RAI Plus Immunotherapy for Recurrent/Metastatic Thyroid Cancers

Inclusion Criteria: * Patients must have histologically or cytologically confirmed thyroid carcinoma of follicular origin (including papillary, follicular, hurthle cell or poorly differentiated subtypes and their respective variants). * Diagnosis of recurrent and/or metastatic thyroid cancer * At least one RAI-avid lesion identified on the most recent radioiodine scan (a diagnostic, post-therapy, or post-ablation scan) OR at least one lesion on the most recent FDG PET scan with an SUV max of 10 or less. (Both RAI-sensitive and RAI-refractory patients are eligible if at least one tumor with RAI avidity of any degree can be identified within one of these parameters.) * 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 with conventional techniques or as ≥ 10 mm with CT scan, MRI, or calipers by clinical exam. See Section 11 for the evaluation of measurable disease. Tumors in previously irradiated fields may be considered measureable if there is evidence of tumor progression after radiation treatment. * ECOG Performance Status (PS) 0 or 1. (or Karnofsky ≥60%) * Age ≥ 18 years at time of study entry * Adequate normal organ and marrow function as defined below: * Hemoglobin ≥ 9.0 g/dL * Absolute neutrophil count (ANC) ≥ 1.5 x 10\^9/L (\> 1500 per mm\^3) * Platelet count ≥ 100 x 10\^9/L (\>100,000 per mm\^3) * Serum bilirubin ≤ 1.5 x institutional upper limit of normal (ULN). (This will not apply to subjects with confirmed Gilbert's syndrome (persistent or recurrent hyperbilirubinemia that is predominantly unconjugated in the absence of hemolysis or hepatic pathology), who will be allowed only in consultation with their physician.) * AST (SGOT)/ALT (SGPT) ≤ 2.5 x institutional upper limit of normal unless liver metastases are present, in which case it must be ≤ 5x ULN * Serum creatinine CL\>40 mL/min by the Cockcroft-Gault formula (Cockcroft and Gault 1976) or by 24-hour urine collection for determination of creatinine clearance: * Males: Creatinine CL (mL/min) = Weight (kg) x (140 - Age) . 72 x serum creatinine (mg/dL) * Females: Creatinine CL (mL/min) = Weight (kg) x (140 - Age) x 0.85 72 x serum creatinine (mg/dL) * Female subjects must either be of non-reproductive potential (i.e., post-menopausal by history: ≥60 years old and no menses for ≥1 year without an alternative medical cause; OR history of hysterectomy, OR history of bilateral tubal ligation, OR history of bilateral oophorectomy) or must have a negative serum pregnancy test upon study entry. * Subject is willing and able to comply with the protocol for the duration of the study including undergoing treatment and scheduled visits and examinations including follow up. * Patients must agree to undergo two research biopsies of (a) malignant lesion(s). Biopsies do not need to be done if the investigator or person performing the biopsy judges there is no tumor accessible for biopsy, the only accessible tumor must be used for RECIST measurement, or the biopsy poses too great a risk to the patient. If the patient has only one RECIST measureable target lesion for response assessment, research biopsies must not be performed on that target lesion. * Availability of archival tumor tissue from the thyroid cancer primary or metastasis (a tissue block or a minimum of 30 unstained slides would be required. Patients with less archival tissue available may still be eligible for the study after discussion with the MSK Principal Investigator.) Exclusion Criteria: * 131I therapy \< 6 months prior to initiation of therapy on this protocol. A diagnostic study using \< 10 mCi of 131I is not considered 131I therapy. * Any previous treatment with a PD1 or PD-L1 inhibitor, including durvalumab. * History of pneumonitis. * External beam radiation therapy \< 4 weeks prior to initiation of therapy on this protocol. * Chemotherapy, immunotherapy, targeted therapy, monoclonal antibodies, tumor embolization, or other investigational agent within 28 days prior to the first dose of study drug. * Current or prior use of immunosuppressive medication within 28 days before the first dose of durvalumab, with the exceptions of intranasal and inhaled corticosteroids or systemic corticosteroids at physiological doses, which are not to exceed 10 mg/day of prednisone, or an equivalent corticosteroid. * Any unresolved toxicity CTCAE grade ≥ 2 from previous anti-cancer therapy. Exceptions include hearing loss, peripheral neuropathy, and alopecia. * Any prior Grade ≥3 immune-related adverse event (irAE) while receiving any previous immunotherapy agent, or any unresolved irAE \>Grade 1. * Active or prior documented autoimmune disease within the past 2 years. NOTE: Subjects with a history of autoimmune thyroid disease are not excluded. Subjects with vitiligo or psoriasis not requiring systemic treatment (within the past 2 years) are not excluded. * Active or prior documented inflammatory bowel disease (e.g., Crohn's disease, ulcerative colitis). * History of primary immunodeficiency. * History of allogeneic organ transplant. * History of hypersensitivity to durvalumab or any excipient. * History of hypersensitivity to thyrotropin alpha (Thyrogen). * Patients unable to follow a low iodine diet or requiring medication with high content in iodide (amiodarone). * Uncontrolled intercurrent illness including, but not limited to, ongoing or active infection, symptomatic congestive heart failure, uncontrolled hypertension, unstable angina pectoris, cardiac arrhythmia, active peptic ulcer disease or gastritis, active bleeding diatheses acute or chronic hepatitis B, hepatitis C or human immunodeficiency virus (HIV), or psychiatric illness/social situations that would limit compliance with study requirements or compromise the ability of the subject to give written informed consent. * Known history of active tuberculosis. * Symptomatic brain metasteses, leptomeningeal carcinomatosis, or spinal cord compression (treated metastatic brain, leptomeningeal carcinomatosis, or spinal cord compression are allowed). Note: Patients must be off steroids used for brain metasteses, leptomeningeal carcinomatosis, or spinal cord compression. * Receipt of live attenuated vaccination within 30 days prior to study entry or within 30 days of receiving durvalumab. * Female subjects who are pregnant, breast-feeding or male or female patients of reproductive potential who are not employing an effective method of birth control. * Any condition that, in the opinion of the investigator, would interfere with evaluation of study treatment or interpretation of patient safety or study results * Subjects with uncontrolled seizures.