• To evaluate the transplantation of two unmanipulated cord blood units versus one unmanipulated and one expanded cord blood unit in patients with hematologic malignancies following high-dose or nonmyeloablative therapy. [ Time Frame: 10 Years ] [ Designated as safety issue: No ]
  

• To evaluate the rate, extent, and durability of hematopoietic reconstitution. [ Time Frame: 10 Years ] [ Designated as safety issue: No ]
  

Inclusion Criteria:

1. Disease-Specific Eligibility Requirements: Patients must have one of the following hematologic malignancies: 1. Acute Myelogenous Leukemia (AML), Myelodysplastic Syndrome (MDS) 2. Acute Lymphoblastic Leukemia (ALL) 3. Chronic Myelogenous Leukemia (CML) 4. Non-Hodgkin's Lymphoma (NHL) 5. Hodgkin's Disease (HD) 6. Chronic Lymphocytic Leukemia (CLL).
2. Greater than 1 month old and <=60 years old for full myeloablative therapy.
3. Patients must have two CB units available which are matched with the patient at 4, 5, or 6/6 HLA class I (serological) and II (molecular) antigens. Each cord must contain at least 1E7 total nucleated cells/Kg recipient body weight in the pre-thawed fraction.
4. Patient must be willing to undergo bone marrow harvest or PBPC collection for use in case of engraftment failure. If the patient is unable or fails to successfully undergo the collection, a family member must be identified to donate hematopoietic stem cells for haploidentical transplant in case of engraftment failure. If autologous hematopoietic stem cells cannot be procured due to marrow contamination by malignancy, or due to harvest failure, and a haploidentical relative is not available or not willing to donate, two cord blood units can be used as the back-up graft.
5. Continuation to Criteria # 4: These units will be identified prior to enrollment in this study.
6. Regimen 1 (Myeloablative mel/thiotepa/fludarabine): 1.Patients with ALL, HD, NHL, AML, MDS, CML, CLL who are candidates for full myeloablative therapy. 2.Performance score of at least 60% by Karnofsky (age >= 12 years), or Lansky Play-Performance Scale (age <12 years). 3.Age >=1 month <=60 years (high-dose). 4.Adequate major organ system function as demonstrated by:a. Left ventricular ejection function of at least 40%. b.Pulmonary function test demonstrating a diffusion capacity of at least 50%. predicted (high-dose). c.Creatinine < 1.6 mg/dL. d.SGPT/bilirubin <= to 2.0 x normal (high-dose).
7. Eligibility for Regimen 2 (Non-myeloablative Cy-Flu-TBI): 1. Patients with ALL, AML, MDS, CML, NHL, CLL, HD who are not candidates for full myeloablative therapy. All patients who received a prior autologous transplant are eligible. 2. Performance score of at least 60% by Karnofsky or PS < 3 (ECOG) (age >= 12 years), or Lansky Play-Performance Scale (age <12 years) 3. Age >= 1 month <=80 years 4. Left ventricular ejection function of at least 30%; 5. Pulmonary function test demonstrating a diffusion capacity of at least 40% predicted; 6. Creatinine < 3.0 mg/dL; 7. SGPT <= to 4.0 x normal.
8. Regimen 3 (Myeloablative VP16-TBI): 1. Patients with ALL who are candidates for myeloablative therapy, and require a TBI-containing regimen. 2. Performance score of at least 60% by Karnofsky or PS < 2 (ECOG) (age >= 12 years), or Lansky Play-Performance Scale (age <12 years). 3. Age >= 1 month <=50 years. 4. Organ function requirements: a. Left ventricular ejection function of at least 50%. b. Pulmonary function test demonstrating a diffusion capacity of at least 50% predicted. c. Creatinine < 1.6 mg/dL. d. SGPT <= 2.0 x normal.

Exclusion Criteria:

1. HIV positive.
2. Pregnancy.
3. Serious medical Condition.
4. Patients with signs & symptoms leading to positive lumbar puncture (malignant cells in the CSF) or to documented metastatic parenchymal disease are ineligible for this study.
5. Availability of appropriate, willing, HLA-matched related donor.