The purpose of this research study is to determine if an experimental agent, LLME can decrease the incidence and severity of Graft-Versus-Host-Disease (GVHD) following blood (hematopoietic) stem cell transplantation

We believe that the risks of allogeneic transplant can be drastically reduced if the following criteria can be met: (1) consistent engraftment, (2) little or no GVHD with the ability to rapidly withdraw immune suppression, (3) rapid recovery of CD4 counts to levels greater than 200 cells/micro liter. Our prior (ongoing) trial attempts to address how LLME treated T cells given as donor lymphocyte infusion (DLI) can address points 2 and 3 above. The current study addresses how treatment of the CD34- fraction of the graft attempts to address points 1 and 2 (and to a lesser extent point 3) above. We believe that if these points can be consistently achieved that the mortality of allogeneic HSCT may be reduced to levels more akin to those of autologous HSCT. We propose to test the hypothesis that LLME-treated T cells will be safe with regard to reducing GVHD or other infusion related toxicities and that their administration as part of the transplant will facilitate engraftment. We believe that this approach will ultimately be an important step in a variety of transplant settings ranging from matched siblings to haplodisparate donors.

Inclusion Criteria:

• Patients must be > 18 years of age, with no upper age limit.
• Patients must have an ECOG performance status of 0 or 1.
• Any patient with a hematologic malignancy which is unlikely to be cured by conventional treatment is eligible for this study.
• Patients for whom a disease specific protocol exists will be transplanted on those protocols as discussed in the introduction.
• Patients who have had prior autografts may be treated on this protocol.
• Patients must have adequate physical function as measured by the following criteria:
• Cardiac: Asymptomatic or, if symptomatic, then left ventricular ejection fraction at rest must be >40%.
• Hepatic: Aspartate transaminase (AST) micro 3x the upper limits of normal and total serum bilirubin < 2.5 mg/dL. Patients with a higher bilirubin from "benign conditions" such as Gilbert's disease may still be eligible for the study.
• Renal: Serum creatinine within the normal range or if creatinine outside normal range then creatinine clearance > 60 ml/min/1.73m2. Serum creatinine must be less than or equal to 2.0 mg/dl.
• Pulmonary: Asymptomatic or, if symptomatic, DLCO (diffusion capacity) > 45% of predicted (corrected for hemoglobin)
• The patient or guardian(s) must be able to give informed consent to the study.
• Patient must have a suitable donor who is identical for HLA (human leukocyte antigens) -A, -B, -C, -DR. Single antigen mismatches for HLA-A, -B, -C, -DR are also permitted. Donors obtained through the National Marrow Donor Program (NMDP) will follow NMDP guidelines.

Exclusion Criteria:

• Patients who are eligible for a standard myeloablative transplant and for whom a standard myeloablative transplant is preferable will not be treated on this protocol.