This phase II clinical trial studies how well personalized natural killer (NK) cell therapy works after chemotherapy and umbilical cord blood transplant in treating patients with myelodysplastic syndrome, leukemia, lymphoma or multiple myeloma. This clinical trial will test cord blood (CB) selection for human leukocyte antigen (HLA)-C1/x recipients based on HLA-killer-cell immunoglobulin-like receptor (KIR) typing, and adoptive therapy with CB-derived NK cells for HLA-C2/C2 patients. Natural killer cells may kill tumor cells that remain in the body after chemotherapy treatment and lessen the risk of graft versus host disease after cord blood transplant.
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Progression free survival (PFS) time in C2C2 patients
Timeframe: From the date of engraftment to disease progression or death, assessed up to 4 years
Progression free survival (PFS) time in C1 patients
Timeframe: From the date of cord blood transplant to disease progression or death, assessed up to 4 years