The goal of this clinical research study is to learn if combining cord blood units will be safe and result in the cells "taking" faster in recipients. The cord blood units will have their cell number increased in the lab using cells from a family member or they will be collected from an unrelated healthy donor.

Umbilical cord blood is a source of blood-forming cells that can be used for transplantation. The major problem with this type of transplant is the small number of blood-forming cells available in each cord unit, which may delay the "take" of the graft in the recipient. A strategy to overcome this problem is to give 2 cord blood units, increasing the number of cord blood cells in one of them in the laboratory before they are transplanted. This is done in order to increase their number. The expansion of 1, but not both, of the cord blood units will allow the research team to decide more effectively whether the laboratory expansion is responsible for the speed and content of the "take." At this time, no proof exists that this expansion technique will improve the performance of the cord blood specimens.

In order to collect a larger expansion of the numbers of cord blood cells, recent research suggests that growing the cord blood cells on a layer of bone marrow stromal cells increases the number of expanded cells which can be collected. In the body, these stromal cells form a matrix or "spider web" in the bone marrow. Blood-forming bone marrow cells (looking like dewdrops) grow on this stromal cell matrix (spider web) and are nurtured by the stromal cells. In this research study, researchers will either collect marrow stromal cells from a family member through a bone marrow aspiration, or they will use "off-the-shelf" marrow stromal cells that have been collected from a healthy donor. These "off-the-shelf" marrow stromal cells were grown and frozen by Mesoblast Systems. These stromal cells from either your family member or "off-the-shelf" will hopefully improve the performance of the cord blood cells after they are given to you.

Placement of central venous catheter for collection of "back-up" peripheral blood progenitor cells (PBPC):

Before you have back-up PBPC collected or you receive chemotherapy, you will require placement of a hollow plastic tube (catheter) into a large vein inside your body. This catheter will be used to draw blood and to give medications and fluids. The catheter is inserted through the skin in the upper chest and extends to a point above the right side of your heart. Your doctor will explain this procedure to you in more detail, and you will be required to sign a separate consent form for this procedure.

Collection of back-up stem cells:

Because collecting additional cells from the donor of the cord blood is not possible if the transplant with cord blood fails, a back-up PBPC or bone marrow sample to ensure recovery of your marrow function will be collected from you and frozen before the high dose chemotherapy begins. This specimen will only be used if doctors think it is necessary.

Peripheral blood progenitor cell collection (Leukapheresis):

Before collection of the PBPC, you will be treated with a drug called granulocyte colony stimulating factor (G-CSF), which will cause the important stem cells in the marrow to move the peripheral blood where they will be collected. This medication is given as a shot under the skin once a day for 3-7 days, at which time your PBPC will be collected from your central catheter during a 3-4 hour outpatient procedure. This procedure is standard, and is called leukapheresis. You will be required to sign a separate consent form for this procedure.

Bone Marrow Collection:

If the leukapheresis cannot be performed successfully, you will receive general anesthesia in the operating room and will have multiple needle sticks of the hip bones in order to collect bone marrow. Less than 5 percent of your bone marrow will be taken. This procedure, if done, will also require you to sign a separate consent form.

Selection of alternate back-up donor:

If your own back-up bone marrow or back-up PBPC cannot be collected, a family member will be asked to serve as a back-up donor or an identified third cord blood unit may be used should both the cord blood units fail to function. A back-up donor or product will be required for you to participate in this research

Selection of a stromal cell donor for cord cultures:

In this study, researchers will identify a donor for the stromal cells. This will either be a family member or from an unrelated healthy donor whose marrow stromal cells were collected, grown, and frozen by Mesoblast Systems (off-the-shelf). If your family member is used, they will collect about 7 tablespoons of bone marrow from them.This family member's tissue type (HLA antigens) does not exactly match yours. That bone marrow will be taken to the laboratory where over 3-4 weeks, marrow stromal cells will be grown. These marrow stromal cells produce vitamin-like growth factors that may help your cord blood cells expand to greater numbers. If a family member is not appropriately HLA matched, does not meet eligibility, is not a candidate due to health reasons, or if the participant's disease is getting worse so that there is not enough time to grow the family member's marrow cells into marrow stromal cells, then "off-the-shelf" marrow stromal cells that have been collected from a healthy donor will be used for the procedure. These "off-the-shelf" marrow stromal cells were grown and frozen by Mesoblast Systems. These cells have been safely given to patients with peripheral vascular disease, but their use in this study is considered investigational.

