Lenvatinib 24 mg/Day Versus 10 mg/Day to Treat Symptomatic or Progressive Radioactive Iodine Resistant (RAIR) Differentiated Thyroid Cancer (DTC)

Inclusion Criteria: * Histologically confirmed DTC, defined as papillary, follicular, or Hurthle Cell thyroid cancer. Papillary has several sub-types such as tall-cell and columnar cell, which are all allowed. * Patient must have incurable RAIR DTC, defined as disease not amenable to cure by surgery AND meeting one or more of the following criteria: * one or more sites of disease that do not take up RAI. * disease progression on RAI (given within the last 12 months). * receipt of cumulative dose of RAI of ≥ 600mCi. * patient declines or is ineligible for surgery and/or RAI. * Measurable or evaluable disease per RECIST 1.1. * No more than 1 prior line of VEGF/VEGFR targeted therapy for DTC. Examples of VEGF/VEGFR therapies include sorafenib, pazopanib, vandetinib, axitinib, sunitinib, and cabozantinib, but others exist. * Symptomatic (defined by usual standard of care clinical criteria) or progressive disease on most recent prior treatment (ex: surgery, RAI, or TKI/targeted therapy) by RECIST 1.1 over the last 16 months. * At least 18 years of age. * ECOG performance status ≤ 2. * Screening blood pressure measurement \<140/90. Retesting is allowed. * Adequate bone marrow and organ function as defined below: * Absolute neutrophil count ≥ 1.0 K/cumm * Platelets ≥ 100 K/cumm * Hemoglobin ≥ 9.0 g/dL * Total bilirubin ≤ 1.5 x IULN * AST(SGOT)/ALT(SGPT) ≤ 3.0 x IULN (5.0 x IULN in the presence of hepatic metastases) * Creatinine clearance \> 30 mL/min by Cockc…