Combination of Temsirolimus and Sorafenib in the Treatment of Radioactive Iodine Refractory Thyroid Cancer

Inclusion Criteria: * Patients must have histopathologically confirmed at MSKCC thyroid carcinoma of follicular cell origin (D-TC-FCO), which includes papillary, follicular, Hürthle cell histology, or anaplastic along with their respective variants. * Available pathology for RAF mutational testing (e.g., paraffin block or 5-10 unstained slides). It is not required that mutational testing be completed before starting the clinical study. * Patients must have surgically inoperable and/or recurrent/metastatic disease. * Patients must have a PET scan prior to the protocol start date and have at least one FDGavid lesion that has not been removed surgically or radiated (unless it has progressed by RECIST criteria after the completion of radiation therapy and is still FDG-avid). FDGavidity will be defined as any focus of increased FDG uptake greater than normal activity with SUV maximum levels greater than or equal to 3. PET scan can have been done at any time prior to the start of therapy, although it is recommended that it be done within 3 months prior to the start of therapy. * Patients must have measurable disease by RECIST criteria, defined as at least one lesion that can be accurately measured in at least one dimension (longest diameter to be recorded) as ≥ 20 mm with conventional techniques or as ≥ 10 mm with spiral CT scan; performed ≤ 4 weeks of protocol start date. * Patients must have progressive disease defined by at least one of the following occurring during or after previous treatment (including RAI treatment): * The presence of new or progressive lesions on CT/MRI. * New lesions on bone scan or PET scan. * Rising thyroglobulin level (documented by a minimum of three consecutive rises, with an interval of \> 1 week between each determination). * Prior RAI therapy is allowed if \> 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. * Patients may have received prior external beam radiation therapy to index lesions ≥ 4 weeks prior to initiation of therapy on this protocol if there has been documented progression by RECIST criteria. Prior external beam radiation therapy to the non-index lesions is allowed if ≥ 4 weeks prior to initiation of therapy on this protocol. * ECOG performance status ≤ 2 (or Karnofsky performance status ≥ 60%). * Patients must have normal organ and marrow function as defined below: * Absolute neutrophil count ≥1,500/mcL * Platelets ≥100,000/mcL * Total bilirubin ≤ 1.5 X institutional ULN * AST(SGOT)/ALT(SGPT) ≤ 2.5 X institutional ULN * Creatinine ≤ 1.5 X institutional ULN OR * Creatinine clearance ≥ 60 mL/min/1.73 m2 for patients with creatinine levels above 1.5 X institutional ULN \[in this circumstance, either of a measured level based on a 24 hour urine collection, or a calculated level using the Cockcroft and Gault equation: (140 - age in years) X (weight in kg) X (0.85 if female)/72 X serum Cr may be used\]. * International normalized ratio (INR) ≤ 1.5 (or in range INR, usually between 2 and 3, if patient is on a stable dose of therapeutic warfarin). * \*ULN = upper limit of normal * \*\*unless liver metastasis present in which AST/ALT should be \< 5 x ULN. * Ability to understand and the willingness to sign a written informed consent document. * Age 21 years old or older. Exclusion Criteria: * Patients may not be receiving any other investigational agents. * Patients with known history of active intraparenchymal brain metastasis within previous 3 months. * Serious or non-healing wound, ulcer, or bone fracture. * History of abdominal fistula, gastrointestinal perforation, or intra-abdominal abscess within 28 days of treatment. * Patients with a reported history of clinically active diverticulosis or diverticulitis in the prior 3 years. * Patients with clinically significant cardiovascular disease as defined by the following: * History of CVA within past 6 months * Myocardial infarction, CABG or unstable angina within past 6 months * New York Heart Association grade III or greater congestive heart failure or Canadian Cardiovascular Class grade III or greater angina within past 6 months (Appendices B\&C) Clinically significant peripheral vascular disease within past 6 months * Pulmonary embolism, DVT, or other thromboembolic event within past 6 months * Uncontrolled coronary artery disease, angina, congestive heart failure, or ventricular arrhythmia requiring acute medical management within past 6 months * History of myocardial infarct, cerebrovascular accident, or transient ischemic event within the past 6 month * Uncontrolled intercurrent illness including, but not limited to, ongoing or active infection or psychiatric illness/social situations that would limit compliance with study requirements. * While the use of Angiotensin-Converting Enzyme (ACE) inhibitors is not absolutely excluded, efforts should be made to see if patients on ACE inhibitors can be taken off the medication or switched to another medication. * Pregnant women will be ineligible; breastfeeding should be discontinued if the mother is treated with study drugs. * The use of agents that inhibit or induce CYP3A metabolism is not strictly prohibited, but should be avoided if possible. Potential CYP3A inducing agents include carbamazepine, phenytoin, barbiturates, rifabutin, rifampicin, and St. John's Wort. Potential CYP3A inhibitors include protease inhibitors, antifungals, macrolide antibiotics, nefazodone, and selective serotonin inhibitors.