Your bone marrow transplant doctors will assign you to one of 3 chemotherapy treatments, which are discussed below.

High-dose chemotherapy treatment (Myeloablative):

If you are between 1 and 55 years of age and can receive high-dose chemotherapy, or you are between 55 and 65 years old and your doctor agrees, you will receive fludarabine, clofarabine, busulfan, antithymocyte globulin (ATG), and total body irradiation.

You will receive a test dose of busulfan by vein over 60 minutes as an outpatient usually during the week before admission. If the test dose cannot be given as an outpatient, you will be admitted to the hospital on Day -9 for IV fluids and the busulfan test dose. This low-level "test" dose of busulfan is to check how the level of busulfan in your blood levels changes over time. This information will be used to decide the next dose needed to reach the target blood level that matches your body size.

About 11 samples of blood will be drawn for pharmacokinetic (PK) testing of busulfan. PK testing measures the amount of study drug in the body at different time points. These blood samples will be drawn at various timepoints starting before the Busulfan infusion and continuing over approximately the next 11 hours. The blood samples will be repeated again with the first day of high-dose busulfan treatment. Each sample will be about 1 teaspoon of blood. A temporary heparin lock will be placed in your vein to lower the number of needle sticks needed for these draws. If it is not possible for the PK tests to be performed for technical or scheduling reasons, you will receive the standard fixed dose of busulfan.

If you received the test dose as an outpatient, you will be admitted to the hospital on Day -8 and will receive fluids by vein. On Days -7 through -4, you will receive by vein fludarabine over 1 hour, clofarabine by vein over 1 hour, and busulfan by vein over 3 hours. You will receive ATG on Days -4 and -3. On Day -3, you will receive a single treatment of low-dose total body irradiation. You will "rest" (not receive chemotherapy drugs) on Days -2 and -1. Day 0 is the day of transplantation, so the negative day numbers are used to label the treatment days before transplant.

All chemotherapy drugs, fluids, and other medications that must be given by vein will be infused through the catheter. Once the back-up cells are collected, all participants will be admitted to the hospital as indicated by their assigned treatment plan schedule. Chemotherapy may be stopped if intolerable side effects occur.

Lower-dose chemotherapy treatment (Non-myeloablative):

If you are older than 55 and 80 years of age or less, or are of any age with a pre-existing medical condition that prevents you from receiving high dose chemotherapy, you will receive fludarabine- cyclophosphamide-ATG-total body irradiation. You will receive cyclophosphamide as a single dose on Day -6. Fludarabine will be given once a day for 4 days in a row (Days -6 through -3). You will receive ATG on Days -4 and -3. You will also receive a single treatment of low-dose total body irradiation on Day -1. Day 0 is the day of transplantation. Rituximab may be given on Day -9 if appropriate for your disease.

All chemotherapy drugs, fluids, and other medications that must be given by vein will be infused through the catheter. Once the back-up cells are collected, all participants will be admitted to the hospital as indicated by their assigned treatment plan schedule. Chemotherapy may be stopped if intolerable side effects occur.

Reduced Intensity chemotherapy treatment:

If you are between the ages of 1 and 80 years old (and the study doctors think this would keep you from receiving the full myeloablative therapy), if you cannot receive low-dose total body irradiation, and if your doctor agrees, you will receive fludarabine-melphalan-ATG. Starting on Day -5 you will receive fludarabine given once a day for 4 days in a row (Days -5 through -2). You will receive Melphalan as a single dose on Day -2. You will receive ATG on Days -3 and -2. Day 0 is the day of transplantation.

All chemotherapy drugs, fluids, and other medications that must be given by vein will be infused through the catheter. Once the back-up cells are collected, all participants will be admitted to the hospital as indicated by their assigned treatment plan schedule. Chemotherapy may be stopped if intolerable side effects occur.

Expansion of cord blood:

On Day -14, one of your two cord blood units will be thawed in the MD Anderson Stem Cell Laboratory and expanded over a layer of marrow stromal cells from your family member or from "off-the-shelf" marrow stromal cells. The expansion will continue for about 2 weeks and when complete, the cells will be given to you on Day 0 as described below. A small amount of cord blood cells (less than 3%) will be used for laboratory procedures that measure the quality of the product.

