Iodine I-131 With or Without Selumetinib in Treating Patients With Recurrent or Metastatic Thyroid Cancer

Exclusion Criteria

• •* For patients with non-measurable, structural disease the following must apply:
• •Undetectable thyroglobulin antibody AND
• •A serum thyroglobulin of 10 ng/ml or greater in the context of suppressed thyroid-stimulating hormone (TSH) (TSH =\< 0.4 mcU/ml) =\< 28 days prior to study registration; use of any thyroglobulin assay is allowed, though all serum thyroglobulin measurements for study purposes must be conducted with the same thyroglobulin assay
• •* Eastern Cooperative Oncology Group (ECOG) performance status (PS) 0, 1 or 2
• •* Able to swallow and retain orally-administered medication with no clinically significant gastrointestinal abnormalities that may alter absorption
• •* Absolute neutrophil count (ANC) \>= 1500/mm\^3 (obtained =\< 28 days prior to randomization)
• •* Platelet count \>= 100,000/mm\^3 (obtained =\< 28 days prior to randomization)
• •* Hemoglobin \> 9.0 g/dL (obtained =\< 28 days prior to randomization)
• •* Total bilirubin =\< 1.5 x upper limit of normal (ULN) (obtained =\< 28 days prior to randomization)
• •* Aspartate transaminase (AST) =\< 2.5 x ULN (or =\< 5 x ULN in presence of liver metastases) (obtained =\< 28 days prior to randomization)
• •* Creatinine =\< 1.5 mg/dL OR calculated creatinine clearance of \>= 50 ml/min by either the Cockcroft-Gault formula or 24-hours urine collection analysis (obtained =\< 28 days prior to randomization)
• •* Negative pregnancy test performed =\< 7 days prior to registration for women of childbearing potential only
• •* Provide informed written consent
• •* Willing to return to enrolling institution for follow-up (during the Active Monitoring Phase of the study)
• •Note: during the Active Monitoring Phase of a study (i.e., active treatment and observation), participants must be willing to return to the consenting institution for follow-up
• •* Willing to provide mandatory archival tumor tissue (block or minimum of 30 unstained slides from a primary or recurrent/metastatic thyroid cancer) for correlative research purposes; NOTE: patients with less archival tumor tissue available may still be eligible for the study after discussion with Academic and Community Cancer Research United (ACCRU); receipt of archival tumor tissue is not required for study registration and initiation of therapy
• •* Willing to provide mandatory blood samples for correlative research purposes
• •131I therapy =\< 6 months prior to registration; Note: 131I administered solely for diagnostic purposes is not considered 131I therapy
• •External beam radiation therapy =\< 28 days prior to registration; note: previous treatment with radiation is allowed if the investigator judges it will not compromise patient safety on the study
• •Having been treated with a total cumulative (lifetime) 131I therapeutic activity \> 800 mCi (excluding 131I activity administered for diagnostic scans)
• •Treatment with chemotherapy or targeted therapy (e.g. tyrosine kinase inhibitor) =\< 28 days prior to registration
• •Prior exposure to mitogen-activated protein kinase kinase (MEK), RAS, or RAF inhibitors (note: previous exposure to sorafenib is allowed) OR history of hypersensitivity to selumetinib, thyrotropin alpha (Thyrogen), or any excipient agents
• •Unresolved toxicity \> Common Terminology Criteria for Adverse Events (CTCAE) grade 2 from previous anti-cancer therapy, except for alopecia
• •Cardiac conditions as follows:
• •* Uncontrolled hypertension (blood pressure \[BP\] \>=150/95 mmHg despite medical therapy)
• •* Left ventricular ejection fraction \< 55% measured by echocardiography
• •* Atrial fibrillation with a ventricular rate \> 100 beats per minute (bpm) on electrocardiogram (ECG) at rest
• •* Symptomatic heart failure (New York Heart Association \[NYHA\] grade II-IV)
• •* Prior or current cardiomyopathy
• •* Severe valvular heart disease
• •* Uncontrolled angina (Canadian Cardiovascular Society grade II-IV despite medical therapy)
• •* Acute coronary syndrome =\< 6 months prior to registration
• •Ophthalmological conditions as follows:
• •* Intra-ocular pressure \> 21 mmHg, or uncontrolled glaucoma (irrespective of intra-ocular pressure)
• •* Current or past history of central serous retinopathy or retinal vein occlusion
• •Symptomatic or untreated leptomeningeal disease, brain metastasis, or spinal cord compression
• •Unable to follow a low iodine diet or requiring medication with high content in iodide (e.g., amiodarone)