Phase II Study of Pembrolizumab and Lenvatinib in Advanced Well-differentiated Neuroendocrine Tumors

Inclusion Criteria: * Diagnosis/Condition for entry into the trial: Metastatic well differentiated neuroendocrine tumors of primary lung, thymic, small bowel and colorectal origin (including unknown primary) * Evidence of radiographic disease progression with scan documenting progression occurring within 8 months of signing informed consent * At least two prior lines of systemic treatment. If the only prior line of treatment was adjuvant or neoadjuvant, patient must have completed treatment within 12 months. There is no limit to number of prior therapies. * Willing and able to provide written informed consent/assent for the trial. * ≥ 18 years of age on day of signing informed consent. * Have measurable disease based on Response Evaluation Criteria in Solid Tumors (RECIST) 1.1. * Have a performance status of 0 or 1 on the Eastern Cooperative Oncology Group (ECOG) Performance Scale. * Demonstrate adequate organ function and laboratory values. All screening labs should be performed within 14 days of treatment initiation. * Females of childbearing potential (FOCBP) should have a negative serum pregnancy within 72 hours prior to receiving the first dose of study medication. * FOCBP must agree to use adequate contraception as outlined in study documentation for the course of the through 120 days after the last dose of study medication. * Male participants of childbearing potential must agree to use an adequate method of contraception as outlined in study documentation, starting with the first dose of study therapy through 120 days after the last dose of study therapy. Exclusion Criteria: * Poorly differentiated neuroendocrine carcinoma * Pancreatic neuroendocrine tumor * Currently participating and receiving study therapy or has participated in a study of an investigational agent and received study therapy or used an investigational device within 4 weeks of the first dose of treatment. * A diagnosis of immunodeficiency or is receiving systemic steroid therapy or any other form of immunosuppressive therapy within 7 days prior to the first dose of trial treatment. * Known history of active TB (Bacillus Tuberculosis) * Hypersensitivity to pembrolizumab or any of its excipients. * Has had a prior anti-cancer monoclonal antibody (mAb) within 4 weeks prior to study Day 1 or who has not recovered (i.e., ≤ Grade 1 or at baseline) from adverse events due to agents administered more than 4 weeks earlier. * Has had prior chemotherapy, targeted small molecule therapy, or radiation therapy within 2 weeks prior to study Day 1 or who has not recovered (i.e., ≤ Grade 1 or at baseline) from adverse events due to a previously administered agent. - Note: Potential participants with ≤ Grade 2 neuropathy are an exception to this criterion and may qualify for the study. Note: If have received major surgery within 3 weeks, must have recovered adequately from the toxicity and/or complications from the intervention prior to starting therapy. Adequate wound healing after major surgery must be assessed clinically, independent of time elapsed for eligibility. * Serious non-healing wound, ulcer or bone fracture * Has pre-existing \>/= Grade 3 gastrointestinal (GI) or non-GI fistula * Has significant cardiovascular impairment within 12 months of the first dose of study drug * Known additional malignancy that is progressing or requires active treatment. Exceptions include basal cell carcinoma of the skin or squamous cell carcinoma of the skin that has undergone potentially curative therapy or in situ cervical cancer. * Known active central nervous system (CNS) metastases and/or carcinomatous meningitis. Potential participants with previously treated brain metastases may participate provided they are stable (without evidence of progression by imaging for at least four weeks prior to the first dose of trial treatment and any neurologic symptoms have returned to baseline), have no evidence of new or enlarging brain metastases, and are not using steroids for at least 7 days prior to trial treatment. This exception does not include carcinomatous meningitis which is excluded regardless of clinical stability. * Has active autoimmune disease that has required systemic treatment in the past 2 years. Replacement therapy is not considered a form of systemic treatment. * History of (non-infectious) pneumonitis that required steroids, or current pneumonitis. * An active infection requiring systemic therapy. * History or current evidence of any condition, therapy, or laboratory abnormality that might confound the results of the trial, interfere with the subject's participation for the full duration of the trial, or is not in the best interest of the subject to participate, in the opinion of the treating investigator. * Known psychiatric or substance abuse disorders that would interfere with cooperation with the requirements of the trial. * Pregnant or breastfeeding, or expecting to conceive or father children within the projected duration of the trial, starting with the pre-screening or screening visit through 120 days after the last dose of trial treatment. * Has received prior therapy with an anti-PD-1, anti-PD-L1, or anti-PD-L2 agent. * Has received prior therapy with a tyrosine kinase inhibitor (TKI) (e.g.; sunitinib, pazopanib, cabozantinib) * Known history of Human Immunodeficiency Virus (HIV) (HIV 1/2 antibodies). * Known active Hepatitis B (e.g., HBsAg reactive) or Hepatitis C (e.g., HCV RNA \[qualitative\] is detected). * Has received a live vaccine within 30 days of planned start of study therapy. Note: Seasonal influenza vaccines for injection are generally inactivated flu vaccines and are allowed; however intranasal influenza vaccines (e.g., Flu-Mist®) are live attenuated vaccines, and are not allowed. * Uncontrolled hypertension defined as systolic blood pressure \>150 mmHg or diastolic pressure \>90 mmHg, despite optimal medical management * Thrombotic or embolic events such as a cerebrovascular accident including transient ischemic, attacks, DVT within the past 6 months * Bleeding or thrombotic disorders or use of anticoagulants, such as warfarin, or similar agents requiring therapeutic international normalized ration (INR) monitoring.(Treatment with low molecular weight heparin (LMWH) is allowed) * Marked baseline prolongation of QT/QTc interval (QTc interval ≥ 480 msec) using the Fridericia method (QTc = QT/RR0.33) for QTc analysis * Clinically significant bleeding within 4 weeks * Medical need for the continued use of potent inhibitors/inducers of CYP3A4 * Creatinine clearance \<30 mL/min * Any condition that impairs patient's ability to swallow whole pills or gastrointestinal malabsorption that, in the investigator's opinion, might affect absorption of lenvatinib