A Study of Ramucirumab (LY3009806) or Merestinib (LY2801653) in Advanced or Metastatic Biliary Tract Cancer

Inclusion Criteria:

* Have an Eastern Cooperative Oncology Group performance status of 0 or 1.
* Have a histologically or cytologically confirmed diagnosis of non-resectable, recurrent, or metastatic biliary tract adenocarcinoma (intrahepatic or extrahepatic cholangiocarcinoma, gallbladder cancer, or Ampulla of Vater) .
* Have measurable disease by Response Evaluation Criteria in Solid Tumors (RECIST).
* Have adequate biliary drainage.
* Have adequate organ function.
* Males and females are sterile, postmenopausal, or compliant with a highly effective contraceptive method.
* Female participants of childbearing potential must have a negative serum pregnancy test within 7 days prior to first dose.
* Are willing to provide blood/serum/plasma and tumor tissue samples for research purposes. Submission of blood/serum/plasma and tumor tissue samples is mandatory for participation in this study, unless restricted per local regulations.

Exclusion Criteria:

* Previous systemic therapy for locally advanced or metastatic disease is not allowed.
* Have a history of or have current hepatic encephalopathy of any grade, or ascites of Grade \>1, or cirrhosis with Child-Pugh Stage B or higher.
* Have ongoing or recent (≤6 months) hepatorenal syndrome.
* Have had a major surgical procedure or significant traumatic injury including nonhealing wound, peptic ulcer, or bone fracture ≤28 days prior to randomization.
* Anticipate having a major surgical procedure during the course of the study.
* Has documented brain metastases, leptomeningeal disease, or uncontrolled spinal cord compression.
* Within 6 months prior to randomization, have had any arterial thrombotic event, including myocardial infarction, unstable angina, cerebrovascular accident, or transient ischemic attack.
* Have an uncontrolled arterial hypertension with systolic blood pressure ≥150 or diastolic blood pressure ≥90 millimeters of mercury (mm Hg) despite standard medical management.
* Have a previous malignancy within 5 years of study entry or a concurrent malignancy.
* Have a history of gastrointestinal perforation and/or fistulae within 6 months prior to randomization.
* Have a known allergy or hypersensitivity reaction to any of the treatment components.
* Have a history of uncontrolled hereditary or acquired thrombotic disorder.
* Have uncontrolled metabolic disorders or other nonmalignant organ or systemic diseases or secondary effects of cancer that induce a high medical risk and/or make assessment of survival uncertain.
* Have mixed hepatocellular biliary tract cancer histology.
* Have a corrected QT interval \>470 milliseconds as calculated by the Fridericia equation.