TCRαβ-depleted Progenitor Cell Graft With Additional Memory T-cell DLI, Plus Selected Use of Blinatumomab, in Naive T-cell Depleted Haploidentical Donor Hematopoietc Cell Transplantation for Hematologic Malignancies

In this study, participants with high-risk hematologic malignancies undergoing hematopoietic cell transplantation (HCT), who lack an available suitable human leukocyte antigen (HLA) matched related/sibling donor (MSD) or matched unrelated donor (MUD), will receive a TCRαβ-depleted haploidentical donor HCT with additional memory cell DLI. One course of blinatumomab will be empirically added for patients with CD19+ malignancy. Primary Objectives * Determine the maximum effective dose for prophylactic CD45RA-depleted DLI when given in the early post-engraftment period. * Assess the efficacy of TCRαβ-depleted progenitor cell graft with additional memory T-cell DLI, plus selected use of blinatumomab, in haploidentical donor hematopoietic cell transplantation for hematologic malignancies as measured by 1 year EFS (events = relapse, death) Secondary Objectives * Assess the safety and feasibility of the addition of blinatumomab in the early post- engraftment period in patients with CD19+ malignancy * Estimate the incidence of neutrophil and platelet engraftment, malignant relapse,event-free survival per disease subgroups (e.g. ALL vs AML), and overall survival at one-year post-transplantation. * Estimate incidence and severity of acute and chronic (GVHD). * Estimate the rate of transplant related mortality (TRM) in the first 100 days after transplantation. * To measure and describe the pharmacokinetics of rabbit ATG in HCT recipients on this study. Exploratory Objectives * Record immune reconstitution parameters, including chimerism analysis, quantitative lymphocyte subsets, T cell receptor excision circle (TREC) analysis, V-beta spectratyping, and lymphocyte phenotype and function. * Describe the use of additional CD45RA-depleted DLI for recipients who have severe viral infections, disease recurrence or progression, or poor immune reconstitution. Assess and record efficacy of CD45RA-depleted DLI for these conditions, and all adverse events that are related to CD45RA-depleted DLI. Blinatumomab will be given to patients with a history of CD19+ malignancy as determined by St. Jude hematopathologist review of current and historical specimens and reports. Blinatumomab dosing will begin no sooner than 1 week after CD45RA-depleted DLI and no later than Day +90. There must be no acute GVHD or it must be quiescent. ALT must be less than 5x ULN, bili less than or equal to 1.5x ULN, and creatinine less than or equal to 1.5x ULN. If more than one family member donor is suitable, then donor selection will be based on several factors including: degree of KIR mismatching, donor-recipient matching of CMV serology, donor-recipient red blood cell compatibility, degree of HLA matching, size of the potential donor, previous use as a donor, presence of donor-specific antibody, and overall health and availability of the potential donor. A G-CSF mobilized peripheral blood progenitor cell product (identified as HPC,A) is the preferred progenitor cell graft source. Our desired target goal will be 5 x 10\^6 CD34+ cells/kg. This number of cells will be necessary to provide an adequate graft, following the various ex vivo manipulations, for prompt reconstitution. More than one collection may be needed to achieve this goal. Donors will undergo a standard hematopoietic progenitor cell mobilization regimen consisting of 5 days of GSF given subcutaneously at 10 micrograms/kilogram. The graft will be collected by leukapheresis on day 5 (and 6 if needed) of G-CSF. The HPC product will typically be collected and infused fresh, however there may be patients or logistical situations that require the HPC product to be collected early, processed, and stored frozen.