Nonmyeloablative Hematopoietic Cell Transplantation (HCT) for Patients With Hematologic Malignancies Using Related, HLA-Haploidentical Donors: A Pilot Trial of Peripheral Blood Stem Cells (PBSC) as the Donor Source

Inclusion Criteria: Subject 1. Age\< 70. 2. Molecular based HLA typing will be performed for the HLA-A, -B, -Cw, DRB1 and -DQB1 loci to the resolution adequate to establish haplo identity. A minimum match of 5/10 is required. An unrelated donor search is not required for a patient to be eligible for this protocol if the clinical situation dictates an urgent transplant. Clinical urgency is defined as 6-8 weeks from referral or low-likelihood of finding a matched, unrelated donor. 3. Subjects must meet one of the disease classifications listed below: Acute leukemias (includes T lymphoblastic lymphoma). Remission is defined as \< 5% blasts with no morphological characteristics of acute leukemia (e.g., Auer Rods) in a bone marrow with \> 20% cellularity, peripheral blood counts showing ANC \>1000/ul, including patients in CRp. Acute Lymphoblastic Leukemia in high risk CR1 as defined by at least one of the following: Adverse cytogenetics such as t(9;22), t(1;19), t(4;11), MLL rearrangements White blood cell counts \>30,000/mcL Patients over 30 years of age Time to complete remission \>4 weeks Presence of extramedullary disease Acute Myelogeneous Leukemia in high risk CR1 as defined by at least one of the following: Greater than 1 cycle of induction therapy required to achieve remission Preceding myelodysplastic syndrome (MDS) Presence of Flt3 abnormalities FAB M6 or M7 leukemia or Adverse cytogenetics for overall survival such as: those associated with MDS Complex karyotype (≥ 3 abnormalities) 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(11;19)(q23;p13.1) Acute Leukemias in 2nd or subsequent remission Biphenotypic/Undifferentiated Leukemias in 1st or subsequent CR. High-risk MDS status-post cytotoxic chemotherapy Myelofibrosis Burkitt's lymphoma: second or subsequent CR. Lymphoma. Chemotherapy-sensitive (complete or partial response; see response criteria Appendix C) large cell, Mantle Cell or Hodgkin's lymphomas that have failed at least 1 prior regimen of multi-agent chemotherapy and are ineligible for an autologous transplant or relapsed/progressed after autologous stem cell transplant. Marginal zone B-cell lymphoma or follicular lymphoma that has progressed after at least two prior therapies (excluding single agent Rituxan) and are ineligible for an autologous transplant or relapsed/progressed after autologous stem cell transplant.. 4. Patients with adequate physical function as measured by: Cardiac: left ventricular ejection fraction at rest must be ≥ 35%. Hepatic: bilirubin ≤ 2.5 mg/dL; and ALT, AST, and Alkaline Phosphatase \< 5 x ULN. Renal: serum creatinine within normal range for age, or if serum creatinine outside normal range for age, then renal function(creatinine clearance or GFR) \> 40 mL/min/1.73m2. Pulmonary: FEV1, FVC, DLCO (diffusion capacity) ≥ 40% predicted (corrected for hemoglobin); if unable to perform pulmonary function tests, then O2 saturation \> 92% on room air. Performance status: Karnofsky/Lansky score ≥ 60%. 5. Patients who have received a prior allogeneic HSCT and who have either rejected their grafts or who have become tolerant of their grafts with no active GVHD requiring immunosuppressive therapy. Donor 1. Donors must be HLA-haploidentical first-degree or second degree relatives of the patient. 2. Age ≥ 18 years 3. Weight ≥ 40 kg Exclusion Criteria: Subject 1. HLA-matched donor able to donate. 2. Pregnancy or breast-feeding. 3. Current uncontrolled bacterial, viral or fungal infection (currently taking medication with evidence of progression of clinical symptoms or radiologic findings). Donor 1\) Positive anti-donor HLA antibody.