Fourth-gen CAR T Cells Targeting CD19/CD22 for Highly Resistant B-cell Lymphoma/Leukemia (PMBCL/CNS-BCL).

Inclusion Criteria: * Diagnosis: ALL In view of the PI and the primary oncologist, there must be no available alternative curative therapies or subject has declined to pursue alternative therapy; and subjects must be either ineligible for allogeneic stem cell transplant (SCT), have refused SCT, recurred after SCT, or have disease activity that prohibits SCT at the time of enrollment. * Chemotherapy refractory disease in subjects with B-ALL is defined as progression or stable disease after two lines of therapies * Recurrence of disease after achieving a complete response (CR). * Subjects with persistent or relapsed minimal residual disease (MRD) (by flow cytometry, PCR, FISH, or next generation sequencing) require verification of MRD positivity on two occasions at least 4 weeks apart. * Subjects with Philadelphia Chromosome positive acute lymphoblastic leukemia (Ph+ALL) subjects are eligible if they progressed, had stable disease or relapsed after two lines of therapy, including tyrosine kinase inhibitors (TKIs). * Subjects with recurrence of isolated CNS relapse after achieving complete remission (CR); if relapsed with MRD, will require verification of MRD positivity on two occasions at least 4 weeks apart. * Diagnosis: Lymphoma Subjects with lymphoma must have progressed, had SD, or recurred after initial treatment regimens that include an anthracycline and an anti CD20 monoclonal antibody. Subjects who relapse ≥12 months after therapy should have progressed after autologous transplant or been ineligible for autologous transplant. * CD19 expression CD19 expression is required at any time since diagnosis. If patient has received anti-CD19 targeted therapy (i.e. Blinatumomab), then CD19 expression must be subsequently demonstrated. CD19 expression. must be detected on greater than 50% of the malignant cells by immunohistochemistry or ≥ 90% by flow cytometry. The choice of whether to use flow cytometry or immunohistochemistry will be determined by what is the most easily available tissue sample in each subject. In general, immunohistochemistry will be used for lymph node biopsies, flow cytometry will be used for peripheral blood and bone marrow samples. * Subjects who have undergone autologous SCT with disease progression or relapse following SCT will be eligible if all other eligibility criteria are met. Subjects who have undergone allogeneic SCT will be eligible if, in addition to meeting other eligibility criteria, they are at least 100 days post-transplant, they have no evidence of active GVHD and have been without immunosuppressive agents for at least 30 days. * Subjects who have undergone prior anti-CD19 or anti-CD22 CAR therapy will be eligible if \< 5% of circulating levels of CD3+ cells express the previous CAR by flow cytometry. * Must have evaluable or measurable disease; subjects with lymphoma must have evaluable or measurable disease according to the revised IWG Response Criteria for Malignant Lymphoma\[66\] must be present. Lesions that have been previously irradiated will be considered measurable only if progression has been documented following completion of radiation therapy. * At least 2 weeks or 5 half-lives, whichever is shorter, must have elapsed since any prior systemic therapy at the time the subject is planned for leukapheresis, except for systemic inhibitory/stimulatory immune checkpoint therapy, which requires 5 half-lives. * Exceptions: * There is no time restriction with regard to prior intrathecal chemotherapy (incl. steroids) provided there is complete recovery from any acute toxic effects of such; g. Subjects receiving hydroxyurea may be enrolled provided there has been no increase in dose for at least 2 weeks prior to starting apheresis; h. Subjects who are on standard ALL maintenance type chemotherapy (vincristine, 6-mercaptopurine or oral methotrexate) may be enrolled provided that chemotherapy is discontinued at least 1 week prior to apheresis. * Subjects receiving steroid therapy at physiologic replacement doses (≤ 5 mg/day of prednisone or equivalent doses of other corticosteroids) only are allowed provided there has been no increase in dose for at least 2 weeks prior to starting apheresis; j. For radiation therapy: Radiation therapy must have been completed at least 3 weeks prior to enrollment, with the exception that there is no time restriction if the volume of bone marrow treated is less than 10% and also the subject has measurable/evaluable disease outside the radiation port. * Toxicities due to prior therapy must be stable and recovered to ≤ Grade 1 (except for clinically non-significant toxicities, such as alopecia, nutritional support measures, electrolyte abnormalities, or those not impacting the investigator's ability to assess treatment emergent toxicities) * Age Greater than or equal to 1 year of age and less than or equal to 30 years of age at time of enrollment; must meet parameters for apheresis per institutional guidelines. NOTE: The first subject in the first dose cohort must be ≥ 18 years of age if an adult has not been treated at that dose cohort on the companion Stanford protocol "Phase 1 Dose Escalation Study of CD19/CD22 Chimeric Antigen Receptor (CAR) T Cells in Adults with Recurrent or Refractory B Cell Malignancies" and undergone safety evaluation at Day 28 without evidence of DLT. * Performance Status: Subjects \> 10 years of age: Karnofsky ≥ 50%; Subjects ≤ 10 years of age: Lansky scale ≥ 50% (See Appendix B Section 14.2) * Normal Organ and Marrow Function (supportive care is allowed per institutional standards, i.e. filgrastim, transfusion) * ANC ≥750/uL\* * Platelet count ≥50,000/uL\* * Absolute lymphocyte count ≥150/uL\* * Adequate renal, hepatic, pulmonary and cardiac function defined as: * Serum ALT/AST ≤10 ULN (unless elevated ALT/AST is attributed to leukemia or lymphoma involvement of the liver, in which case this criterion will be waived and not disqualify a patient). * Total bilirubin ≤1.5 mg/dl, except in subjects with Gilbert's syndrome. * Cardiac ejection fraction ≥ 