Donor Umbilical Cord Blood Transplant With or Without Ex-vivo Expanded Cord Blood Progenitor Cells in Treating Patients With Acute Myeloid Leukemia, Acute Lymphoblastic Leukemia, Chronic Myelogenous Leukemia, or Myelodysplastic Syndromes

This randomized phase II trial studies how well donor umbilical cord blood transplant with or without ex-vivo expanded cord blood progenitor cells works in treating patients with acute myeloid leukemia, acute lymphoblastic leukemia, chronic myelogenous leukemia, or myelodysplastic syndromes. Giving chemotherapy and total-body irradiation before a donor umbilical cord blood transplant helps stop the growth of cancer cells. It may also stop the patient's immune system from rejecting the donor's cells. When the healthy stem cells and ex-vivo expanded cord blood progenitor cells are infused into the patient they may help the patient's bone marrow make stem cells, red blood cells, white blood cells, and platelets. It is not yet known whether giving donor umbilical cord blood transplant plus ex-vivo expanded cord blood progenitor cells is more effective than giving a donor umbilical cord blood transplant alone.

PRIMARY OBJECTIVES:

I. Compare the time to neutrophil engraftment (absolute neutrophil count \[ANC\] \>= 500) in patients receiving a standard of care myeloablative cord blood transplant (CBT) augmented with an off-the-shelf pre-expanded and cryopreserved cord blood product to those who do not receive the product.

SECONDARY OBJECTIVES:

I. Provide initial data on clinical and economic benefit, such as time to platelet engraftment, duration of initial hospitalization, transplant-related mortality (TRM), death without engraftment, and incidence of severe infections in the first 100 days post-transplant.

II. The kinetics of immune system recovery will also be evaluated in both arms.

OUTLINE: Patients are randomized to 1 of 2 treatment arms.

Standard of Care Arm:

CONDITIONING REGIMEN: One of two possible conditioning regimens is chosen by the attending physician: Patients receive fludarabine phosphate intravenously (IV) over 30 minutes on days -8 to -6, cyclophosphamide IV on days -7 to -6., and undergo high dose total-body irradiation (TBI) twice daily (BID) on days -4 to -1 OR, patients receive fludarabine phosphate IV over 30-60 minutes on days -6 to -2, cyclophosphamide IV on day -6, thiotepa IV over 4 hours on days -5 to -4, and undergo middle intensity TBI once daily (QD) on days -2 to -1.

TRANSPLANT: Patients undergo single-unit or double-unit unmanipulated umbilical cord blood (UCB) transplant on day 0.

GRAFT-VERSUS-HOST DISEASE (GVHD) PROPHYLAXIS: Patients receive cyclosporine IV over 1 hour BID (adults) or thrice daily (TID) (children) or orally (PO) on days -3 to 100 with taper beginning on day 101. Patients also receive mycophenolate mofetil (MMF) IV TID on days 0-7 then may receive MMF PO TID. Patients remain on MMF TID for a minimum of 30 days, and then may begin taper if there is no evidence of GVHD and are well-engrafted from one donor unit.

Experimental Arm:

CONDITIONING REGIMEN: Patients receive the conditioning regimen chosen by the attending physician as in Standard of Care Arm.

TRANSPLANT: Patients undergo single-unit or double-unit unmanipulated UCB transplant on day 0. Patients also receive an infusion of ex vivo-expanded cord blood progenitors at least 4 hours after completion of UCB transplant.

GVHD PROPHYLAXIS: Patients receive cyclosporine IV or PO and mycophenolate mofetil IV or PO as in Standard of Care Arm.

After completion of study treatment, patients are followed up periodically for 2 years.

