Infusion of Expanded Cord Blood Cells in Addition to Single Cord Blood Transplant in Treating Patients With Acute Leukemia, Chronic Myeloid Leukemia, or Myelodysplastic Syndromes

This phase II trial studies how well donor umbilical cord blood transplant with ex-vivo expanded cord blood progenitor cells (dilanubicel) works in treating patients with blood cancer. Before the transplant, patients will receive chemotherapy (fludarabine, cyclophosphamide and in some cases thiotepa) and radiation therapy. Giving chemotherapy and total-body irradiation before a donor umbilical cord blood transplant helps stop the growth of cells in the bone marrow, including normal blood-forming cells (stem cells) and cancer cells. It may also stop the patient's immune system from rejecting the donor's stem cells. When the healthy stem cells from a donor are infused into the patient they may help the patient's bone marrow make stem cells, red blood cells, white blood cells, and platelets. The donated stem cells may also replace the patient's immune cells and help destroy any remaining cancer cells.

OUTLINE:

Patients receive either regimen A or regimen B.

REGIMEN A: Patients (10 through 45 years old) receive fludarabine intravenously (IV) over 30 minutes on days -8 to -6 and cyclophosphamide IV on days -7 and -6. Patients undergo total body irradiation (TBI) twice daily (BID) on days -4 to -1. Patients receive unmanipulated cord blood unit IV followed by dilanubicel IV within the next 24 hours on day 0.

REGIMEN B: Patients (10 through 65 years old) receive fludarabine IV over 30-60 minutes on days -6 to -3 and IV over 30 minutes on day -2, cyclophosphamide IV on day -6, and thiotepa IV over 2-4 hours on days -5 and -4. Patients undergo TBI once daily (QD) on days -2 and -1. Patients receive unmanipulated cord blood unit IV followed by dilanubicel IV within the next 24 hours on day 0.

All patients undergo bone marrow aspirate and biopsy as clinically indicated during screening and on study. Patients undergo multigated acquisition scan (MUGA) or echocardiography (ECHO), and computed tomography (CT) during screening. Patients also undergo blood sample collection on study.

After completion of study treatment, patients are followed up at 180 days, 1 year, and 2 years.

Eligibility Criteria:
Inclusion Criteria:

* Patients 10 to 65 years old with a hematologic malignancy in need of hematopoietic cell transplant who are \> 30 kg and without a suitable related donor
* Patient must have hematologic malignancy that meets institutional eligibility requirements for cord blood transplant
* Malignancies included are:

  * Acute leukemia, including acute myeloid leukemia (AML), biphenotypic acute leukemia or mixed-lineage leukemia, acute lymphoblastic leukemia (ALL); all patients must be in complete response (CR) as defined by \< 5% blasts by morphology/flow cytometry in a representative bone marrow sample with adequate cellularity to assess remission status
  * 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% in a representative bone marrow aspirate
  * Chronic myeloid 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
* High dose TBI regimen: 10 to =\< 45 years
* Intermediate intensity regimen: 10 to =\< 65 years
* Patients 10 to =\< 45 years: Lansky (\< 16 years old) or Karnofsky (\>= 16 years old) \>= 70 or Eastern Cooperative Oncology Group (ECOG) 0-1
* Patients \> 45 to =\< 65 years: Karnofsky \>= 70 or ECOG 0-1 and non-age adjusted comorbidity index =\< 5
* 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 treating institution
* Carbon monoxide diffusing capability (DLCO) corrected \>= 60% normal (may not be on supplemental oxygen)
* For pediatric patients unable to perform pulmonary function tests, O2 saturation \> 92% on room air
* Left ventricular ejection fraction \>= 50% OR
* Shortening fraction \> 26%
* Ability of participant or legally authorized representative to understand and the willingness to sign a written informed consent form
* DONOR: Minimum requirement: The cord blood (CB) unit must be matched at a minimum at 4/6 HLA-A, B antigens and DRB1 allele with the recipient; therefore, 0-2 mismatches at the A or B or DRB1 loci based on intermediate resolution at HLA-A, B and high resolution allele level typing at HLA- DRB1 are allowed
* DONOR: Institutional guidelines for HLA-match may be followed as long as the minimum criteria for HLA-matching as above are met
* DONOR: The CB unit selected for transplant must have a MINIMUM of 2.5 x 10\^7 TNC/kg
* DONOR: The minimum recommended CD34/kg cell dose is 1.7 x 10\^5 CD34/kg
* DONOR: A backup unit must be identified and reserved prior to the start of the treatment plan for possible infusion in the unlikely event of poor post-thaw viability of the primary CB unit. A suitable back up unit will be considered, as follows:

  * Must be matched at a minimum at 4/6 HLA-A, B, DRBl loci with the recipient. Therefore 0-2 mismatches at the A or B or DRBl loci based on intermediate resolution A, B antigen and DRBl allele typing for determination of HLA-match is allowed (Fred Hutch Protocol 2010).
  * Must contain a MINIMUM of 1.5 x 10\^7 TNC/kg to ensure the same requirement we use for a standard double CBT per CB selection guideline (Fred Hutch Protocol 2010).

Exclusion Criteria:

* Uncontrolled viral or bacterial infection at the time of study enrollment
* Active or recent (prior 6 month) invasive fungal infection unless cleared by infectious disease (ID) consult
* History of human immunodeficiency virus (HIV) infection
* Pregnant or breastfeeding
* Prior allogeneic transplant
* Central nervous system (CNS) leukemic involvement not clearing with intrathecal chemotherapy; diagnostic lumbar puncture is to be performed
* \< 30 kg