The CliniMACS System is a medical device that is used to separate types of blood cells from blood that is removed from the body during leukapheresis. These separated cells are processed for use in treatments such as stem cell transplants.

Transplantation of cord blood:

Two (2) days after completion of high or lower-dose therapy (Day 0), both units of cord blood (the expanded and the unexpanded cord) will be infused into you (one at a time) through your catheter. The unexpanded cord will be thawed and infused first, followed by the infusion of the cells that were expanded in the lab. Each unit will take about 30 minutes to infuse. You will then be hospitalized until your marrow function is restored enough. This usually takes between 3-6 weeks. During this time, you will receive a variety of medications, transfusions, and other standard procedures aimed at decreasing the risks of this procedure, such as graft vs. host disease (GVHD).

Graft versus host disease (GVHD) preventive therapy:

GVHD results from a reaction of the transplanted cord blood cells against certain tissues in your body. In an attempt to prevent or decrease the severity of GVHD, you will receive 2 drugs. Mycophenolate will be given through your catheter 2 times per day (morning and evening) on Days -3 through Day 100. Tacrolimus will be given as a 24-hour continuous infusion over 3-6 weeks. Around Day 30 or 40 (after engraftment), the tacrolimus will be changed to pills given once a day for 180 days (6 months). If GVHD is present, tacrolimus may then be continued longer. The number of tacrolimus pills may vary according to the blood levels of the drug, but usually are between 1-5 pills. Your dose of tacrolimus can be gradually lowered around Day 180 if no GVHD is present. This medicine is used for 6-9 months (longer if chronic GVHD occurs).

You will remain on study as long as your disease does not return. If your disease returns, you will be taken off study and you may be offered participation in another study or other standard treatments.

Follow-up after transplant:

After you leave the hospital, you will be seen regularly in the Department of Blood and Marrow Transplantation at MD Anderson. The frequency of the visits may vary, but may be as often as daily. Routine blood (1-2 tablespoons) and urine tests will be performed. The frequency of blood tests may also vary, but may be performed daily. You will have bone marrow samples collected before transplant and then as needed during the first 100 days after transplant, every 3 months during the first year after transplant, and then once a year while you are on study. After that, bone marrow samples will be collected once a year, indefinitely. These samples are being collected to look for tumor (as a staging test) and to looFk for chimerism (the percent of tumor cells detectable that can predict graft failure and/or relapse). To collect a bone marrow sample, an area of the hip or chest bone is numbed with anesthetic and a small amount of bone marrow is withdrawn through a large needle.

Participants with lymphomas and Hodgkin's disease will need a computed tomography (CT) scan of the chest, abdomen, and pelvis performed before transplant, then as needed during the first 100 days after transplant, and then every 3 months for the first year after transplant. After that, these scans will be done yearly. You will be on study for 1 year but then will be followed yearly as is standard of care.

This is an investigational study. All treatment drugs given to you are FDA approved and commercially available. The CliniMACS device is not FDA approved. At this time, it is being used in research only. Up to 125 patients may take part in this study. All will be enrolled at MD Anderson.

Inclusion Criteria:

