Durvalumab and Tremelimumab in Treating Patients With Recurrent Stage IV Lung Cancer

Inclusion Criteria:

* Patients must have been assigned to S1400F
* Patients must have progressed during or after prior platinum-based chemotherapy; patients whose only prior platinum-based chemotherapy regimen was for stage I-III disease (i.e. patient has not received any platinum-based chemotherapy for stage IV or recurrent disease), disease progression on platinum-based chemotherapy must have occurred within one year from the last date that patient received that therapy; patients must have experienced disease progression during or after prior anti-PD-1 or anti-PD-L1 antibody monotherapy as their most recent line of treatment; prior PD-1/PD-L1 combination therapy is not permitted
* Prior exposure to CTLA-4 inhibitors (ipilimumab and tremelimumab) is not permitted; prior exposure to the following is allowed: attenuated vaccines, anti-EGFR agents, and granulocyte-macrophage colony-stimulating factor (GM-CSF)
* Patients must not have received nitrosoureas or mitomycin-C within 42 days prior to sub-study registration
* Patients must not have any prior documented autoimmune or inflammatory disease (including inflammatory bowel disease, diverticulitis with the exception of diverticulosis, celiac disease, irritable bowel disease; Wegner syndrome; Hashimoto syndrome) within 3 years prior to sub-study registration; patients with vitiligo, immune-mediated alopecia, Grave?s disease, or psoriasis requiring systemic treatment within the past 2 years are not eligible; patients with hypothyroidism (e.g. post Hashimoto syndrome) who are stable on hormone replacement therapy are eligible
* Patients must not have any history of primary immunodeficiency
* Patients must not have received any immunosuppressive medication within 28 days prior to sub-study registration and must not be planning to receive these medications while on protocol therapy; systemic corticosteroids must be stopped at least 24 hours prior to sub-study registration; however, intranasal and inhaled corticosteroids are allowed at any time before and during protocol therapy
* Patients must not have experienced a grade 3 or worse immune-related adverse event (irAE) (except asymptomatic nonbullous/nonexfoliative rash) or any unresolved irAE grade 2, nor have experienced a toxicity that led to permanent discontinuation of prior anti-PD-1/PD-L1 immunotherapy
* Patients must not have any history of organ transplant that requires use of immunosuppressives
* Patients must not have any known allergy or reaction to any component of the durvalumab (MEDI4736) and/or tremelimumab formulation
* Patients must not have clinical signs or symptoms of active tuberculosis infection
* Patients must not have received a live attenuated vaccination within 28 days prior to sub-study registration
* Patients must not have known human immunodeficiency virus (HIV), or a known positive test for hepatitis B virus surface antigen (HBV sAg), or hepatitis C virus ribonucleic acid (HCV antibody) indicating current acute or chronic infection; patients with a positive hepatitis C antibody with a negative viral load are allowed
* Patients must have a thyroid-stimulating hormone (TSH) with reflex free T3/free T4 (if TSH is out of normal range) and electrocardiogram (EKG) obtained within 7 days prior to sub-study registration
* Patients must also be offered participation in banking and in the correlative studies for collection and future use of specimens