Aflibercept in Treating Patients With Recurrent and/or Metastatic Thyroid Cancer That Did Not Respond to Radioactive Iodine Therapy

Inclusion Criteria: * Histopathologically confirmed differentiated thyroid carcinoma of follicular cell origin, including any of the following histologies and their respective variants: * Papillary * Follicular * Hürthle cell * Must have surgically inoperable and/or recurrent or metastatic disease * At least one fludeoxyglucose F 18 (FDG)-PET-avid lesion, defined as any focus of increased FDG uptake \> normal mediastinal activity with standard uptake variable (SUV) maximum levels ≥ 3, as documented by baseline PET scan * Measurable disease, defined as ≥ 1 unidimensionally measurable lesion ≥ 20 mm by conventional techniques or ≥ 10 mm by spiral CT scan * Progressive disease, defined by ≥ 1 of the following occurring during or after prior treatment (e.g., radioactive isotope \[RAI\] treatment): * Presence of new or progressive lesions on CT scan or MRI * New lesions on bone scan or PET scan * Rising thyroglobulin level documented by a minimum of 3 consecutive rises, with an interval of \> 1 week between each determination * No known history of brain metastasis * ECOG performance status (PS) 0-2 OR Karnofsky PS 60-100% * ANC ≥ 1,500/mcL * Platelet count ≥ 75,000/mcL * WBC ≥ 3,000/mcL * Total bilirubin ≤ 1.5 times upper limit of normal(ULN) * AST and ALT ≤ 2.5 times ULN (≤ 5 times ULN for liver metastases) * Creatinine ≤ 1.5 times ULNOR creatinine clearance ≥ 60 mL/min * INR ≤ 1.2 (≤ 1.5 times ULN if on prophylactic-dose anticoagulation) * Urine protein: creatinine ratio \< 1 OR 24-hour urine protein \< 500 mg * Not pregnant or nursing * Negative pregnancy test * Fertile patients must use effective contraception during and for ≥ 6 months after completion of study therapy * Documentation of systolic blood pressure ≤150 mm Hg and diastolic blood pressure ≤100 mm Hg * No known hypersensitivity to Chinese hamster ovary cell products or other recombinant human antibodies * No history of allergic reactions attributed to compounds of similar chemical or biologic composition to other agents used in the study * No serious or non-healing wound, ulcer, or bone fracture * No abdominal fistula, gastrointestinal perforation, or intra-abdominal abscess in the past 28 days * No significant traumatic injury within the past 28 days * No clinically significant cardiovascular disease, defined as any of the following: * Cerebrovascular accident within the past 6 months * Myocardial infarction within the past 6 months * Coronary artery bypass grafting or unstable angina within the past 6 months * NYHA grade III-IV congestive heart failure * Canadian Cardiovascular Class grade III or greater angina within the past 6 months * Clinically significant peripheral vascular disease within the past 6 months * Pulmonary embolism, deep-vein thrombosis, or other thromboembolic event within the past 6 months * Uncontrolled coronary artery disease, angina, congestive heart failure, or ventricular arrhythmia requiring acute medical management * Myocardial infarction, cerebrovascular accident, or transient ischemic attack within the past 6 months * No evidence of bleeding diathesis or coagulopathy within the past 12 months * No uncontrolled intercurrent illness including, but not limited to, ongoing or active infection or psychiatric illness or social situation that would limit study compliance * No known HIV positivity * See Disease Characteristics * Recovered from prior therapy * No prior VEGF-targeted antibody therapy (e.g., bevacizumab or aflibercept) * More than 4 weeks since prior systemic therapy or radiotherapy * More than 7 days since prior core biopsy * Up to 1 prior targeted biologic agent (e.g., small-molecule tyrosine kinase inhibitor or histone deacetylase inhibitor) allowed provided treatment was stopped ≥ 4 weeks prior to initiation of therapy on this study * Up to 1 prior cytotoxic chemotherapy (e.g., doxorubicin hydrochloride) allowed provided treatment was stopped ≥ 4 weeks prior to initiation of therapy on this study * Prior systemic chemotherapy administered as part of initial definitive treatment (e.g., as a radiation sensitizer or as initial adjuvant therapy) allowed provided treatment was stopped ≥ 3 months prior to initiation of therapy on this study and does not count in the determination of prior targeted or cytotoxic therapy * At least 2 weeks since prior cyclooxygenase-2 (COX-2) inhibitors, cis-retinoic acid, or complementary medications if given with anti-cancer intent * Medications given for a specific clinical indication (e.g., daily aspirin status post myocardial infarction or COX-2 inhibitors at standard anti-inflammatory/pain doses) may be continued based on the clinical judgment of the involved investigator * Prior RAI therapy allowed provided it was stopped \> 3 months prior to initiation of therapy on this protocol and evidence of progression (as defined above) has been documented in the interim * A diagnostic study using \< 10 mCi of RAI is not considered RAI therapy * Prior external-beam radiotherapy to index lesions allowed provided there has been documented progression by RECIST criteria and at least 4 weeks have elapsed * At least 4 weeks since prior external-beam radiation therapy to non-index lesions * At least 4 weeks since prior surgery * Concurrent therapeutic-dose anticoagulants (e.g., warfarin) with PT INR \> 1.5 allowed provided that both of the following criteria are met: * In-range INR appropriate to the treatment indication (e.g., between 2 and 3 for atrial fibrillation) AND on a stable dose of oral anticoagulant or on a stable dose of low molecular weight heparin * No active bleeding or pathological condition that carries a high risk of bleeding (e.g., tumor involving major vessels or known varices) * Patients receiving concurrent antihypertensive agents must have documentation of the date of the last change in dosage * No other concurrent investigational agents * No major surgical procedure or open biopsy within the past 28 days * No anticipation of need for major surgical procedures during the course of the study