Intra-Osseous Co-Transplant of UCB and hMSC

Inclusion Criteria: * Patients must have one of the following malignancies: * Acute myelogenous leukemia (AML): high-risk AML including: * Antecedent hematological disease (e.g., myelodysplasia \[MDS\]) * Treatment-related leukemia * Complete remission (first complete remission \[CR1\]) with poor-risk cytogenetics or molecular markers (e.g. fms-related tyrosine kinase 3 \[Flt 3\] mutation, 11q23, del 5, del 7, complex cytogenetics) * Second complete remission (CR2) or third complete remission (CR3) * Induction failure or first relapse with either * ≤ 10% blasts in the marrow and/or * ≤ 5% blasts in the peripheral blood * Acute lymphoblastic leukemia (ALL) * High-risk CR1 including: * Poor-risk cytogenetics (e.g., Philadelphia chromosome t(9;22)or 11q23 rearrangements) * Presence of minimal disease by flow cytometry after 2 or more cycles of chemotherapy * No complete remission (CR) within 4 weeks of initial treatment * Induction failure * CR2 or CR3 with either: * ≤ 10% blasts in the marrow and/or * ≤ 5% blasts in the peripheral blood * Myelodysplastic syndromes (MDS), Intermediate-1 (INT-1), intermediate-2 (INT-2) or high Revised International Prognostic Scoring System (IPSS-R) score that has failed at least 1 first line therapy * Myelofibrosis (MF): * Intermediate-2 or high risk by Dynamic International Prognostic Scoring System (DIPSS)-plus * Monosomal karyotype * Presence of inv(3)/i(17q) abnormalities * Other unfavorable karyotype OR leukocytes ≥40 X 10\^9/L AND * Circulating blasts ≤ 9% * Relapsed or refractory lymphoid malignancies (including non-Hodgkin lymphoma, Hodgkin lymphoma and chronic lymphocytic leukemia) meeting the following criteria: * Disease status: stable disease, partial remission or 2nd and 3rd complete remission * Chronic myelogenous leukemia (CML) in second chronic phase after accelerated or blast crisis; blast crisis defined as: * Blast count ≥ 20% in the peripheral blood or bone marrow * Large foci of blasts on bone marrow * Presence of extra-medullary blastic infiltrate (myeloid sarcoma or chloroma) * Recipients of prior autologous or allogeneic transplant are eligible, as long as at least 3 months have passed since the transplant, and the patient fulfills other eligibility criteria * Eastern Cooperative Oncology Group (ECOG) performance status ≤2 * Candidates for reduced intensity conditioning regimens * Patients who do not have HLA-matched (defined as matched in HLA A, B, C, and DRB1) related or unrelated donors, those who elect to undergo UCB even if they have a MRD or MUD, or patients who require a UCB either for emergency indications such as primary graft failure. * Cord Blood Units available through NMDP with the following minimal criteria: * HLA Match: 4/6 or better match (HLA A, B, DRB1) * Cell dose: Minimum of 2.0x107TNC/kg pre thaw * Concurrent therapy for extramedullary leukemia or central nervous system (CNS) lymphoma: concurrent therapy or prophylaxis for testicular leukemia, CNS leukemia, and CNS lymphoma including standard intrathecal chemotherapy and/or radiation therapy will be allowed as clinically indicated; such treatment may continue until the planned course is completed; subjects must be in CNS remission at the time of protocol enrollment if there is a history of CNS involvement * Subjects must have a back-up umbilical cord on the registry in addition to the umbilical cord being used in this study * Subjects must have the ability to understand and the willingness to sign a written informed consent document Exclusion Criteria: * Patients with inadequate Organ Function as defined by: * Creatinine clearance \< 30 ml/min * Bilirubin ≥ 2 x institutional upper limit of normal * Aspartate aminotransferase (AST) (serum glutamic oxaloacetic transaminase \[SGOT\]) ≥ 2 x institutional upper limit of normal * Alanine aminotransferase (ALT) (serum glutamate pyruvate transaminase \[SGPT\]) ≥ 2 x institutional upper limit of normal * Corrected diffusing capacity of the lung for carbon monoxide (DLCOcorr) \< 40% normal * Left ventricular ejection fraction \< 35% * Patients with uncontrolled inter-current illness including, but not limited to ongoing or active infection, symptomatic congestive heart failure, unstable angina pectoris, cardiac arrhythmia, or psychiatric illness/social situations that would limit compliance with study requirements * Pregnant or breastfeeding women are excluded from this study