Cord Blood Transplantation in Children and Young Adults With Blood Cancer

Inclusion Criteria: A patient cannot be considered eligible for this study unless ALL of the following conditions are met. ° Disease type Cohort 1, High Risk Disease: Patients with age ≤ 26 years at the time of informed consent with no available and suitably matched related or unrelated donor within 4 weeks, with one of the following diagnoses: I. Acute myelogenous leukemia (AML): * Complete first remission (CR1) with blast count \< 5% by bone marrow morphology at high risk for relapse such as any of the following: * Known prior diagnosis of myelodysplasia (MDS) * High risk cytogenetics (e.g., those associated with MDS, abnormalities of 5, 7, 8, complex karyotype) and/or high-risk molecular abnormalities (e.g., TP53) * Requirement for 2 or more inductions to achieve CR1 * Therapy-related AML (t-AML) or therapy-related myeloid neoplasm (t-MN) (including after therapy for other malignancy, and/or gene therapy or cell therapy) * Presence of Minimal/Measurable Residual Disease (MRD+) by cytogenetics, flow cytometry or molecular methods (at End of Induction or End of Consolidation) * Other high-risk features not defined above. * Complete second remission (CR2) or subsequent remission, with blast count \< 5% by bone marrow morphology * Presence of MRD by multiparameter flow cytometry at pre-transplant evaluation is acceptable. II. Acute lymphoblastic leukemia (ALL): * Complete first remission (CR1) with MRD negative status by multicolor flow cytometry, at high risk for relapse such as any of the following: * Presence of any high risk cytogenetic abnormalities such as t(9;22), t(1;19), t(4;11) or other, KMT2A (11q23) or other high risk molecular abnormality * Failure to achieve complete remission (CR) after four weeks of induction therapy (transplant to follow antibody therapy and/or CAR T cells) * Persistence or recurrence of MRD on therapy (Transplant to follow antibody therapy and/or CAR T cells) * T-ALL in CR even with presence of MRD * Other high-risk features not defined above * Complete second remission (CR2) or subsequent remission with MRD negative status by multiparameter flow cytometry. * Relapse in less than 36 months from CR1 * Relapse for T-ALL * Patients after antibody therapy (e.g., blinatumomab, inotuzumab, other) and/or CAR-T cell therapy that resulted in MRD negative status by multiparameter flow cytometry. III. Other acute leukemias: * Leukemias of ambiguous lineage or of other types with \< 5% blasts by bone marrow morphology. * Patients with persistent/relapsed disease with cytogenetic, flow cytometric or molecular aberrations in \< 5% of cells. * Chronic myelogenous leukemia: Patients with history of blast crisis or accelerated phase. * Any leukemia that developed after gene therapy or cell therapy IV. Myelodysplastic Syndrome (MDS): * Any IPSS risk category with life-threatening cytopenia(s). * Any IPSS risk category with high risk cytogenetic/molecular findings (5, 7, 8, complex karyotype, or TP53) V. Non-Hodgkin lymphoma (NHL) or Hodgkin lymphoma (HL) at high risk of relapse or progression if not in remission: * Patients with aggressive histology (such as, but not limited to, diffuse large B-cell NHL, mantle cell NHL, and T-cell NHL) in CR. * Patients with indolent B cell NHL (such as, but not limited to, follicular, small cell or marginal zone NHL) will have 2nd or subsequent progression with stable disease/ CR/ PR with no single lesion equal to or more than 5 cm. * Patients with HL without progression of disease (POD) after salvage chemotherapy with no single lesion ≥5 cm. Cohort 2: Very High-Risk disease: 1. Patients in CR (bone marrow blasts \<5% by morphology) who had prior allogeneic transplant and disease recurrence. The second transplant will take place at least 4 months after the first. * Acute myelogenous leukemia (AML) or Myelodysplastic Syndrome (MDS): Relapse after previous transplant, in CR after induction therapy. MRD positive status by multi-parameter flow cytometry is accepted. * Acute lymphoblastic leukemia (ALL): Relapse after previous transplant, in CR after induction therapy and/or antibody therapy/CAR T cells. MRD positive status after targeted therapy, as evaluated by multi-parameter flow cytometry is accepted. * Other: patients with leukemia or lymphoma, who, in the opinion of their physician, are not likely to have reduction in disease burden with further chemotherapy. 2. Patients with relapsed/refractory disease at either first or second allogeneic transplant, with up to 30% bone marrow blasts by multiparameter flow cytometry or morphology. ° Relapse after previous transplant with \< 30% blasts by bone marrow morphology, or with cytogenetic, flow cytometric, or molecular abnormalities in \< 30% of bone marrow cells, after induction therapy. ° Primary refractory or relapsed AML with \< 30% blasts by bone marrow morphology or with cytogenetic, flow cytometric, or molecular abnormalities in \< 30% of bone marrow cells. ° Age 0-26 years at the time of informed consent ° Performance: Karnofsky (≥16 years) or Lansky score (\<16 years) of ≥70% (see Appendix A). ° Not Pregnant and Not Nursing ° Required Organ Function * Bilirubin ≤ 1.5 mg/dL (unless benign congenital hyperbilirubinemia). * ALT ≤ 3 x upper limit of normal. * Pulmonary function (FVC, FEV1 and DLCO corrected for hemoglobin) ≥ 50% predicted. * In young children unable to perform pulmonary function testing: pulse oximetry \>92% in room air, and a normal CT of the chest (if CT is not normal, the child needs to be evaluated and cleared by pediatric pulmonary physician). * Left ventricular ejection fraction \> 50%. * Age-adjusted Hematopoietic Cell Transplantation-Comorbidity Index (aaHCT-CI) ≤ 7. * Female patients of childbearing potential must have a negative serum pregnancy test within 7 days of enrolment and must be willing to use an effective contraceptive method while enrolled in the study.