Additional Relevant Details The purpose of this clinical trial is to determine whether BPX-501 infusion can enhance immune reconstitution and retain the graft versus leukemia (GVL) effect, with the potential for reducing the severity and duration of severe acute GvHD. The trial will evaluate the treatment of GvHD by the infusion of dimerizer drug (AP1903) in those subjects who present with Grade III-IV acute GVHD, as well as those subjects with Grade II gut/liver GvHD or with Grade I/II GvHD (skin only) who progress or do not respond within 7 days to standard of care treatment.