Haplo-identical Viral-Specific T-cells for Treatment of Cytomegalovirus and Adenovirus Infections After Hematopoietic Cell Transplantation

The investigators want to learn if CMV- and ADV-specific T-cells (cells that fight infections) isolated (selected) from a donor using an automated medical device can be a safe treatment for treating patients with CMV, and ADV after transplant.This study will test the effects and safety of giving VSTs produced here at St. Jude in treating the participant's infection. Primary objective To determine the efficacy of VSTs to achieve a ≥1 log10 reduction in CMV and/or ADV viral load in the peripheral blood 4 weeks after VST infusion. When the initial viral load is \<1 log10 above the threshold of detection, the objective is to achieve a reduction to below the threshold of detection. Secondary objectives * Determine the safety of VSTs when used to treat CMV and/or ADV viremia post-HCT. * Determine the proportion of patients who achieve a negative viral load at 3 months post-infusion. * Assess the persistence of response for 6 months post-infusion.