Selumetinib in Treating Patients With Papillary Thyroid Cancer That Did Not Respond to Radioactive Iodine

Inclusion Criteria: * Histologically or cytologically confirmed papillary thyroid cancer or papillary thyroid cancer with follicular elements * No longer amenable to radioactive iodine therapy or curative surgical resection * Tumor is no longer iodine avid * Tumor did not respond to the most recent radioactive iodine treatment * Patient is ineligible for further radioactive iodine therapy due to medical contraindications (e.g., lung toxicity) * Measurable disease, defined as at least one lesion that can be accurately measured in at least one dimension (longest diameter to be recorded) as ≥ 20 mm with conventional techniques or as ≥ 10 mm with spiral CT scan * Evidence of disease progression (objective growth of existing tumors) * New or enlarging measurable lesions within the past 12 months * If the most recent imaging study is older than 12 months, patients will still be eligible if objectively measurable disease progression is associated with clinical symptoms * Archival tumor tissue available for mutational analysis * No known brain metastases * ECOG performance status (PS) 0-2 OR Karnofsky PS 60-100% * Life expectancy \> 12 weeks * WBC ≥ 3,000/µL * ANC ≥ 1,500/µL * Platelet count ≥ 100,000/µL * Total bilirubin normal * AST and ALT \< 2.5 times upper limit of normal * Creatinine normal OR creatinine clearance ≥ 60 mL/min * Not pregnant or nursing * Negative pregnancy test * Fertile patients must use effective contraception prior to, during, and for 4 weeks after completion of study treatment * Able to understand and willing to sign a written informed consent document * History of allergic reactions attributed to compounds of similar chemical or biologic composition to selumetinib (AZD6244) or its excipient Captisol® * QTc interval \> 450 msec or other factors that increase the risk of QT prolongation * Arrhythmic events (e.g., heart failure, hypokalemia, family history of long QT interval syndrome), including heart failure that meets NYHA class III and IV definition * Refractory nausea and vomiting, chronic gastrointestinal diseases (e.g., inflammatory bowel disease), or significant bowel resection that would preclude adequate absorption * Concurrent uncontrolled illness including, but not limited to, ongoing or active infection or psychiatric illness/social situations that would limit compliance with study requirements * At least 4 weeks since prior radiotherapy or chemotherapy (6 weeks for nitrosoureas or mitomycin C) * Prior treatment with tyrosine kinase inhibitors that target RET or RAF * Prior treatment with MEK inhibitors * Concurrent combination antiretroviral therapy for HIV-positive patients * Concurrent medication that can prolong the QT interval * Other concurrent investigational agents