Modified Immune Cells (CD19-CD22 CAR T Cells) in Treating Patients With Recurrent or Refractory CD19 Positive, CD22 Positive Leukemia or Lymphoma

PRIMARY OBJECTIVES: I. To determine the safety of infusion with chimeric antigen receptor T cells targeting CD19 and CD22. II. To find the recommended phase II dose for recurrent/refractory CD19+CD22+ B cell malignancies. SECONDARY OBJECTIVES: I. To describe the overall response rate and complete response rate of relapsed B cell malignancies treated with CAR-T cells targeting CD19 and CD22. II. To assess other response variables including minimal residual disease (MRD) negative remission, overall survival (OS), and event free survival (EFS). EXPLORATORY OBJECTIVES: I. To evaluate the immune reconstitution and persistence of CAR T cells for one year post infusion. OUTLINE: This is a phase I, dose escalation study of autologous CD19/CD22 chimeric antigen receptor T-cells (CD19-CD22 CAR T cells) followed by a phase II study. Patients receive standard of care cyclophosphamide intravenously (IV) over 30 minutes and fludarabine IV over 30 minutes on days -5, -4, and -3, and then receive CD19-CD22 CAR T cells IV on day 0. Patients with relapsed or persistent disease after a protocol assessment may receive a second infusion of CD19-CD22 CAR T cells. After completion of study treatment, patients are followed up at 1, 2, 3, 6, and 12 months.