Donor Umbilical Cord Blood Transplant in Treating Patients With Hematologic Cancer

PRIMARY OBJECTIVES: I. Estimate probability of one year survival. II. Demonstrate equivalent or improved engraftment rates with a non-anti-thymocyte globulin (ATG) based conditioning regimen. Patients will be considered graft failure/rejections provided they meet any of the criteria listed below: * Absence of 3 consecutive days with neutrophils \>= 500/ul combined with host cluster of differentiation (CD)3 peripheral blood chimerism \>= 50% at day 42 * Absence of 3 consecutive days with neutrophils \>= 500/ul under any circumstances at day 55 * Death after day 28 with neutrophil count \< 100/ul without any evidence of engraftment (\< 5% donor CD3) * Primary autologous count recovery with \< 5% donor CD3 peripheral blood chimerism at count recovery and without relapse SECONDARY OBJECTIVES: I. Six month non-relapse mortality. II. Overall incidence of graft failure/rejection. Patients will be considered graft failure/rejections provided they meet any of the criteria listed below: * Absence of 3 consecutive days with neutrophils \>= 500/ul combined with host CD3 peripheral blood chimerism \>= 50% at day 42 * Absence of 3 consecutive days with neutrophils \>= 500/ul under any circumstances at day 55 * Death after day 28 with neutrophil count \< 100/ul without any evidence of engraftment (\< 5% donor CD3) * Primary autologous count recovery with \< 5% donor CD3 peripheral blood chimerism at count recovery and without relapse III. Kinetics of chimeric reconstitution. IV. Incidence of neutrophil engraftment by day 42. V. Incidence of platelet engraftment by six months. VI. Incidence of grade II-IV and III-IV acute graft-versus-host disease (GvHD) at day 100. VII. Incidence of one year chronic GvHD. VIII. Incidence of clinically significant infections at 6 months, 1 year, 2 years. IX. Probability of one and two year survival. X. Incidence of one and two year relapse or disease progression. XI. Fred Hutchinson Cancer Research Center (FHCRC) patients: Kinetics of immune reconstitution, with both functional and quantitative assays. XII. FHCRC patients: Examination of possible immunologic factors leading to emergence of a dominant unit. OUTLINE: CONDITIONING REGIMEN: Patients receive fludarabine phosphate intravenously (IV) over 1 hour on days -6 to -2 and cyclophosphamide IV over 1-2 hours on day -6. Patients undergo a lower dose of total-body irradiation (TBI) on day -1. UMBILICAL CORD BLOOD TRANSPLANT: Patients undergo donor umbilical cord blood infusion on day 0. IMMUNOSUPRESSIVE THERAPIES: Patients receive cyclosporine IV over 1 hour every 8-12 hours on days -3 to +180 and mycophenolate mofetil IV or orally (PO) every 8 hours on days 0 to +96. After completion of study treatment, patients are followed periodically for up to 2 years.