Eligibility Criteria:
Inclusion Criteria:

* Age criteria:

  * High dose TBI regimen: 6 months to =\< 45 years
  * Middle intensity TBI regimen: 6 months to =\< 65 years
  * Conditioning regimen selection should be based on the underlying disease, presence of minimal residual disease (MRD), age, co-morbidities, attending physician, and site preference; conditioning regimen will not require stratification of the randomization due to heterogeneity in the cohort of eligible patients.
* Acute myeloid leukemia, including biphenotypic acute leukemia or mixed-lineage leukemia

  * All patients must have acute myeloid leukemia (AML) that is considered best treated by stem cell transplant by the referring physician and the attending transplant physician
  * All patients must be in complete remission (CR) as defined by \< 5% blasts by morphology/flow cytometry in a representative bone marrow sample with cellularity \>= 15% for age
  * Patients in which adequate marrow/biopsy specimens cannot be obtained to determine remission status by morphologic assessment, but have fulfilled criteria of remission by flow cytometry, recovery of peripheral blood counts with no circulating blasts, and/or normal cytogenetics (if applicable) may still be eligible; reasonable attempts must be made to obtain an adequate specimen for morphologic assessment, including possible repeat procedures; these patients must be discussed with the principal investigator prior to enrollment
* Acute lymphoblastic leukemia, including biphenotypic acute leukemia or mixed-lineage leukemia

  * High risk first complete remission (CR1) (for example, but not limited to: t(9;22), t(1;19), t(4;11) or other mixed-lineage leukemia \[MLL\] rearrangements, hypodiploid); or high risk (HR) as defined by referring institution treatment protocol greater than 1 cycle to obtain CR; second complete remission (CR2) or greater
  * All patients must be in CR as defined by \< 5% blasts by morphology/flow cytometry in a representative bone marrow sample with cellularity \>= 15% for age
  * Patients in which adequate marrow/biopsy specimens cannot be obtained to determine remission status by morphologic assessment, but have fulfilled criteria of remission by flow cytometry, recovery of peripheral blood counts with no circulating blasts, and/or normal cytogenetics (if applicable) may still be eligible; reasonable attempts must be made to obtain an adequate specimen for morphologic assessment, including possible repeat procedures; these patients must be discussed with the principal investigator prior to enrollment
* Chronic myelogenous leukemia excluding refractory blast crisis; to be eligible in first chronic phase (CP1) patient must have failed or be intolerant to tyrosine kinase inhibitor therapy
* Myelodysplasia (MDS) International Prognostic Scoring System (IPSS) intermediate (Int)-2 or high risk (i.e., refractory anemia with excess blasts \[RAEB\], refractory anemia with excess blasts in transformation \[RAEBt\]) or refractory anemia with severe pancytopenia or high risk cytogenetics; blasts must be \< 10% by a representative bone marrow aspirate morphology
* Karnofsky (\>= 16 years old) \>= 70 or Eastern Cooperative Oncology Group (ECOG) 0-1
* Lansky (\< 16 years old) \>= 60
* Adults: calculated creatinine clearance must be \> 60 mL and serum creatinine =\< 2 mg/dL
* Children (\< 18 years old): calculated creatinine clearance must be \> 60 mL/min
* Total serum bilirubin must be \< 3 mg/dL unless the elevation is thought to be due to Gilbert's disease or hemolysis
* Transaminases must be \< 3 x the upper limit of normal per reference values of referring institution
* Diffusing capacity of the lung for carbon monoxide (DLCO) corrected \> 60% normal
* For pediatric patients unable to perform pulmonary function tests, oxygen (O2) saturation \> 92% on room air
* May not be on supplemental oxygen
* Left ventricular ejection fraction \> 45% OR
* Shortening fraction \> 26%
* Ability to understand and the willingness to sign a written informed consent document

Exclusion Criteria:

* Uncontrolled viral or bacterial infection at the time of study enrollment
* Active or recent (prior 6 month) invasive fungal infection without infectious disease (ID) consult and approval
* History of human immunodeficiency virus (HIV) infection
* Pregnant or breastfeeding
* Prior myeloablative transplant containing full dose TBI (greater than 8 Gy)
* Central nervous system (CNS) leukemic involvement not clearing with intrathecal chemotherapy and/or cranial radiation prior to initiation of conditioning; diagnostic lumbar puncture is to be performed per protocol
* Patients \>= 45 years: comorbidity score of 5 or higher