The purpose of this study is to evaluate the safety of progressively substituting day +3 and +4 post-transplant cyclophosphamide (PT-CY) with post-transplant bendamustine (PT-BEN) in myeloablative (MAC) haploidentical hematopoietic cell transplantation (HHCT) for patients with hematological malignancies.

The goal of the Phase 1 component of the study is to evaluate the safety of progressively substituting post-transplant cyclophosphamide (PT-CY) given on Days +3 and +4 with bendamustine (PT-BEN). The Phase I component of the study has been completed.

The Phase Ib component of the study will continue to evaluate the safety and efficacy of subjects who receive PT-BEN on Days +3 and +4 at the maximum tolerated dose determined by Phase I. The Phase Ib component of the study has been completed.

Approximately, 18-36 subjects will be treated as part of Phase I and 15 as part of Phase Ib. Approximately 18 subjects will be used as controls, subjects that receive no PET-BEN, for direct comparison. Total, approximately 38-56 treatment and control patients and 38-56 donor subjects will be enrolled.

This study will follow the standard-of-care bone marrow transplant (BMT), with the only exception being to progressively substitute post-transplant cyclophosphamide (on Days +3 and +4 after BMT) with bendamustine. Six dose levels were planned for the Phase I component of the study, consisting of a combination of sequentially reduced doses of cyclophosphamide (PT-CY) and increased doses of bendamustine (PT-BEN) initially on Day +4 after BMT, followed by the same sequential reduction and increase on Day +3. An interim analysis was performed after cohort 3 was completed in Phase I and included a preliminary comparison between treatment and control groups. Phase Ib will evaluate patients treated with PT-CY on day +3 and PT-BEN on day +4.

Control patients will be patients that have declined to participate in the main trial but will receive haploidentical BMT with the current standard of two days of PT-CY (and no PT-BEN) and will be consented for the immune monitoring studies only.

Inclusion Criteria:

• Be willing and able to provide written consent/assent for the trial.
• Diagnosed with one of the following high-risk malignancies, which require hematopoietic cell transplantation (HCT) but do not have an available Human Leukocyte Antigen (HLA)-matched related or unrelated donor or acceptable cord blood
• High risk acute lymphoblastic leukemia (ALL) in 1st complete remission (CR1) or greater
• High risk acute myelogenous leukemia (AML) in CR1 or greater
• High risk undifferentiated acute leukemia
• High risk myelodysplastic syndrome (MDS)
• Chronic Myelogenous Leukemia (CML) failing or intolerant to Tyrosine Kinase Inhibitors (TKIs) or in accelerated, blastic phase, or in second or subsequent chronic phase
• Lymphoma, (Hodgkin and Non-Hodgkins Lymphoma including marginal zone, follicular lymphoma, chemotherapy-sensitive large-cell, mantle cell lymphoma, gray zone, and Burkitt's lymphoma in remission).
• At least one haploidentical related donor is available for bone marrow harvest.
• Molecular based HLA typing for the HLA-A, -B, -Cw, beta chain (-DRB1) and - DQ Beta 1 Locus (DQB1loci) to the resolution is needed to establish haploidentity.
• A minimum match of 5/10 is required.
• No availability of an 8/8 HLA-matched related or unrelated donor or clinical urgency for transplant (e.g., needed within 4-8 weeks) at which time an acceptable unrelated donor will not be available.

Exclusion Criteria:

• Refractory acute leukemia (>5% blasts) or progressive disease
• Untreated or progressive central nervous system leukemia
• Refractory to chemotherapy lymphoma
• Co-morbidities precluding patient's ability to tolerate BMT
• Aspartate Aminotransferase (AST)/ Alanine Aminotransferase (ALT) > 5 x upper limit of normal (ULN)
• Bilirubin > 2 x ULN
• Creatinine greater than >2 x ULN for age or creatinine clearance/glomerular filtration rate (GFR) <40 ml/min/1.73m2
• Pulmonary function: Diffusing capacity of the lung for carbon monoxide (DLCO) < 40% of normal or O2 Sat <92%
• Cardiac: left ventricular ejection fraction <35%
• Active infection at time of hospital admission of Haplo BMT
• Documented fungal infection or highly suspected and receiving treatment for presumed fungal infection within 3 months of BMT
• HIV positive
• Karnofsky score (adults) < 60% or Lansky score < 50% (pediatrics).
• Positive pregnancy test for girls post menarche or women of childbearing age.
• Severe psychiatric illness or mental deficiency making compliance to treatment unlikely and/or informed consent impossible.
• Any reason, at the investigator's discretion, that the participation of the patient in this protocol would not be in patient's best interest, or where the patient would be unable to adhere to the study requirements.