CD19/CD22 Bicistronic Chimeric Antigen Receptor (CAR) T Cells in Children and Young Adults With Recurrent or Refractory B Cell Malignancies

Background: Acute lymphoblastic leukemia (ALL) is the most common cancer in children. About 90% of children and young adults who are treated for ALL can now be cured. But if the disease comes back, the survival rate drops to less than 50%. Better treatments are needed for ALL relapses. Objective: To test chimeric antigen receptor (CAR) therapy. CARs are genetically modified cells created from each patient s own blood cells. his trial will use a new type of CAR T-cell that is targeting both CD19 and CD22 at the same time. CD19 and CD22 are proteins found on the surface of most types of ALL. Eligibility: People aged 3 to 39 with ALL or related B-cell lymphoma that has not been cured by standard therapy. Design: Participants will be screened. This will include: Physical exam Blood and urine tests Tests of their lung and heart function Imaging scans Bone marrow biopsy. A large needle will be inserted into the body to draw some tissues from the interior of a bone. Lumbar puncture. A needle will be inserted into the lower back to draw fluid from the area around the spinal cord. Participants will undergo apheresis. Their blood will circulate through a machine that separates blood into different parts. The portion containing T cells will be collected; the remaining cells and fluids will be returned to the body. The T cells will be changed in a laboratory to make them better at fighting cancer cells. Participants will receive chemotherapy starting 4 or 5 days before the CAR treatment. Participants will be admitted to the hospital. Their own modified T cells will be returned to their body. Participants will visit the clinic 2 times a week for 28 days after treatment. Follow-up will continue for 15 years....

* INCLUSION CRITERIA: * Diagnosis * Participant must: * Have pathology confirmed B cell ALL (not isolated to the testis or CNS), CML with ALL transformation, or high-grade lymphoma (e.g., Burkitt's lymphoma, B-lymphoblastic lymphoma, diffuse large B-cell lymphoma, inclusive of low-grade lymphoma that has transformed to high grade disease); and * Have relapsed or been refractory after at least one standard chemotherapy regimen and at least one salvage treatment. Participants with Philadelphia chromosome + ALL must have failed prior tyrosine kinase inhibitor; and * Be ineligible for allogeneic stem cell transplant (SCT), have refused SCT, or have recurred after SCT; and * Be unable to access (in a timely manner), ineligible for, or have relapsed/failed after or not responded to a commercially available CD19 CAR T-cell construct; and * Have evidence of at least minimal residual disease or PET-avid disease (lymphoma) at the time of enrollment. * CD22/CD19 expression * Cohorts A1b, B1b, C2b * CD19 must be detected on \>15% of the malignant cells by immunohistochemistry or \> 80% by flow cytometry. * CD22 positivity must be confirmed. * Cohorts D1b, 2 B-ALL * CD19 or CD22 positivity must be confirmed * Age \>= 3 years of age and \= 16 years of age: Karnofsky \>= 50%; Participants \< 16 years of age: Lansky scale \>= 50%. * Participants must have adequate organ and marrow function as defined below: * leukocytes \>= 750/mcL\* * platelets \>= 50,000/mcL\* * total bilirubin \