ADI-PEG 20 in Combination With Gemcitabine and Docetaxel After Progression on Frontline Therapy in Non-small Cell and Small Cell Lung Cancers

Inclusion Criteria: * Histologically or cytologically confirmed extensive stage small cell or metastatic non-small cell lung cancer that has progressed on frontline therapy who are fit for treatment with gemcitabine and docetaxel in the opinion of the treating physician. Phase II enrollment will occur separately to the SCLC and NSCLC cohorts, with up to 36 enrolled in each cohort. * Measurable disease per RECIST 1.1. * Treated with at least one previous line of systemic therapy. Specific prior treatment requirements include: * Patients with ES-SCLC must have been treated with platinum doublet and anti-PD(L)1 therapy, if eligible. * Patients with NSCLC without a driver mutation must have been treated with platinum doublet and anti-PD(L)1 therapy, if eligible. * Patients with NSCLC with a driver mutation (EGFR, ALK, ROS1) must have been treated with an FDA approved targeted therapy and platinum doublet therapy, if eligible. * If the most recent prior line included immunotherapy, patient must have experienced progression by CT scan after cessation of immunotherapy and prior to starting study therapy. * Patient must have archival tissue available for correlatives. If tissue is not available, approval of enrollment may be granted on a case-by-case basis. Sponsor-investigator approval is required in these instances. * At least 18 years of age. * ECOG performance status ≤ 1. * Adequate bone marrow and organ function as defined below: * Absolute neutrophil count ≥ 1.5 K/cumm * Platelets ≥ 100 K/cumm * Hemoglobin ≥ 9 g/dL * Total bilirubin ≤ 2 x IULN, patients with Gilberts must be below 3xIULN * AST(SGOT)/ALT(SGPT) ≤ 3 x IULN (or ≤ 5 x IULN if liver metastases are present) * Creatinine clearance \> 60 mL/min by MDRD or by 24 hour urine * Serum uric acid ≤ 8 mg/dL (with or without medication control) * If patient is on oxygen, must require 3L O2 at rest or less. * The effects of ADI-PEG 20 on the developing human fetus are unknown. For this reason and because chemotherapeutics are known to be teratogenic, women of childbearing potential must agree to use adequate contraception (hormonal or barrier method of birth control, abstinence) prior to study entry, for the duration of study participation, and for one month after completion of study treatment. Should a woman become pregnant or suspect she is pregnant while participating in this study, she must inform her treating physician immediately. Men treated or enrolled on this protocol must also agree to use adequate contraception prior to the study, for the duration of the study, and for one month after completion of study treatment. * Ability to understand and willingness to sign an IRB approved written informed consent document. Exclusion Criteria: * A history of other malignancy with the exception of: * Malignancies for which all treatment was completed at least 2 years before registration and the patient has no evidence of disease * Basal cell or squamous cell carcinoma of the skin which was treated with local resection only * Carcinoma in situ of the cervix * Other tumors discussed with the study PI * Receipt of any of the following therapies within the below time frames: * 21 days for chemotherapy * 21 days or 5 half-lives for an oral small molecule inhibitor (whichever is shorter) * 21 days for immunotherapy * 21 days for RT, or 7 days for SBRT or any palliative radiation * 21 days for surgery * 28 days for an investigational agent. * Prior treatment with ADI-PEG 20 or gemcitabine (prior docetaxel is allowed). * Presence of untreated or unstable brain metastases. Patients with treated/stable brain metastases, defined as patients who have received prior therapy for their brain metastases and whose CNS disease is radiographically stable at study entry, are eligible. * A history of allergic reactions attributed to compounds of similar chemical or biologic composition to ADI-PEG 20, gemcitabine, pegylated compounds, or other agents used in the study. * Uncontrolled intercurrent illness including, but not limited to, ongoing or active infection, symptomatic congestive heart failure, unstable angina pectoris, or cardiac arrhythmia. * History of seizure disorder not related to underlying cancer. * Grade 2 or higher neuropathy * Pregnant and/or breastfeeding. Women of childbearing potential must have a negative pregnancy test within 14 days of study entry. * Patients with known active Hepatitis B or C or HIV.