 days from initiation of ADT are allowed * The 120-day window commences at the start of either the antiandrogen agent or LHRH analogue, whichever is earlier * Antiandrogens (e.g., bicalutamide or flutamide) may be used in addition to LHRH analogue ≤60 days before initiation of LHRH analogue to cover the testosterone surge associated with certain LHRH agonists but must be discontinued prior to study registration * Second-generation hormonal agents (e.g., abiraterone acetate) are not allowed * Subjects must not have undergone prostatectomy * Prostate radiation is allowed before or after study enrollment and may be delivered concurrently with study chemoimmunotherapy, per provider discretion, assuming adequate prostate biopsy tissue is collected before prostatic radiation * Metastasis-directed radiation is allowed before or after study enrollment and may be delivered concurrently with study chemoimmunotherapy, per provider discretion * Subjects who are receiving any other investigational agents * Any previous treatment with a PD-1 or PD-L1 inhibitor * Subjects with known brain metastases should be excluded from this clinical trial because of their poor prognosis and because they often develop progressive neurologic dysfunction that would confound the evaluation of neurologic and other AEs * History of allergic reactions attributed to compounds of similar chemical or biologic composition to docetaxel (including any drugs formulated with polysorbate 80), nivolumab, or LHRH analogue (e.g., leuprolide, goserelin acetate, degarelix) * History of another primary malignancy, except for: * Malignancy treated with curative intent and with no known active disease for ≥2 years before the first dose of study treatment and of low potential risk for recurrence * Adequately treated non-melanoma skin cancer or lentigo maligna without evidence of disease * 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 that would limit compliance with study requirements * Major surgical procedure as defined by the Site Investigator within 28 days prior to the first dose of chemoimmunotherapy * Known history of testing positive for human immunodeficiency virus (HIV) or known acquired immunodeficiency syndrome. HIV-positive subjects on combination antiretroviral therapy are ineligible because of the potential for nivolumab to be less clinically active in this population. In addition, these subjects are at increased risk of lethal infections when treated with marrow-suppressive chemotherapy * History of allogeneic bone marrow or organ transplantation * Active or prior documented autoimmune or inflammatory disorders, including= inflammatory bowel disease (e.g., Crohn's disease), systemic lupus erythematosus, Sarcoidosis syndrome, Grave's disease, rheumatoid arthritis, hypophysitis, uveitis, with the following exceptions: * Vitiligo or alopecia * Hypothyroidism stable on hormone replacement * Chronic skin condition that does not require systemic therapy * Celiac disease controlled by diet alone * Subjects with inactive disease in the last 5 years may be included but only after consultation with the study physician * Active infection including tuberculosis, hepatitis B (known positive HBV surface antigen \[HBsAg\]), or hepatitis C (HCV) * Subjects with a past or resolved HBV infection (defined as the presence of hepatitis B core antibody \[anti-HBc\] and absence of HBsAg) are eligible * Subjects with positive HCV antibody are eligible if polymerase chain reaction is negative for HCV RNA * Concurrent or prior use of immunosuppressive medication within 14 days before the first dose of study chemoimmunotherapy, with the following exceptions: * Premedication for docetaxel with oral dexamethasone (See Section 5.1) * Intranasal, inhaled, topical steroids, or local steroid injections (e.g., intraarticular injection) * Systemic corticosteroids at physiologic doses not exceeding 10mg/day of prednisone or its equivalent * Steroids as premedication for hypersensitivity reactions (e.g., premedication for iodinated contrast allergy before CT scan) Inclusion of Minorities • Men of all races and ethnic groups are eligible for this trial.