Modified Immune Cells (CD19/CD20 CAR-T Cells) in Treating Patients With Recurrent or Refractory B-Cell Lymphoma or Chronic Lymphocytic Leukemia

PRIMARY OBJECTIVES: I. To evaluate the safety of the autologous anti-CD19/anti-CD20 CAR-expressing naive/memory T cells (CART19/20), including determination of the maximum tolerated dose and assessment for replication competent lentivirus (RCL). SECONDARY OBJECTIVES: I. Clinical response. Ia. Overall response rate. Ib. Duration of remission. Ic. Progression-free survival. Id. Overall survival. II. CD19/CD20 bispecific CAR transgenic T-cell persistence. IIa. T-cell monitoring and analyses. IIb. Evidence of B-cell aplasia. EXPLORATORY OBJECTIVES: I. To determine the serum levels of cytokines associated with cytokine release syndrome (CRS) in subjects exhibiting \> grade-2 CRS following CART19/20 cell treatment. OUTLINE: This is a dose-escalation study of CD19/CD20 CAR-T cells. CONDITIONING CHEMOTHERAPY: Patients receive fludarabine phosphate intravenously (IV) over 30 minutes and cyclophosphamide IV over 60 minutes 5, 4, and 3 days before cell infusion. T-CELL INFUSION: Patients receive CD19/CD20 CAR-T cells IV on day 0. Patients with cytokine release syndrome may also receive tocilizumab IV on day 2 at the discretion of the clinical investigator. After completion of study treatment, patients are followed up daily for 14 days, on days 30, 45, 60, 70, 90, and 120, every 3 months for 2 years, every 6 months for 3 years, and then annually for a minimum of 15 years.