Infusion of Expanded Cord Blood Cells

This phase II trial studies how well donor umbilical cord blood transplant with ex-vivo expanded cord blood progenitor cells (dilanubicel) works in treating patients with blood cancer. Before the transplant, patients will receive chemotherapy (fludarabine, cyclophosphamide and in some cases thiotepa) and radiation therapy. Giving chemotherapy and total-body irradiation before a donor umbilical cord blood transplant helps stop the growth of cells in the bone marrow, including normal blood-forming cells (stem cells) and cancer cells. It may also stop the patient's immune system from rejecting the donor's stem cells. When the healthy stem cells from a donor are infused into the patient they may help the patient's bone marrow make stem cells, red blood cells, white blood cells, and platelets. The donated stem cells may also replace the patient's immune cells and help destroy any remaining cancer cells.