Treatment repeats every 14 days for up to 12 months in the absence of disease progression or unacceptable toxicity. Patients experiencing clear clinical benefit with aflibercept may continue treatment beyond 12 months, at the discretion of the study sponsor. Patients undergo fludeoxyglucose F 18 (FDG)-PET scans at baseline and after 8 weeks of study therapy to evaluate changes in FDG avidity on FDG-PET scan. Blood samples are obtained at baseline and periodically during study for laboratory correlative studies. Samples are examined for pretreatment serum VEGF concentration, thyroglobulin levels (when elevated), serum pharmacokinetics of aflibercept by ELISA, and anti-aflibercept antibodies.