This study will evaluate patients with blood cell cancers who are going to have an allogeneic (donor) blood stem cell transplant from a partially matched relative. The research study will test whether immune cells, called T cells, which come from the donor relative and are specially grown in the laboratory and then given back to the patient along with the stem cell transplant (T cell addback), can help the immune system recover faster after the transplant. As a safety measure, these T cells have been "programmed" with a "self-destruct switch" so that if, after they have been given to the patient, the T cells start to react against the tissues (called "graft versus host" disease, GVHD), the T cells can be destroyed.

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 (Rimiducid) in those subjects who present with GvHD that does not adequately respond to standard of care therapy.

Inclusion Criteria:

1. Signed informed consent

2. Age ≥ 18 years and ≤ 65 years

3. Deemed eligible for allogeneic stem cell transplantation

4. Lack of suitable conventional donor (i.e. 8/8 related or unrelated donor) or presence of rapidly progressive disease not permitting time to identify an unrelated donor

5. HLA typing will be performed at high resolution (allele level) for the HLA-A, -B, Cw, and DRBl, and loci

   • A minimum genotypic identical match of 4/8 is required.
   • The donor and recipient must be identical, as determined by high resolution typing, at least one allele of each of the following genetic loci: HLA-A, HLA-B, HLA-Cw, and HLA-DRB1

6. Subjects with adequate organ functions as measured by:

   1. Cardiac: Left ventricular ejection fraction at rest must be ≥ 45%
   2. Hepatic: Bilirubin ≤ 2.5 mg/dL and ALT, AST and Alkaline Phosphatase < 5 x ULN
   3. Renal: Serum creatinine within normal range for age or creatinine clearance, or with a recommended GFR ≥ 50 mL/min/1.73m2
   4. Pulmonary: FEV 1, FVC and DLCO (diffusion capacity) ≥ 50% predicted (corrected for hemoglobin); or O2 saturation > 92% on room air

7. Clinical diagnosis of one of the following:

   a. Acute Leukemia (includes T lymphoblastic lymphoma) in 2nd or subsequent complete remission (CR) i. Acute Lymphoblastic Leukemia (ALL) in 2nd or subsequent CR. ALL shall be morphologic remission at the time of transplant. Morphologic remission is defined that subjects with normal neutrophil and platelet counts, less than 5% blast cells in a bone marrow (BM) smear and no extramedullary disease ii. Acute Myeloid Leukemia (AML) in 2nd or subsequent CR with or without persistent minimal residual disease b. High-risk ALL in 1st CR (including features such as those in i-iii) i. Adverse cytogenetics such as t(9;22), t(1;19), t(4;11), MLL rearrangements ii. Subjects over 30 years of age, or iii. Time to complete remission was greater than 4 weeks. c. High-risk AML in 1st CR (including features such as those listed in i-vii) i. Greater than 1 cycle of induction therapy required to achieve remission ii. Preceding myelodysplastic syndrome (MDS) iii. Presence of FLT3 abnormalities iv. FAB M6 or M7 leukemia v. Adverse cytogenetics for overall survival such as those associated with MDS vi. Complex karyotype (>3 abnormalities), or vii. Any of the following: inv(3) or t(3;3), t(6;9), t(6;11), + 8 [alone or with other abnormalities except for t(8;21), t(9;11), inv(16) or t(16;16)], t(l1;19)(q23;p13.1) d. High risk Myelodysplastic Syndrome e. Non-Hodgkin's Lymphoma relapsed after autologous transplantation f. Non-Hodgkin's Lymphoma with insufficient autologous hematopoietic stem cells to undergo autologous transplantation g. CML i. in first chronic phase that has not attained at least a complete cytogenetic remission after exposure to at least 3 tyrosine kinase inhibitors ii. in accelerated phase that has not attained at least a complete cytogenetic remission iii. in second chronic phase

8. Performance status: Karnofsky score ≥60%.

9. Patient with hematologic malignancy not responding to /or not eligible for conventional therapy and are approved by Sponsor

Exclusion Criteria:

1. HLA 8/8 allele matched (HLA-A,-B,-Cw,-DRBl) related or unrelated donor able to donate.
2. Autologous hematopoietic stem cell transplant < 3 months prior to enrollment.
3. Pregnancy or breast-feeding.
4. Evidence of HIV infection or known HIV positive serology.
5. Current uncontrolled bacterial, viral or fungal infection (currently taking medication with evidence of progression of clinical symptoms or radiologic findings). The treating physician will make final determination.
6. Non-hematologic malignancy within prior three (3) years, with the exception of squamous cell or basal cell skin carcinoma.
7. Prior allogeneic hematopoietic stem cell transplant.
8. Subjects with a history of primary idiopathic myelofibrosis.
9. Bovine product allergy.