The trial treatment involves collecting the patient's own WBCs from the blood by a standard well established and safe process called apheresis. The cells are then cultured in a specialized laboratory (under Good Manufacturing Practice conditions, similar to standards under which pharmaceuticals are produced) over 12 days to convert the cells to specialized tumor-attacking T cells. Early in that culture process the cells are exposed to a virus (that is modified so that it cannot infect or replicate outside the special culture conditions) that contains a special gene. Via the virus, this gene inserts into the patient's T cells in culture and gets incorporated into the T cell's genetic machinery. As the T cells replicate, the new gene produces a protein receptor that becomes part of the patient's T cells. This protein receptor on the T cells has the capacity to specifically recognize and bind to a protein on the leukemia or myeloma cells called the "Lewis Y" antigen.