A Two Step Approach to Allogeneic Hematopoietic Stem Cell Transplantation for Patients With Hematologic Malignancies-Increasing GVT Effects Without Increasing Toxicity

Inclusion Criteria: This treatment is for patients with high risk hematologic malignancies. High risk is defined as: * Any patient with a hematologic malignancy in which allogeneic HSCT is pursued with the expectation of cure. Patients may have post-treatment residual disease, but the disease should be stable or minimally progressive and must be responsive to chemotherapy. * Any patient with an untreated hematologic malignancy in which allogeneic HSCT is thought to be the sole or the best option for cure and in Patients without morphologic evidence of disease but with high risk features which would predict for relapsed despite remission at HSCT such as adverse cytogenetics, 3rd or greater CR, or failure to recover peripheral blood counts to normal ranges. While these patients do not have detectable disease by current methods, like all patients they have non-detectable disease which in their case is highly aggressive. * Patients with uncommon diagnoses in which allogeneic HSCT is thought to be beneficial but are no comparable to the majority of patients on this protocol will not be counted in the statistical aims of the study and will be reported descriptively. The PI and at least one Co-I must document this exception in the study binder and the rationale for descriptive report. An example of a patient who may meet this criteria is someone with a malignancy that is an overlap of two different diagnoses or one whose malignancy is difficult to categorize. While this circumstance is expected to be rare, it will prevent patients with rare diagnoses to be treated off study and it will help maintain homogeneity of the study population. * Patients must have one related donor who is HLA mismatched in the GVHD direction at two or more HLA loci (except as described below) * Patients must have adequate organ function: * Left Ventricular Ejection Fraction (LVEF) of ≥50% * DLCO (adjusted for hemoglobin) ≥50% of predicted and FEV-1 ≥50% * Adequate liver function as defined by a serum bilirubin ≤1.8, Aspartate aminotransferase (AST) or alanine aminotransferase (ALT) ≤ 2.5 x Upper Limit of Normal (ULN) * Creatinine clearance of ≥ 60ml/min * Karnofsky Performance Status (KPS) of ≥80% on the modified KPS tool (see Appendix) * Patients must be willing to use contraception if they have childbearing potential * Able to give informed consent * Age ≥ 18 years of age Exclusion Criteria: * Modified KPS of \<80% * \> 5 Comorbidity Points on the Hematopoietic Cell Transplant Co-Morbidity Index (HCT CI) (See Appendix) (Patients with greater than 5 points will be allowed for trial with approval of the PI and at least 1 Co-I not on the primary care team of the patient.) this is an adjustment to account for healthy patients who meet the spirit of this protocol but have histories that result in higher than HCT-CI 5 points. An example is a patient with a solid tumor malignancy in their remote history (adds 3 points to HCT-CI total) where the treatment for the malignancy occurred years to decades before and there has been complete recovery of toxicities. * Human Immunodeficiency Virus (HIV) positive * Active involvement of the central nervous system with malignancy * Psychiatric disorder that would preclude patients from signing an informed consent * Pregnancy, or unwillingness to use contraception if they have childbearing potential * Patients with life expectancy of ≤ 6 months for reasons other than their underlying hematologic/oncologic disorder * Alemtuzumab treatment within 8 weeks of HSCT admission * ATG within 8 weeks of HSCT administration * Inability to tolerate cyclophosphamide or undergo total body irradiation at the doses specified in the treatment plan.