Brief Summary: The goal of this clinical research study is to learn if treating umbilical cord blood with growth factors before a transplant can help to improve the body's ability to accept the cord blood transplants.

Detailed Description: Umbilical cord blood is a source of stem cells that can be used for transplantation. The major problem with this type of transplant is the small number of stem cells that are available in the cord blood. This delays the recipients body's ability to accept the cord blood. This may also cause the immune system to slow, making infection more likely. One method to help fix this problem is to treat a sample of the cord blood with growth factors, which may promote the growth of the needed T cells before they are given to the recipient as a transplant. For this study, participants will receive high-dose chemotherapy, then an untreated cord blood cells transplant, and then 14 days later receive the treated cord blood cells transplant. The Study Drugs: Melphalan is designed to damage the DNA (the genetic material) of cells, which may cause cancer cells to die. Thiotepa is designed to the damage DNA, which may cause cancer cells to die Fludarabine is designed to make cancer cells less able to repair damaged DNA. This may increase the likelihood of the cells dying. Rituximab is designed to attach to a specific protein on the surface of tumor cells, which may cause them to die. Etoposide is designed to block cell growth. Filgrastim is a drug that helps cells in the bone marrow to divide Anti-thymocyte globulin (ATG) is a drug that helps the immune system by attaching to and inactivating T cells. Mycophenolate mofetil (MMF) is a drug that suppresses the immune system. Tacrolimus is a drug that helps the immune system to prevent GVHD. IL-2 is a protein that is used to help the growth of T cells. Placement of Central Venous Catheter (CVC): Before you can have back-up stem cells collected or you receive chemotherapy, you will be required to have a CVC, which is a sterile flexible tube, placed into a large vein in the upper chest while you are under local anesthesia. Children are usually unable to hold still for this procedure and will be given drugs to make them sleep for the procedure. 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. If you are donating back-up cells (as described below) the catheter will be placed the day of the collection. If you are not donating back up cells, the CVC will be placed a few days before admission to the hospital. This catheter will be used to draw blood, give fluids, and give the study drugs. Collection of "back up" stem cells: Collection of additional stem cells from the donor of the cord blood will not be possible if the transplant with the cord blood fails. Back-up blood or additional bone marrow will be collected from you and frozen about a week before the high-dose chemotherapy begins. The study doctor will decide to collect or not collect this sample based on the status of the disease. The sample(s) will be collected by leukapheresis or with bone marrow aspirates. Selection of another donor as the alternate source of stem cells If the cord blood transplant is not successful, you will receive the back-up cells collected before admission as described above. If you are unable to donate these stem cells, for example it they are contamination with tumor cells, a family member or a second cord blood transplant will be used. The potential family member donor will have screening tests performed to find out if they are eligible to donate stem cells. Leukapheresis: If you are donating back-up stem cells, before the collection of the blood stem cells, you will be treated with a drug called filgrastim, which will make the stem cells in the marrow easier for the study doctor to collect and help to increase the number of white blood cells. This drug is given through a needle under the skin 1-2 times a day for 3-7 days. When the white cell count is high enough (usually around day 4), your blood stem cells will be collected from your CVC over 3-4 hours/day. The collection process will be repeated daily until a enough stem cells are collected. You will continue to receive filgrastim until all the stem cells are collected. Bone Marrow Collection: If the leukapheresis cannot be performed successfully, for example the white blood cell count does not increase enough with the filgrastim, you will have bone marrow collected. You will go to the operating room and be asleep for this. Multiple small bone marrow aspirates will be performed and the marrow collected. Only a small sample of your bone marrow (less than 5%) will be taken. To collect a bone marrow aspirate, an area of the hip is numbed with anesthetic, and a small amount of bone marrow is withdrawn through a large needle. High-Dose Chemotherapy Administration: Day 0 is the day of the stem cell transplant, so the negative day numbers are used to describe the treatment days before the transplant. All chemotherapy, fluids, and other drugs that must be given by vein will be infused through the CVC. After the "backup" stem cells are collected, you will be admitted to the hospital on Day -9 to begin receiving fluids. Chemotherapy may be stopped if intolerable side effects occur. You will be placed in 1 of 2 treatment groups. The study doctors will decide which treatment group will be the best treatment for you. If you are in Group 1, you will receive chemotherapy and the cord blood transfusion. If you are in Group 2, you will receive treatment with chemotherapy, radiation, and the cord blood