Inclusion Criteria:
* 1\. Patients with age ≧20 years old.
* 2.Histologically confirmed biliary tract cancer which is locally advanced, recurrent or metastatic disease. The disease entities include intrahepatic cholangiocarcinoma, perihilar cholangiocarcinoma, distal bile duct cholangiocarcinoma, Ampulla of Vater cancer, and gallbladder cancer.
3.Documented disease progression during or within 6 months after gemcitabine-based (regimens containing gemcitabine plus cisplatin, gemcitabine plus S-1, or gemcitabine plus oxaliplatin) chemotherapy. Patient who has received antiangiogenetic agent (bevacizumab, ramucirumab, lenvatinib), taxane-based chemotherapy, or more than 1 line of chemotherapy for locally advanced or metastatic biliary tract cancer is ineligible.
4\. Documented measurable disease as defined by RECIST v1.1. 5. Baseline Eastern Cooperative Oncology Group performance status score 0-1. 6. Patient has life expectancy of at least 12 weeks. 7. Adequate hematologic parameters, and hepatic and renal functions defined as 7.1: Hepatic: alanine aminotransferase (ALT) and aspartate aminotransferase (AST) 2.5 x upper limit of normal (ULN) ( 5.0 x ULN if attributable to liver metastases), total bilirubin 3 mg/dL.
7.2: Renal: serum creatinine level 1.5 x ULN or creatinine clearance \> 30 ml/min \[calculated by either Cockcroft-Gault equation \[(140-age) x body weight (kg) x (1 if male or 0.85 if female) / (72 x serum creatinine level, mg/dl)\] or 24-hour urine test\].
7.3: Hematological: white blood cell 3,000/ul, absolute neutrophil count (ANC) 1,500/ul, hemoglobin 9 g/dl and platelet count 90,000/ul.
8\. Adequate controlled blood pressure (BP), defined as BP≦140/90 mmHg at screening and no change in antihypertensive medication within 1 week prior to the cycle1/day 1.
9\. Adequate blood coagulation function, defined as prothrombin time international normalized ratio (PT INR)≦ 2.3.
10\. Normal ECG or ECG without any clinical significant findings 11. Able to understand and sign an informed consent (or have a legal representative who is able to do so).
12\. Women or men of reproductive potential should agree to use an effective contraceptive method
Exclusion Criteria:
* 1\. Patients who have major abdominal surgery, radiotherapy or other investigating agents within 2 weeks are not eligible. Patients who have palliative radiotherapy will be eligible if the irradiated area does not involve the only lesion of measurable / evaluable disease.
2\. Patients having liver dysfunction with Child-Pugh score ≧7. 3. Patients with gastrointestinal malabsorption or condition that might affect the absorption of lenvatinib in the opinion of the investigator.
4\. Bleeding or thrombotic disorders or subjects at risk for severe hemorrhage. Patients with tumor invasion/infiltration of major blood vessels should be excluded because of the potential risk of severe hemorrhage associated with tumor shrinkage/necrosis following therapy.
5\. Uncontrolled blood pressure (systolic BP\>140 mmHg or diastolic BP\>90 mmHg) in spite of an optimized regimen of antihypertensive medication.
6\. Significant cardiovascular impairment: history of congestive heart failure greater than New York Heart Association (NYHA) Class II, unstable angina, myocardial infarction or stroke within 6 months, or cardiac arrhythmia requiring medical treatment.
7\. Patients having \> 1+ proteinuria on urine dipstick testing will undergo 24 h urine collection for quantitative assessment of proteinuria. Subjects with urine protein≧ 1 g/24 h will be ineligible.
8\. Patients with electrolyte abnormalities that have not been corrected. 9. Patients with metastatic lesion in central nervous system. 10. Patients with active infection. 11. Subjects who have not recovered adequately from any toxicity from other anti- cancer treatment regimens and/or complications from major surgery prior to starting therapy.
12\. Patients who have peripheral neuropathy \> grade I of any etiology 13. Patients who have serious concomitant systemic disorders incompatible with the study, i.e. poorly controlled diabetes mellitus, auto-immune disorders, or other condition that in the opinion of the investigator would preclude the subject's participation in the study.
14\. Patients who have other prior or concurrent malignancy except for adequately treated in situ carcinoma or basal cell carcinoma of skin, or any malignancy which remains disease-free for 3 or more years after curative treatment.
15\. Females who are breastfeeding or pregnant at screening or baseline. 16. Patients with psychiatric illness which would preclude study compliance.
