In the proposed study, escalating doses of inotuzumab ozogamicin will be added to a standard pediatric inspired re-induction regimen and administered to patients with relapsed or refractory B-cell acute lymphoblastic leukemia (B-ALL). Two re-induction regimens will be tested (one without pegaspargase and one including pegaspargase) and participants will be followed for disease status, allogeneic hematopoietic cell transplant (allo HCT), veno-occlusive disease following allo HCT, and overall survival.

Inotuzumab ozogamicin has been studied as a single agent in refractory and relapsed ALL. In the relapsed setting, inotuzumab ozogamicin has been shown to achieve complete remission (CR) in 81% of patients and minimal residual disease (MRD) negativity in 78% of patients who achieve CR. In the proposed study, escalating doses of inotuzumab ozogamicin will be added to a standard pediatric inspired re-induction regimen and administered to patients with relapsed or refractory B-cell ALL. Two re-induction regimens will be tested. The first regimen is a 3-drug regimen comprised of prednisone, vincristine, and daunorubicin. The second is a 4-drug regimen comprised of prednisone, vincristine, daunorubicin, and pegaspargase. Intrathecal methotrexate (IT-methotrexate) and intrathecal cytarabine (IT-ARA-C) will be included for central nervous system (CNS) prophylaxis with both the 3-drug and 4-drug regimens. We hypothesize that combining inotuzumab ozogamicin with these regimens is safe and will improve CR rates, successful transition to allo HCT, and overall survival in patients with relapsed or refractory B-ALL.

Inclusion Criteria:

1. Provision of signed and dated informed consent form
2. Stated willingness to comply with all study procedures and availability for the duration of the study
3. Diagnosed with CD-22 positive* B-cell Acute Lymphoblastic Leukemia or B-cell Lymphoblastic Lymphoma (Philadelphia chromosome negative) * For the purposes of this study, CD-22 positive will be defined based on the analysis completed for diagnostic purposes.
4. Male or female, aged 16-60 years
5. ECOG performance status of 0-2
6. Left ventricular ejection fraction ≥ 50% measured by echocardiogram or MUGA
7. Either relapsed following remission after initial induction therapy or refractory to induction therapy
8. Adequate organ function, including serum creatinine ≤ 1.6 mg/dL OR creatinine clearance >50 ml/min by Cockgroft-Gault formula, bilirubin ≤ 1.5 mg/dL (except in patients with Gilbert's disease), AST, ALT and alkaline phosphatase ≤ 3 x upper limit of normal (elevation exceeding this threshold of either AST OR ALT would not meet eligibility)
9. For females of reproductive potential: negative pregnancy test
10. For females and males of reproductive potential: agreement to use adequate contraception during study participation and for an additional 1 year after the end of study treatment
11. Agreement to adhere to Lifestyle Considerations throughout study duration and for 1 year following last study treatment.

Exclusion Criteria:

1. Past receipt of a total of ≥ 300 mg/m^2 doxorubicin equivalents (600 mg/m^2 daunorubicin, 60 mg/m^2 idarubicin, 75 mg/m^2 mitoxantrone)
2. Current or past history of pancreatitis
3. QT interval on electrocardiogram (ECG) > 0.45 by Framingham formula
4. Known congestive heart failure
5. Known allergy to asparaginase (only an exclusion criteria for participants enrolling in part 2)
6. Presence of central nervous system (CNS) disease
7. Pregnancy or lactation
8. Chronic liver disease including chronic active hepatitis and/or cirrhosis
9. Active Hepatitis B virus (HBV) by core antibody, surface antigen (HBsAg) or viral load
10. Active Hepatitis C virus (HCV) (positive antibody test confirmed by viral load if antibody test is positive)
11. Known history of infection with Human Immunodeficiency Virus (HIV)
12. Active or uncontrolled infections
13. Abnormal baseline hepatic ultrasound (including Dopplers)
14. Prior allogeneic stem cell transplant
15. Prior use of inotuzumab ozogamicin
16. Known diagnosis of hemochromatosis with iron overload
17. Treatment with steroids or hydroxyurea for more than 7 days with each within the 2 weeks prior to registration -that is, each is allowed for up to 7 days
18. Gastrointestinal tract disease causing the inability to take oral medication, malabsorption syndrome, a requirement for intravenous (IV) alimentation, prior surgical procedures affecting absorption, uncontrolled inflammatory GI disease, or inability to swallow medications.
19. Philadelphia chromosome positive B-cell ALL