In this trial, the investigators will begin to explore the possibility that, as in mice, JAK1/2 inhibition with hematopoietic cell transplantation (HCT) may mitigate graft-versus-host-disease (GVHD) while retaining engraftment and Graft-versus-Leukemia (GVL). Both preclinical and clinical data suggest that inhibition of IFNy and IL-6, directly and using downstream JAK Inhibitors, may be an effective strategy to decrease toxicities and improve disease control for patients undergoing Allogeneic HSCT. Baricitinib, as a JAK1/2 inhibitor, has shown superiority to other JAK inhibitors in preclinical GVHD models. The purpose of this phase I clinical trial is to determine the safety of baricitinib with HSCT measured by the effect on engraftment and grade III-IV acute graft-versus-host-disease (aGVHD).

Inclusion Criteria:

Patients must meet the following criteria within 30 days prior to Day 0 unless otherwise noted.

• Diagnosis of a hematological malignancy listed below:

  • Acute myelogenous leukemia (AML) in complete morphological remission (based on IWG Criteria).
  • Acute lymphocytic leukemia (ALL) in complete morphological remission (MRD negative, based on IWG Criteria).
  • Myelodysplastic syndrome with less than 10% blasts in bone marrow.
  • Non-Hodgkin's lymphoma (NHL) or Hodgkin's disease (HD) in 2nd or greater complete or partial remission.

• Planned treatment is myeloablative or reduced intensity conditioning followed by peripheral blood HLA matched donor transplantation

• Available HLA-identical donor who meets the following criteria:

  • At least 18 years of age.
  • HLA-identical donor/recipient match by high-resolution typing per institutional standards.
  • In the investigator's opinion, is in general good health, and medically able to tolerate leukapheresis required for harvesting HSC.
  • No active hepatitis.
  • Negative for HTLV and HIV.
  • Not pregnant.
  • Donor selection will be in compliance with institutional standards
  • Safety Lead-In Phase: For the first three patients at each dose level, related donors must consent to a second product collection should it prove necessary.

• Eastern Cooperative Oncology Group (ECOG) performance status ≤ 2

• Adequate organ function as defined below:

  • Total bilirubin must be within normal range at baseline.
  • AST (SGOT) and ALT (SGPT) ≤ 3.0 x IULN.
  • Estimated creatinine clearance ≥ 60 mL/min/1.73 m2 by Cockcroft-Gault Formula.
  • Oxygen saturation ≥ 90% on room air.
  • LVEF ≥ 40%.
  • FEV1 and FVC ≥ 40% predicted, DLCOc ≥ 40% predicted. If DLCO is < 40%, patients will still be considered eligible if deemed safe after a pulmonary evaluation.

• At least 18 years of age at the time of study registration

• Able to understand and willing to sign an IRB approved written informed consent document (or that of legally authorized representative, if applicable).

• Must be able to receive GVHD prophylaxis with tacrolimus, mini-methotrexate with or without ATG or post transplant Cy with MMF and tacrolimus as outlined in the protocol

Exclusion Criteria:

• Must not have undergone a prior allogeneic donor (related, unrelated, or cord) transplant. Prior autologous transplant is not exclusionary.
• Known HIV or active hepatitis B or C infection.
• Known latent tuberculosis infection, or at high risk for latent TB infection, or a positive t-spot tuberculosis test
• Known hypersensitivity to one or more of the study agents, including baricitinib.
• Must not have myelofibrosis or other disease known to prolong neutrophil engraftment to > 35 days after transplant.
• Currently receiving or has received any investigational drugs within the 14 days prior to the first dose of study drug (Day -3).
• Pregnant and/or breastfeeding.
• Uncontrolled intercurrent illness including, but not limited to, ongoing or active infection, autoimmune disease, symptomatic congestive heart failure, unstable angina pectoris, unstable cardiac arrhythmias, or psychiatric illness/social situations that would limit compliance with study requirements.
• Immunosuppressive doses of steroids. Subjects with steroids for adrenal insufficiency will not be excluded.
• History of unprovoked thrombosis or known thrombophilia. Provoked and/or superficial DVTs are eligible provided they are treated and resolved at the time of screening.
• Recent (less than 1 year from screening) myocardial infarction or embolic stroke