The Lowest Effective Dose of Post-Transplantation Cyclophosphamide in Combination With Sirolimus and Mycophenolate Mofetil as Graft-Versus-Host Disease Prophylaxis After Reduced Intensity Conditioning and Peripheral Blood Stem Cell Transplantation

Background: Blood cancers (such as leukemias or lymphomas) often do not respond to standard treatments. A transplant of blood stem cells from a healthy donor can help people with these cancers. Sometimes these transplants cause serious side effects, including a common immunologic problem called graft-versus-host disease. A drug called cyclophosphamide given early after the transplant (post-transplantation cyclophosphamide, PTCy) can reduce these complications. But sometimes this drug has its own negative effects. Furthermore, studies in mice suggest that an intermediate, rather than very high, dose of this drug may best protect against graft-versus-host disease. Objective: To find out if a lower dose of PTCy is more helpful for people who undergo blood stem cell transplants. Eligibility: People aged 18 and older who have a blood cancer and are eligible for a transplant of blood stem cells from another person. Healthy donors are also needed but must be related to the individual needing the transplant. Design: Participants will undergo screening. Transplant recipients will have imaging scans and tests of their heart and lung function. They will be assessed for the status of their cancer, including bone marrow taken from their pelvis and possibly also scans and/or fluid drawn from the spine depending on the disease type. Donors will be screened for general health. They will give several tubes of blood. They will give an oral swab and saliva and stool samples for research. Recipients will be in the hospital at least 4 to 6 weeks. They will have a temporary catheter inserted into a vein in the chest or neck. Medications will be given and blood will be drawn through the catheter. The transplanted stem cells will be given through the catheter. Participants will receive medications both before and after the transplant. Participants will return to the clinic at least once a week for 3 months after leaving the hospital. Follow-up visits will continue periodically for 5 years.

Background: * Post-transplantation cyclophosphamide (PTCy) reduces rates of severe acute and chronic graft-versus-host disease (GVHD) after allogeneic hematopoietic cell transplantation (HCT) and safely facilitates human leukocyte antigen (HLA)-haploidentical HCT * When clinically translated, the dose (50 mg/kg) and timing (days +3 and +4) of PTCy used were partly extrapolated from murine major histocompatibility complex (MHC)-matched skin allografting models and were partly empirical * In both MHC-haploidentical and MHC-disparate murine HCT models, a dose of 25 mg/kg/day was superior to 50 mg/kg/day on days +3 and +4 in terms of protection against GVHD severity and mortality. Lower dosing of PTCy also was associated with less broad reduction of T-cell numbers after PTCy and lower toxicity than higher dosing. * In patients on an NIH study using myeloablative conditioning and bone marrow as the graft source, a dose of 25 mg/kg/day on days +3/+4 has been associated with more rapid engraftment, less toxicity, and potentially better immune function without an increase in acute GVHD. * The optimal dosing of PTCy potentially may differ depending on the graft source (bone marrow versus peripheral blood stem cells) and HLA disparity (HLA-matched vs. HLA-partially mismatched). Objective: * Phase I: Determine the lowest effective dose of post-transplantation cyclophosphamide (PTCy) in combination with sirolimus and mycophenolate mofetil (MMF) as graft-versus-host disease (GVHD) prophylaxis after reduced intensity conditioning and peripheral blood stem cell transplantation (PBSCT), as assessed by primary graft failure AND Grade III-IV acute GVHD as the dose limiting toxicities (DLTs). This lowest effective dose will be evaluated in parallel for HLA-matched and HLA-haploidentical HCT in different arms of the study. * Phase II: Evaluate the efficacy of PTCy, at the lowest dose determined for each HLA-matching arm from phase I, as assessed by 1-year GVHD-free relapse-free survival (GRFS) rate. Eligibility: -Recipient Participant: * Histologically or cytologically confirmed hematologic malignancy with standard indication for allogeneic hematopoietic cell transplantation * Age \>= 50 years or 18-49 years but considered ineligible for myeloablative conditioning. * At least one potentially suitable HLA-haploidentical or 10/10 (HLA-A, B, C, DR, DQ) related or unrelated donor. * Karnofsky performance score \>= 70 * Adequate organ function Design: * Open-label, multi-center, non-randomized, phase I/II study. * All recipient participants will receive reduced intensity conditioning, peripheral blood stem cell (PBSC) HCT, and GVHD prophylaxis with PTCy, MMF, and sirolimus. * There will be two parallel arms: one using HLA-haploidentical donors and one using HLA-matched related or unrelated donors. * A small pilot of 10 evaluable participants per arm will receive the standard PTCy 50 mg/kg/day on days +3/+4 to obtain a limited amount of comparative clinical, pharmacokinetic, and T-cell immunophenotyping data. * The study will proceed to a novel phase I time-to-event Bayesian optimal interval (TITE-BOIN) trial design to find the lowest acceptable dose of PTCy for each arm. Primary graft failure and grade III-IV aGVHD at day +100 post-transplant are defined PTCy dose-limiting toxicities. * Three dose levels of PTCy: 35, 25, and 15 mg/kg/day on days +3 and +4 are planned in each arm of phase I. * Recipient participants will be evaluated for development of grade III-IV acute GVHD (aGVHD) and primary graft failure at day +100 as the dose-limiting toxicities. Once the optimal PTCy dose for PBSC transplantation is determined for each arm, we will conduct a phase II expansion for each arm to estimate the efficacy of PTCy in combination with sirolimus and mycophenolate mofetil as GVHD prophylaxis. 1-year GRFS rate will be the primary endpoint during the phase II part.