1. Patients must have one of the following hematologic malignancies: Acute Myelogenous Leukemia (AML), induction failure, high-risk for relapse first remission (with intermediate-risk or high-risk cytogenetics, flt3 mutation positive and/or evidence of minimal residual disease by flow cytometry), secondary leukemia from prior chemotherapy and/or arising from MDS, Langerhan's cell histiocytosis, any disease beyond first remission; or,
2. Myelodysplastic Syndrome (MDS): Primary or therapy related; or,
3. Acute Lymphoblastic Leukemia (ALL): induction failure, primary refractory to treatment (do not achieve complete remission after first course of therapy) or are beyond first remission including second or greater remission or active disease.
4. #3, continued: Patients in first remission are eligible if they are considered high risk, defined as any of the following detected at any time: with translocations 9;22 or 4;11, hypodiploidy, complex karyotype, secondary leukemia developing after cytotoxic drug exposure, and/or evidence of minimal residual disease, or acute biphenotypic leukemia, or double hit non-Hodgkin's lymphoma; or,
5. Non-Hodgkin's Lymphoma (NHL) in second or third complete remission, or relapse (including relapse post autologous hematopoietic stem cell transplant). Double hit lymphomas in first remission or more advanced disease; or,
6. Small Lymphocytic Lymphoma (SLL), or Chronic Lymphocytic Leukemia (CLL) with progressive disease following standard therapy; or,
7. CML second chronic phase or accelerated phase; or,
8. Hodgkin's Disease (HD): Induction failures, second or third complete remission, or relapse (including relapse post autologous hematopoietic stem cell transplant); or,
9. Multiple Myeloma: stage II or III, symptomatic, secretory Multiple Myeloma requiring treatment.
10. Age greater than or equal to 1 year but less than or equal to 55 years (Myeloablative Regimen 4). Eligibility for pediatric patients will be determined in conjunction with an MD Anderson Cancer Center (MDACC) pediatrician. Patients >55 but < 65 years who have a Performance Status of 0 or 1 and no comorbidities may receive the myeloablative regimen 4 at the discretion of the investigator(s).
11. Age greater than 55 years and less than or equal to 80 years (Nonmyeloablative Regimen 2)
12. Age greater than or equal to 1 but less than or equal to 80 years old that in the opinion of the investigator(s) would preclude myeloablative therapy and who cannot receive Total Body Irradiation (TBI) may receive reduced intensity regimen 3.
13. Performance score of at least 60% by Karnofsky or PS less than 3 (ECOG) (age greater than or equal to 12 years), or Lansky Play-Performance Scale of at least 60% or greater (age <12 years)
14. Left ventricular ejection fraction of at least 40% (Myeloablative Regimen 4, Reduced Intensity Regimen 3) or 30% (Nonmyeloablative Regimen 2)
15. Pulmonary function test demonstrating a diffusion capacity of least 50% predicted (Myeloablative Regimen 4, Reduced Intensity Regimen 3) or at least 40% predicted (Nonmyeloablative Regimen 2). For children < 7 years of age who are unable to perform pulmonary function test (PFT), oxygen saturation > 92% on room air by pulse oximetry.
16. Creatinine < 1.6 mg/dL (Myeloablative Regimen 4, Reduced Intensity Regimen 3) or < 3.0 mg/dL (Nonmyeloablative Regimen 2).
17. Serum glutamate pyruvate transaminase (SGPT)/bilirubin < / = to 2.0 x normal (Myeloablative Regimen 4, Reduced Intensity Regimen 3) or < / = 4.0 x normal (Nonmyeloablative Regimen 2)
18. Negative Beta HCG test in a woman with child bearing potential defined as not post-menopausal for 12 months or no previous surgical sterilization and willing to use an effective contraceptive measure while on study.
19. Unrelated Cord Blood will be used as a source of hematopoietic support if a 5 or 6/6 related or 6/6 unrelated bone marrow donor is not available, or if the tempo of a patient's disease dictates it is not in the patient's best interest to wait for an unrelated marrow donor to be procured.
20. Patients must have two Cord Blood 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 10 million total nucleated cells/Kg recipient body weight (pre-thaw)
21. Patients must have a family member who is matched at 2, 3, or 4 HLA antigens typed as described above and willing to donate 80-100 ml or bone marrow for MSC generation or the Angioblast Mesenchymal Precursor Cells will be used for the cord blood co-cultures. Patients that are high risk for relapse are eligible to use the Angioblast "off-the-shelf" Mesenchymal Precursor Cells.
22. Have identified a back-up cell source in case of engraftment failure. The source can be autologous, related, or unrelated.

Exclusion Criteria:

1. HIV positive
2. Positive beta HCG in female of child-bearing potential defined as not post-menopausal for 12 months or no previous surgical sterilization or breast-feeding.
3. Uncontrolled serious medical condition such as persistent septicemia despite adequate antibiotic therapy, decompensated congestive heart failure despite cardiac medications or pulmonary insufficiency requiring intubation. (excluding primary disease for which CB transplantation is proposed), or psychiatric condition that would limit informed consent.
4. Active central nervous system (CNS) disease in patient with history of CNS malignancy.
5. Availability of appropriate, willing, HLA-matched related marrow donor.