Alternating Chemo-immunotherapy and FGFR Inhibitor for FGFR-positive Biliary or Cholangiocarcinoma

Inclusion Criteria: * Adults 18 years of age or older with histologically or cytologically confirmed unresectable locally advanced or metastatic carcinoma of the biliary tract, including intrahepatic and extrahepatic cholangiocarcinoma, gallbladder, and ampulla of Vater, at the time of diagnosis * Patients must have tumors classified as having FGFR2 gene fusion/rearrangements or other gain-of function alterations involved in FGFR as detected by any analytically validated, Clinical Laboratory Improvement Act (CLIA)-certified molecular testing, including commercial tests (Foundation Medicine, Caris, Tempus, Guardant360, and others) or other platforms of next generation sequencing. Gene rearrangements are structural variants that can include inversions, translocations, duplications, and truncations. In addition, all patients will have tumor specimens sent and stored centrally * Patients are permitted to have received one 21-day or 28-day cycle of either gemcitabine/cisplatin or gemcitabine/cisplatin with immunotherapy (either durvalumab or pembrolizumab) prior to inclusion in trial, and this will count as the first cycle in the schematic * One or more measurable lesions per Response Evaluation Criteria in Solid Tumors version 1.1 (RECIST v 1.1) * Eastern Cooperative Oncology Group (ECOG) performance status of 0-1. Patients with ECOG performance status of 2 may be considered on a case-by-case basis by Principal Investigator * Life expectancy of greater than 4 months * Ability to understand and participate voluntarily, sign informed consent, and follow the study treatment plan and scheduled visits * Absolute neutrophil count (ANC) ≥ 1.5 × 109/L (1500/mm3) * Note: Patients must not have required a blood transfusion or growth factor support ≤ 14 days before sample collection * Platelets ≥ 75 × 109/L * Note: Patients must not have required a blood transfusion or growth factor support ≤ 14 days before sample collection * Hemoglobin ≥ 90 g/L (9 g/dL) * Note: Patients must not have required a blood transfusion or growth factor support ≤ 14 days before sample collection * Prothrombin time (PT), international normalized ratio (INR), and activated partial thromboplastin time (APTT) are all ≤ 1.5 × upper limits of normal (ULN), with the exception of patients taking blood thinners with coagulation factor elevation associated with these drugs * Serum bilirubin ≤ 1.5 × ULN (≤ 5 × ULN if the tumor involves the liver), unless associated with patient's primary cancer and/or metastases and with Principal Investigator's approval; biliary drains and stents are acceptable * Aspartate aminotransferase (AST) and alanine aminotransferase (ALT) are ≤ 3 × ULN (if with liver metastases, AST and ALT ≤ 5 × ULN), unless associated with patient's primary cancer and/or metastases and with Principal Investigator's approval; biliary drains and stents are acceptable * Creatinine clearance ≥ 50 mL/min (calculated according to Cockcroft-Gault formula) * Patients with an inherited cancer syndrome or a medical/family history suggestive of an inherited cancer syndrome are eligible * Recovery from adverse events of previous systemic anti-cancer therapies to baseline or grade 1, except for: * alopecia * stable neuropathy of ≤ grade 2 due to prior cancer therapy * Able to swallow and retain oral medication * Patients who are human immunodeficiency virus (HIV) positive may participate IF they meet the following eligibility requirements: * They must be stable on their anti-retroviral regimen with evidence of at least two undetectable viral loads within the past 6 months on the same regimen; the most recent undetectable viral load must be within the past 12 weeks. * They must have a CD4 count of greater than 250 cells/mcL over the past 6 months on this same anti-retroviral regimen and must not have had a CD4 count \< 200 cells/mcL over the past 2 years, unless it was deemed related to the cancer and/or chemotherapy induced bone marrow suppression. * For patients who have received chemotherapy in the previous one month, a CD4 count \< 250 cells/mcL during chemotherapy is permitted as long as viral loads were undetectable during this same chemotherapy. * They must have an undetectable viral load and a CD4 count ≥ 250 cells/mcL within 7 days of enrollment. * They must not be currently receiving prophylactic therapy for an opportunistic infection and must not have had an opportunistic infection within the past 6 months. HIV-infected patients will be monitored every 12 weeks for viral load and CD4 counts * For patients with evidence of chronic hepatitis B virus (HBV) infection, the HBV viral load must be undetectable on suppressive therapy, if indicated. Patients with a past or resolved HBV infection (defined as the presence of hepatitis B core antibody \[anti-HBc\] and absence of hepatitis B surface antigen \[HbsAg\]) are eligible * Patients with a history of hepatitis C virus (HCV) infection must have been treated and cured. For patients with HCV infection who are currently on treatment, they are eligible if they have an undetectable HCV viral load. Patients positive for HCV antibody are eligible only if polymerase chain reaction (PCR) is negative for HCV ribonucleic acid (RNA) Exclusion Criteria: * Received any definitive radiotherapy, targeted therapy, immunotherapy, or any investigational therapies within 13 days of the first study drug administration. Patients are permitted to have received one 21- or 28-day cycle of either gem/cis or gemcitabine/cisplatin/durvalumab prior to inclusion in trial, and this will count as the first cycle in the schematic. Localized palliative radiation therapy (should not include radiation to target lesions) and ongoing bisphosphonates and denosumab, are permitted * Patients who have not recovered from reversible toxicity of prior anti-tumor therapy (except toxicities which are not clinically significant such as alopecia, grade 0-2 neuropathy) * Patients who received prior FGFR-targeted therapy * Within 2 weeks before the first dose of study drug, the subject's calcium and phosphate level continuing to exceed the ULN despite medical treatment * Current evidence of endocrine alterations of calcium/phosphate homeostasis, e.g., parathyroid disorders, history of parathyroidectomy, tumor lysis, tumoral calcinosis, or medical conditions that increase their