45%, no evidence of physiologically significant pericardial effusion as determined by an ECHO, and no clinically significant ECG findings * No clinically significant pleural effusion * Baseline oxygen saturation \>92% on room air at rest * creatinine: within age adjusted normal institutional limits (see table below) OR * creatinine clearance ≥60 mL/min/1.73 m2 (as estimated by Cockcroft Gault Equation) for subjects with creatinine levels above institutional normal. * Age (Years) Maximum Serum Creatinine (mg/dL) -≤5 0.8 5 \< age ≤ 10 1.0 \>10 1.2 * if these cytopenias are not judged by the investigator to be due to underlying disease (i.e. potentially reversible with anti-neoplastic therapy); A subject will not be excluded because of pancytopenia ≥ Grade 3 if it is due to disease, based on the results of bone marrow studies. * CNS Status * Subjects with ALL * Subjects with the following CNS status are eligible only in the absence of neurologic symptoms suggestive of CNS leukemia, such as cranial nerve palsy: * CNS 1, defined as absence of blasts in cerebral spinal fluid (CSF) on cytospin preparation, regardless of the number of WBCs; * CNS 2, defined as presence of \< 5/µL WBCs in CSF and cytospin positive for blasts, or \> 5/µL WBCs but negative by Steinherz/Bleyer algorithm: CNS 2a: \<10/µL RBCs; \< 5/µL WBCs and cytospin positive for blasts; CNS 2b: ≥10/µL RBCs; \< 5/µL WBCs and cytospin positive for blasts; CNS 2c: ≥10/µL RBCs; ≥5/µL WBCs and cytospin positive for blasts but negative by Steinherz/Bleyer algorithm. * Subjects with lymphoma * Subjects must have no signs or symptoms of CNS disease or detectable evidence of CNS disease on MRI at the time of screening. Subjects who have previously been treated for CNS disease and who have the following CNS status will be eligible: * CNS 1, defined as absence of blasts in cerebral spinal fluid (CSF) on cytospin preparation, regardless of the number of WBCs; * CNS 2, defined as presence of \< 5/µL WBCs in CSF and cytospin positive for blasts, or \> 5/µL WBCs but negative by Steinherz/Bleyer algorithm: * CNS 2a: \< 10/µL RBCs; \< 5/µL WBCs and cytospin positive for blasts; * CNS 2b: ≥ 10/µL RBCs; \< 5/µL WBCs and cytospin positive for blasts; * CNS 2c: ≥ 10/µL RBCs; ≥ 5/µL WBCs and cytospin positive for blasts but negative by Steinherz/Bleyer algorithm. * Females of childbearing potential must have a negative serum or urine pregnancy test (females who have undergone surgical sterilization or who have been postmenopausal for at least 2 years are not considered to be of childbearing potential) * Contraception Subjects of child-bearing or child-fathering potential must be willing to practice birth control from the time of enrollment on this study and for four (4) months after receiving the preparative regimen. * Females of child-bearing potential must have a negative pregnancy test because of the potentially dangerous/unknown effects on the fetus. * Ability to give informed consent. All subjects ≥ 18 years of age must be able to give informed consent. For subjects \<18 years old their legal authorized representative (LAR) (i.e. parent or guardian) must give informed consent. Pediatric subjects will be included in age appropriate discussion and verbal assent will be obtained for those \> 7 years of age, when appropriate. Exclusion Criteria: Subjects meeting any of the following criteria are not eligible for participation in the study: * Recurrent or refractory ALL limited to isolated testicular. * Subjects with radiologically-detected CNS lymphoma or CNS 3 disease (presence of ≥ 5/µL WBCs in CSF and cytospin positive for blasts \[in the absence of a traumatic lumbar puncture\] and/or clinical signs of CNS leukemia). * Hyperleukocytosis (≥ 50,000 blasts/µL) or rapidly progressive disease that in the estimation of the investigator and sponsor would compromise ability to complete study therapy. * History of malignancy other than non-melanoma skin cancer or carcinoma in situ (e.g. cervix, bladder, breast) unless disease free for at least 3 years. * Presence of fungal, bacterial, viral, or other infection that is uncontrolled or requiring IV antimicrobials for management. Simple UTI and uncomplicated bacterial pharyngitis are permitted if responding to active treatment. * Ongoing infection with HIV or hepatitis B (HBsAg positive) or hepatitis C virus (anti-HCV positive) as the immunosuppression contained in this study will pose unacceptable risk. A history of hepatitis B or hepatitis C is permitted if the viral load is undetectable per quantitative PCR and/or nucleic acid testing. * CNS disorder such as cerebrovascular ischemia/hemorrhage, dementia, cerebellar disease, or autoimmune disease with CNS involvement that in the judgment of the investigator may impair the ability to evaluate neurotoxicity. * History of myocardial infarction, cardiac angioplasty or stenting, unstable angina, or other clinically significant cardiac disease within 12 months of enrollment, or have cardiac atrial or cardiac ventricular lymphoma involvement. * Subjects receiving anticoagulation therapy. * Any medical condition that in the judgement of the principal investigator is likely to interfere with assessment of safety or efficacy of study treatment * History of severe immediate hypersensitivity reaction to any of the agents used in this study. * Women of child-bearing potential who are pregnant or breastfeeding because of the potentially dangerous effects of the conditioning lymphodepletion chemotherapy on the fetus or infant. Females who have undergone surgical sterilization or who have been postmenopausal for at least 2 years are not considered to be of childbearing potential. * In the investigator's judgment, the subject is unlikely to complete all protocol-required study visits or procedures, including follow-up visits, or comply with the study requirements for participation. * May not have primary immunodeficiency or history of systemic autoimmune disease (e.g. Crohns, rheumatoid arthritis, systemic lupus) resulting in end organ injury or requiring systemic immunosuppression/systemic disease modifying agents within the last 2 years.