Phase I/II Study of PEGylated Arginine Deiminase (ADI-PEG20) With Carboplatin and Cabazitaxel in Men With Aggressive Variant Prostate Cancers (AVPC)

Inclusion Criteria: 1. Completion of informed consent prior to any study specific procedures. 2. Patients must agree to tissue collection for correlative studies at the specified timepoints. 3. Patients must consent to the MD Anderson Immunotherapy Platform laboratory protocol PA13-0291. 4. Male aged 18 years and above. 5. Histologically or cytologically confirmed prostate carcinoma. 6. Presence of metastatic disease documented on imaging studies (bone scan, CT and/or MRI scans). 7. Patients must meet at least one of the following AVPC criteria: i. Histologically proven small cell (neuroendocrine) prostate carcinoma ii. Exclusive visceral metastases. iii. Predominantly lytic bone metastases identified by plain x-ray or CT scan. iv. Bulky (≥5cm in longest dimension) lymphadenopathy or high-grade tumor mass in prostate/pelvis. v. Low PSA (≤ 10ng/mL) at initial presentation (prior to androgen ablation or at symptomatic progression in the castrate-setting) plus high volume (≥ 20) bone metastases. vi. Elevated serum LDH (≥2 x ULN) or elevated serum CEA (≥2 x ULN) in the absence of other etiologies. vii. Short interval (≤180 days) to castrate-resistant progression following initiation of hormonal therapy. viii. Known loss or mutation (by CLIA certified molecular testing, IHC and/or DNA sequencing) in at least 2 of Tp53, RB1 and PTEN defined as: * AVPC determination by immunohistochemistry. Tumor samples are considered negative (and thus abnormal) for RB1 and PTEN if their labeling index is ≤ 10% and positive (and thus aberrant) for Tp53 if their labeling index is ≥ 10%, where the labeling index is defined as the percentage of positive cells, and calculated as the number of positively stained epithelial cells divided by the total number of epithelial cells, at X200 magnification. * AVPC determination by DNA sequencing. The TP53, RB1 and PTEN genes will be considered aberrant if they contain exonic nonsynonymous missense or stop-gain mutations, frameshift or non-frameshift indels (insertions or deletions), and/or copy number losses. ix. Patients who have castration-resistant disease progression per RECIST in the absence of PSA values rising to ≥ 1.0ng/mL as per PCGW3 PSA progression criteria 8. Patients must have documented evidence of progressive disease as defined by any of the following: I. PSA progression: minimum of 2 rising values (3 measurements) obtained a minimum of 7 days apart with the last result being at least ≥ 1.0 ng/mL; II. New or increasing non-bone disease (RECIST); III. Positive bone scan with 2 or more new lesions (PCWG3); IV. Increasing symptoms unequivocally attributed to disease progression as judged by the treating physician and the PI; V. Biopsy proven new transformation to small cell carcinoma in a patient previously diagnosed with an adenocarcinoma of the prostate. 9. Surgically or ongoing medically castrated, with baseline testosterone levels of ≤ 50 ng/dL (≤2.0 nM). Exception: Patients with de novo primary small cell carcinoma of the prostate may begin chemotherapy on study once treatment with an LHRH agonist or antagonist has been initiated, even if testosterone levels have not reached ≤ 50ng/dL. 10. Eastern Cooperative Oncology Group (ECOG) Performance Status of ≤2. 11. Patients must have adequate organ and bone marrow function measured within 7 days prior to treatment registration as defined below: I. Hemoglobin ≥ 9.0 g/dL (unless deemed by the treating physician to be due to bone marrow infiltration by tumor, in which case hemoglobin ≥8gdL is allowed). Patient may have blood transfusions prior to study enrollment. II. Absolute neutrophil count (ANC) ≥1.5 x 109/L (unless deemed by the treating physician to be due to bone marrow infiltration by tumor, in which case ANC ≥1,000/mm3 is allowed) III. White blood cells (WBC) ≥3x109/L (unless deemed by the treating physician to be due to bone marrow infiltration by tumor, in which case WBC ≥2x109/L is allowed) IV. Platelet count ≥ 100 x 109/L (unless deemed by the treating physician to be due to bone marrow infiltration by tumor, in which case platelet ≥75,000/ mm3 is allowed) V. Total bilirubin ≤ 1.5 x institutional upper limit of normal (ULN) (except for patients with known Gilbert's disease). VI. AST (SGOT) and ALT (SGPT) ≤ 2.5 x institutional upper limit of normal (unless liver metastases are present, in which case it must be ≤ 5x ULN) VII. Calculated creatinine clearance (Cockcroft-Gault Equation) ≥ 30 mL/min. 12. Patients who have partners of childbearing potential (e.g., female that has not been surgically sterilized or who are not amenorrheic for ≥ 12 months) must be willing to use a method of birth control in addition to adequate barrier protection as determined to be acceptable by the investigator during the study and for 3 months after last dose of ADI-PEG 20 administration. In addition, men should not donate sperm during this period. 13. Patient is willing and able to comply with the protocol for the duration of the study including undergoing treatment and scheduled visits and examinations including follow up. Exclusion Criteria: 1. Any prior treatment for CRPC with carboplatin, cisplatin or cabazitaxel. 2. Patients who have received more than one line of chemotherapy for prostate cancer. Any number of prior hormonal or targeted therapies are allowed. 3. Patients who have not recovered from adverse events secondary to systemic therapy (except for LHRH agonist or antagonist treatment for prostate cancer, and bisphosphonates or RANK ligand inhibitors for bone strengthening), major surgery or radiotherapy for the treatment of prostate cancer to a grade \</= 2. 4. Any unresolved toxicity (CTCAE Grade ≥2) from previous anti-cancer therapy. Subjects with irreversible toxicity that is not reasonably expected to be exacerbated by the investigational product may be included (e.g., hearing loss, peripherally neuropathy). 5. Active uncontrolled infection (patients completing a course of antibiotic or antiviral therapy whose infection is deemed to be controlled may be allowed on study after discussion with the PI; the PI will serve as the final arbiter regarding eligibility). 6. Active or symptomatic viral hepatitis or chronic liver disease. 7. A history of extensive bilateral lung disease of non-malignant etiology. 8. A malignancy \[other than the one treated in this study\] which has a '≥ 30% probability of recurrence within 24 months (except for adequately treated non-melanoma skin cancer, curatively treated in situ cancer of the cervix or Ta urothelial carcinomas). 9. Any underlying medical or psychiatric condition, which in the opinion of the Investigator, will make the administration of study drug hazardous or obscure the interpretation of adverse events. Examples: include, but are not limited to, uncontrolled ventricular arrhythmia, recent (within 3 months) myocardial infarction, superior vena cava syndrome, extensive bilateral lung disease on HRCT scan, uncontrolled seizures, history of allogeneic organ transplant, history of primary immunodeficiency or any psychiatric disorder that prohibits obtaining informed consent. 10. Patients with symptomatic uncontrolled brain metastases or spinal cord compressions. A scan to confirm the absence of brain metastases is not required. 11. Prisoners or subjects who are involuntarily incarcerated. 12. Subjects who are compulsorily detained for treatment of either a psychiatric or physical (e.g., infectious disease) illness. 13. History of allergic reactions attributed to compounds of similar chemical or biologic composition to ADI-PEG 20, pegylated compounds, or other agents used in this study.