Pilot Study of Reduced-Intensity Umbilical Cord Blood Transplantation in Adult Patients Wtih Advanced Hematopoietic Malignancies

Inclusion Criteria: * Age 55-70 years, or \< 55 years if deemed ineligible for conventional high dose chemotherapy by the UCSF SCT Committee or by protocol eligibility requirements for myeloablative therapy. Reasons for ineligibility for myeloablative therapy include: * Poor cardiac function (i.e. LVEF \< 40%) * Poor pulmonary function (i.e. DLCO \< 50%) * Hepatic dysfunction * Prior myeloablative therapy * Availability of donor cord blood (one to three units) matching at \> or equal to 4 of 6 HLA antigens (A, B, and DR). HLA class I antigens will be determined by serologic methods, and Class II antigens will be determined by high-resolution DNA typing. Typing will be confirmed by the UCSF Immunogenetics Department. The target UCB TNC dose is \> or equal to 3.5 x 10(7) TNC/kg recipient weight, however the absolute minimum TNC requirement is \> 2.5 x 10(7) TNC per kilogram) based on cell counts prior to cryopreservation. Cord blood units will be obtained from all available international banks. * HLA identical or 1 antigen mismatched related donors or potential HLA-matched unrelated donors (MUD) must NOT be readily available * Disease types include: * Acute myeloid leukemia not expected to be curable with chemotherapy. This will include patients with high-risk cytogenetics (-7, -7q, -5, -5q, t(6, 9), t(9, 11), complex (\> or equal to 3 abnormalities), Ph(+), evolution from prior myelodysplasia or AML secondary to prior chemotherapy, failure to achieve remission, second, or subsequent remission or refractory relapse. Marrow blasts must be \< or equal to 10%. This may be achieved using standard chemotherapy treatment. * Myelodysplasia with high-risk features. This will include patients with IPSS category INT2 or HI-risk MDS or CMML. Marrow blasts must be \< or equal to 20%. If required, chemotherapy may be given to achieve target levels of blasts. Patients are expected to have disease control or not rapidly progressive disease regardless of blast count (but must be \< or equal to 20%). * Acute lymphoblastic leukemia not expected to be cured with chemotherapy. This will include patients with high-risk cytogenetics (Ph+, t(4,11), 11q23 abnormalities, and monosomy 7), patients requiring more than one induction course to achieve remission, as well as patients failing to enter remission or in second or subsequent remission. Marrow blasts must be \< or equal to 10%. If required, chemotherapy may be given to achieve target level of blasts. * Chronic myelogenous leukemia- chronic phase failing imatinib mesylate therapy (either progressive disease or failure to achieve a major cytogenetic response at 1 year following initiation of therapy), accelerated phase failing to achieve a complete cytogenetic remission at 1 year following initiation of therapy, accelerated phase failing to achieve any cytogenetic response at 6 months of therapy, accelerated phase with progressive disease as demonstrated by worsening cytogenetic response in two consecutive analyses separated by 4 weeks or CML in blast crisis. Patients with blast phase of CML must have \< 10% blasts in bone marrow. * Multiple myeloma stage I-III with relapse following autologous transplant, with disease refractory to at least two prior conventional; myeloma therapies or with chromosome 13 abnormalities. Patients with chromosome 13 abnormalities are eligible either at diagnosis or after initial progression. * Lymphomas including diffuse or follicular large cell lymphoma, mantle cell lymphoma, peripheral T-cell lymphoma, T-NK cell lymphoma, or Hodgkin's disease which has failed to respond to primary therapy, progressed, or recurred after prior therapy. Patients who have relapsed following autologous transplant are eligible. * Low-grade NHL and Chronic Lymphocytic Leukemia with relapse or refractory disease following, at least, two chemotherapy-based treatment regimens; or with relapse following autologous transplantation. * Patients must have an ECOG PS \< or equal to 2. * Laboratory requirements: * Creatinine \< 2.0mg/dL and creatinine clearance \> 40/m/min (calculated or based on 24-hour urine collection) * Bilirubin \<2.0mg/dL, AST/alkphos \<3x upper limit of normal * Patients with hepatitis C and active hepatitis B (hepatitis B DNA detectable) are eligible only if a liver biopsy is performed and there is grade \< or equal to 2 fibrosis and/or inflammation. Patients with a history of HBV infection should be tested for HBeAg, anti-HBe and HBV DNA (quantitative). Patients with active HBV viral replication should receive anti-viral therapy. * Cardiac ejection fraction \> 30%, DLCO \> or equal to 40%. * Negative pregnancy test (for females of reproductive age only). * Signed informed consent Exclusion Criteria: * Active infection requiring ongoing antibiotic treatment * HIV infection * Poor performance status (ECOG \> 2) * Opinion of BMT Committee that autologous SCT or conventional therapy would be a preferable form of treatment * Organ function is below requirements * Pregnancy, or breast-feeding