Additional Relevant Details This is a Phase1/2 dose escalation study evaluating the safety and feasibility of BPX-501 infused after partially mismatched, related (haploidentical), T cell-depleted HSCT. The purpose of this clinical trial is to determine whether BPX-501 infusion can facilitate engraftment, enhance immune reconstitution and potentially improve the graft versus leukemia (GVL) effect, with the potential for reducing the severity and duration of severe acute graft versus host disease (GvHD). The trial will evaluate the treatment of GvHD by the infusion of dimerizer drug (AP1903) in those subjects who present with severe GvHD (Grades III and IV) as well as those subjects with Grade I and II who progress or do not respond to corticosteroid therapy within 4 days.