Lenvatinib and Iodine Therapy in Treating Patients With Radioactive Iodine-Sensitive Differentiated Thyroid Cancer

Inclusion Criteria: * Prior treatment with therapeutic dose of radioactive iodine (\> 50 mCi) with evidence of RAI uptake on delayed scan and with progression (biochemical or anatomic) within 12 months of RAI * Eastern Cooperative Oncology Group (ECOG) performance status 0 or 1 (Karnofsky ≥ 80%) * Leukocytes ≥ 3,000/µL * Absolute neutrophil count ≥ 1,500/µL * Platelets ≥ 100,000/µL * Total bilirubin within normal institutional limits * Aspartate aminotransferase (AST) (serum glutamic-oxaloacetic transaminase \[SGOT\])/alanine aminotransferase (ALT) (serum glutamate pyruvate transaminase \[SGPT\]) ≤ 2.5 x institutional upper limit of normal * Creatinine within normal institutional limits OR * Creatinine clearance ≥ 60 mL/min/1.73 m² for patients with creatinine levels above institutional normal * Confirmed diagnosis of differentiated thyroid cancer (follicular or papillary thyroid cancer and their variants) * Ability and willingness to use appropriate contraception; women of child-bearing potential and men must agree to use adequate contraception (hormonal or barrier method of birth control; abstinence) prior to study entry and for the duration of study participation; should a woman become pregnant or suspect she is pregnant while she or her partner is participating in this study, she should 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 study participation, and for 2 weeks after completion of lenvatinib administration * Patients must have measurable disease, defined as at least one lesion that can be accurately measured in at least one dimension (longest diameter to be recorded for non-nodal lesions and short axis for nodal lesions) as ≥ 20 mm (≥ 2 cm) by chest x-ray or as ≥ 10 mm (≥ 1 cm) with computed tomography (CT) scan, magnetic resonance imaging (MRI), or calipers by clinical exam * Ability to understand and the willingness to sign a written informed consent document Exclusion Criteria: * Patients who have received RAI within 12 weeks of planned retreatment * Prior receipt of cumulative RAI doses in excess of 1000 mCi * Patients who have not recovered from adverse events due to prior anti-cancer therapy (i.e., have residual toxicities \> grade 1) * Patients who are receiving any other investigational agents * Patients with previously untreated and or symptomatic brain metastases are excluded from this clinical trial because of the risk of intracranial bleeding with angiogenic agents and tumoral swelling from RAI * History of allergic reactions attributed to compounds of similar chemical or biologic composition to lenvatinib * Uncontrolled intercurrent illness including, but not limited to, ongoing or active infection, symptomatic congestive heart failure, unstable angina pectoris, cardiac arrhythmia, or psychiatric illness/social situations that would limit compliance with study requirements * Patients with uncontrolled hypertension (requirement for more than 2 blood pressure \[BP\] medications or grade 2 or higher BP elevation while on adequate doses of not more than 2 antihypertensive agents) are excluded from the study because one of the significant adverse events of lenvatinib is worsening hypertension * Fridericia's corrected QT (QTcF) interval prolongation greater than 500 ms * Recent arterial thromboembolic event within the previous 6 months * Urine dipstick proteinuria ≥ 2+ or nephrotic range proteinuria on ≥ 2 gram in 24-hour urine * History of gastrointestinal perforation, abscess or fistula * History of and or medical condition (e.g. diverticular disease; aneurysm) that predisposes to risk of major hemorrhage * Pregnant women are excluded from this study because lenvatinib is a tyrosine kinase inhibitor agent with the potential for teratogenic or abortifacient effects; because there is an unknown but potential risk for adverse events in nursing infants secondary to treatment of the mother with lenvatinib, breastfeeding should be discontinued if the mother is treated with lenvatinib * Human immunodeficiency virus (HIV)-positive patients on combination antiretroviral therapy are ineligible because of the potential for pharmacokinetic interactions with lenvatinib