transfusion. * If you are in Group 1, you will receive melphalan by vein over 30 minutes as a single dose on Day -8, thiotepa by vein over 4 hours as a single dose on Day -7, followed by fludarabine by vein over about 30 minutes 1 time each day on Days -6 through -3. Rituximab may be given by vein over 4 - 6 hours on Day -9 if the doctor thinks it is necessary. * If you are in Group 2, you will receive radiation on Days -7,-6, -5 and -4. You will sign a separate consent for the total body irradiation. Then you will be given etoposide by vein over 1-2 hours as a single dose on Day -3. Rituximab may be given by vein over 4-6 hours on Day -8 if the doctor thinks it is necessary. * Group 1 will receive anti-thymocyte globulin (ATG) by vein over 4-6 hours on the Days -4 and -3 of treatment to lower the chances of the cord blood infusion being rejected. * Group 2 will receive anti-thymocyte globulin (ATG) by vein over 4-6 hours on the Days -3, -2, and -1 of treatment to lower the chances of the cord blood infusion being rejected. Cord Blood Infusion: After your chemotherapy treatment, and total body irradiation (only for Group 2), your cord blood unit will be thawed. The larger portion of the cord blood will be infused through your CVC. A small portion of the cells will be expanded as described below. Expansion of Cord Blood T Cells: On Day 0, the smaller unit of your cord blood will be treated at the M. D. Anderson Stem Cell Laboratory with IL-2 and vitamin-like growth factors to produce the needed T cells. At this time, clinical beads will be used to help separate and expand the T cells. This will take 2 weeks. On Day 14, you will then be given the treated cord blood as an infusion through your CVC. Infusion of Back-Up Cells: In case the stem cell transplant is rejected by your immune system, and if researchers were unable to collect a sample of your own stem cells, an unrelated donor or a second cord transplant will be used. The cord blood will be treated using a CliniMACs processing device. Mouse protein antibodies are used in CliniMACs processing procedures. Recipients who have pre-existing immunity to these proteins may be at risk for allergic reactions during the infusion of the processed cells. Epinephrine and antihistamines will be available at the recipient's bedside during the peripheral blood progenitor cell (PBSC) infusion to help treat any allergic reactions. Graft Versus Host Disease (GVHD) Preventive Therapy: GVHD may result from the transplanted cord blood cells reacting against certain tissues in your body. In an attempt to prevent or decrease the severity of GVHD, you will receive 2 drugs (MMF and tacrolimus). Mycophenolate mofetil (MMF) pills will be given starting 3 days before your cord blood transplant, and will be continued until Day 100 after the transplant. If you cannot take pills, the drug can be given through your CVC. If you develop GVHD, you may have to continue to take MMF after Day 100. Tacrolimus will be given 2 days before your transplant as a 24 hour continuous infusion for several weeks. After you have the stem cell transplant and are able to eat and drink, tacrolimus will be given by mouth 2 times a day for up to 6 months. The number of tacrolimus pills may vary depending on the levels of the drug in the blood, but usually you will be given between 1-3 pills each time. If you are not able to swallow pills, a liquid form of this drug is available. This drug is used for about 6-9 months or longer if chronic GVHD occurs. Length of Study: You will be monitored closely for the first 100 days after receiving the transplant and then periodically after that. You may remain on study as long as the disease does not return and you do not experience any intolerable side effects. Your participation in this clinical trial may be ended at any time for any reason. Follow-Up Visits After Transplant: After you leave the hospital, you will come back for regular visits in the clinic at M. D. Anderson. How often you have follow-up visits may vary, but may be as often as daily. Blood (about 1-2 tablespoons) will be collected for routine tests. The number of blood and/or urine tests may also vary, but they may be performed daily. You will have a bone marrow aspirate collected before the transplant, about 30 days after the transplant, and then every 3 months for the first year after the transplant. After that, bone marrow samples will be collected 1 time a year for as long as the doctor thinks is necessary. To collect a bone marrow biopsy, an area of the bone is numbed with anaesthetic and a small amount of bone marrow is withdrawn through a large needle. Patients with lymphomas and Hodgkin's disease will need CT scans of the chest, abdomen, and pelvis performed within 30 days before the transplant, about 100 days after transplant, and then about every 3 months for the first year. After that, it will be done 1 time every year for as long as the doctor thinks it is necessary. This is an investigational study. The clinical beads that will be used for this study may be produced by either the University of Pennsylvania or Invitrogen Corporation. All of the study drugs are all FDA approved and commercially available for treatment in adults. These drugs alone or in any combination are not FDA approved for this treatment in children and their use in this study is considered to be experimental. The CliniMACS processing device used to process the cells for backup infusion is not FDA approved for this use. Up to 18 patients will take part in this study. All will be enrolled at M. D. Anderson.