risk of developing hyperphosphatemia or hypercalcemia * History and/or current evidence of extensive tissue calcification including, but not limited to, the soft tissue, kidneys, intestine, myocardium, and lung, with the exception of calcified lymph nodes, minor pulmonary parenchymal calcifications, and asymptomatic coronary calcification * Patients with clinically significant gastrointestinal dysfunction that may affect drug intake, transport or absorption (e.g., ulcerative diseases, uncontrolled nausea, vomiting, diarrhea, malabsorption syndrome, small bowel resection) * Current evidence of corneal or retinal abnormalities that may increase eye toxicity, including but not limited to: * Currently suffering from central serous retinopathy (CSR) or retinal vein occlusion (RVO), or with relevant history; * Active wet age-related macular degeneration (wAMD); * Diabetic retinopathy with macular edema; * Uncontrollable glaucoma; * Keratopathy, such as keratitis, keratoconjunctivitis, keratopathy, corneal abrasion, inflammation or ulceration * Consumed or anticipate consuming diet or drugs known to be strong or moderate cytochrome P450 (CYP) 3A4 inhibitors, strong CYP3A4 inducers or strong inhibitors of efflux transports including P-gp and BCRP for 28 days (or 5 half-lives, whichever is shorter) before the first dose of study drug or during the study drug treatment * Patients with active or prior documented autoimmune or inflammatory disorders * Participants must not have an active, known or suspected autoimmune disease which may affect vital organ function or has/may require systemic immunosuppressive therapy for management. Participants with inflammatory disorders (including inflammatory bowel disease \[e.g., colitis or Crohn's disease\], diverticulitis \[with the exception of diverticulosis\], systemic lupus erythematosus, Sarcoidosis syndrome, or Wegener syndrome \[granulomatosis with polyangiitis, Graves' disease, rheumatoid arthritis, hypophysitis, uveitis, etc.\]) are also excluded with the exception of the following: participants with type I diabetes mellitus, hypothyroidism (e.g. following Hashimoto syndrome) only requiring and stable on hormone replacement, skin disorders (such as vitiligo, psoriasis, or alopecia) not requiring systemic treatment, or conditions not expected to recur in the absence of an external trigger (such as celiac disease controlled by diet) are permitted to enroll. Patients without active disease for 5 years may also be enrolled after consultation with the study physician * Current or prior use of immunosuppressive medication within 14 days before the first dose of durvalumab. The following are exceptions to this criterion: * Intranasal, inhaled, topical steroids, or local steroid injections (e.g., intra articular injection); * Systemic corticosteroids at physiologic doses not to exceed 10 mg of prednisone or its equivalent. * Steroids as premedication for hypersensitivity reactions (e.g., CT scan premedication) * Receipt of live attenuated vaccine within 30 days of planned start of study therapy. * Note: Patients, if enrolled, should not receive live vaccine while receiving immunotherapy and up to 30 days after the last dose of immunotherapy * Must not have prior history of organ transplantation, allogeneic transplantation, or double umbilical cord transplantation * Active hepatitis B virus (HBV DNA \< ULN is required if HBs-Ag or HBc-Ab is positive), active hepatitis C, or uncontrolled HIV infection * Known allergy or hypersensitivity to any study treatment * Major surgery (thoracotomy, laparotomy, etc.) within 4 weeks or minor surgery (superficial skin surgery, lymphadenectomy, hernia repair, etc.) within 2 weeks before the first dose of study drug * Patients with brain metastases or metastases elsewhere within the central nervous system (CNS) are excluded * Patients with unresolved spinal cord compression * Clinically significant, uncontrolled intercurrent illness including, but not limited to: * Acute symptomatic or active infection requiring systemic therapy and medical intervention (e.g., IV antibiotics and/or hospitalization); * Psychiatric illness and/or social situations that would limit compliance with completion of study requirements or ability to give informed consent * Has a history of uncontrolled cardiovascular diseases including: * Known New York Heart Association (NYHA) grade II or higher congestive heart failure, unstable angina pectoris, or myocardial infarction within 6 months before the first dose of study drug; * Arrhythmias requiring treatment at screening; * Left ventricular ejection fraction (LVEF) \< 50% at screening; * Clinically significant prolonged QTc interval, or QTc interval \> 470 ms in women and \> 450 ms in men at screening; * Cerebrovascular accident within 6 months before the first dose of study drug * History of active bleeding within 6 months, signs of portal hypertension leading to gastric esophageal venous bleeding within 2 months before the first dose of study drug, or the investigator believes that there is uncontrolled bleeding (such as bleeding esophageal varices, local active ulcer lesions, etc.) * At the investigator's discretion, with evidence of severe or uncontrolled systemic disease (such as unstable or non-compensatory lung, liver, or kidney disease); or any unstable systemic disease (including active clinically severe infections, uncontrolled hypertension, or liver, kidney, or metabolic disease) * Pregnant or lactating women, as well as women with childbearing potential who are unwilling or unable to perform contraception from screening to 6 months after the last study drug administration; fertile men who are unwilling or unable to perform contraception from screening to at least 6 months after the last study drug administration * History of another primary malignancy except adequately treated in situ carcinoma of the cervix or non-melanoma carcinoma of the skin or any other curatively treated malignancy that is not expected to require treatment for recurrence during the course of the study or affect survival * Any other medical condition that would, in the investigator's judgment, prevent the patient's participation in the clinical study due to safety concerns or compliance with clinical study procedures * History of hypovitaminosis D requiring supraphysiologic doses (e.g., 50,000 IU/weekly) to replenish the deficiency. Vitamin D supplements are allowed