Trial Evaluating MGTA-456 in Patients With High-Risk Malignancy

Age, Unit Cell Dose and HLA Match Criteria * Subjects must be ≤55 years of age * Subjects must weigh \>11 kg * Subjects must have a partially HLA matched UCB unit with a pre-cryopreserved TNC dose \>1.0 x 107 per kilogram recipient weight. HLA matching is initially based on a minimum of 5 of 8 HLA alleles at high resolution A, B, C, DRB1 typing; searches will be performed according to the current Magenta Cord Blood Search Algorithm. Eligible Diseases: * Acute myelogenous leukemia (AML) in morphological complete remission with: * Minimal residual disease (MRD) by flow cytometry, or * Intermediate to high risk leukemia in first (CR1) based on institutional criteria, eg. not favorable risk AML which is defined as having one of the following: * t(8,21) without cKIT mutation * inv(16) or t(16;16) without cKIT mutation * Normal karyotype with mutated NPM1 but FLT3-ITD wild type * Normal karyotype with double mutated CEBPA * Acute promyelocytic leukemia (APL) in first molecular remission at the end of consolidation * Any second or subsequent CR, or * Secondary AML with prior malignancy that has been in remission for at least 12 months. * Acute lymphocytic leukemia (ALL) at the following stages: * High risk first morphological, cytogenetic and molecular CR with: * MRD by flow cytometry, or * Diagnosis of Philadelphia chromosome (Ph)+ ALL, or * MLL rearrangement at diagnosis with slow early response at Day 14, or * Hypodiploidy (\< 44 chromosomes or DNA index \< 0.81) at diagnosis, or * End of induction M3 bone marrow, or * End of induction M2 with M2-3 at Day 42. * High risk second CR based on institutional criteria (eg, for children, bone marrow relapse \<36 months from induction or T-lineage bone marrow relapse or very early isolated central nervous system (CNS) relapse \<6 months from diagnosis, or slow re-induction (stage M2-3 at day 28 after induction) regardless of length remission. All patients with MRD by flow cytometry. * Any third or subsequent CR. * Secondary ALL * Biphenotypic/undifferentiated leukemia in morphological, cytogenetic and molecular CR . * Chronic Myelogenous Leukemia (CML) in high risk first chronic phase (failure of two tyrosine kinase inhibitors (TKI) or TKI intolerance), accelerated phase or second chronic phase. * Myelodysplasia (MDS) IPSS Int-2 or High risk (i.e. RAEB, RAEBt \<5% blasts) or other high risk features, including multiple cytopenias, high risk cytogenetics or lack of response to standard therapy.. * Relapsed large-cell lymphoma, mantle-cell lymphoma and Hodgkin lymphoma that is chemotherapy sensitive and ineligible for an autologous transplant. * Burkitt's lymphoma in CR2 or subsequent CR. * Relapsed T-cell lymphoma that is chemotherapy sensitive in CR/PR that is ineligible for an autologous transplant. Organ Specific Inclusion Criteria * Karnofsky score ≥70 (16 years and older), Lansky play score \>50 (children 2-16 years, or 'adequate' score for children \<2 years, as detailed in Appendix II. * Adequate organ function defined as: * Renal: Serum creatinine within normal range for age, or if serum creatinine outside normal range for age, then creatinine clearance \>40 ml/min or GFR ≥70 mL/min/1.73 m2.normal for age * Hepatic: Bilirubin \<3x upper limit of normal (ULN) and AST, ALT and alkaline phosphatase \<5x ULN. * Pulmonary function: DLCO, FEV1, FEC (diffusion capacity) \>5030% of predicted (corrected for hemoglobin); if unable to perform pulmonary function tests, then O2 saturation \>95% on room air. * Cardiac: No uncontrolled arrhythmia and left ventricular ejection fraction at rest must be \>3545%. * Available 'back-up' HSPC graft (e.g, second UCB unit, haploidentical related donor). * Females of child bearing potential and sexually active males must agree to use adequate birth control during study treatment. * Voluntary written consent signed (adult or parental) before performance of any study-related procedure not part of normal medical care. Exclusion Criteria * Patients with a HLA matched sibling donor or a HLA matched unrelated donor who is available for marrow or peripheral blood stem cell collection at the desired time of transplant. * Pregnant or breast feeding. The agents used in this study may be teratogenic to a fetus and there is no information on the excretion of agents into breast milk. Females of childbearing potential must have a blood test or urine study within 14 days prior to study enrollment to rule out pregnancy. * Evidence of human immunodeficiency virus (HIV) infection or known HIV positive serology. * Active bacterial, viral or fungal infection (currently taking medication and persistence of clinical signs and symptoms) with a minimum of 4 weeks of anti-fungal treatment * Prior autologous or allogeneic transplant. * Other active malignancy. * Subjects \>2 3 years of age unable to receive TBI 1320 cGy due to extensive prior therapy including \>12 months alkylator therapy or \>6 months alkylator therapy with extensive radiation, or prior Y-90 ibritumomab (Zevalin) or I-131 tostumomab (Bexxar), as part of their salvage therapy.