A Study of Inotuzumab and Blinatumomab in People With B-cell Acute Lymphoblastic Leukemia

Inclusion Criteria: * Age ≥ 18 years of age. * Newly diagnosed CD19+ and CD22+ B-ALL with the following characteristics * Patients ≥55 years old, OR * Patients 18-54 years old who decline or are deemed unfit for conventional chemotherapy with at least one of the following criteria: * ECOG performance status of 2 or more * Severe cardiac comorbidity (including congestive heart failure requiring treatment) * Known pulmonary comorbidity (including DLCO ≤65% or FEV1 ≤65%) * Renal comorbidity (including creatinine clearance 30-45 mL/min) * Relapsed or refractory CD19+ and CD22+ B-ALL * Patients with extramedullary disease will be allowed as long as they have detectable disease by flow cytometry in the bone marrow * Peripheral absolute lymphoblast count of ≤ 10,000/ml after pre-phase (not required for enrollment but required to proceed with first dose of inotuzumab). * Philadelphia chromosome negative by FISH/karyotype for t(19;22) or RT PCR for bcr-abl transcript. * CD19 and CD22 expression will be confirmed by enrolling institutions prior to study registration by flow cytometry and/or IHC. * Creatinine clearance ≥30 mL/min * Total bilirubin ≤ 1.5 x upper limit of normal, AST and ALT ≤3.0x upper limit of normal (ULN) * QTcF ≤ 480 * Ejection fraction ≥ 50% Exclusion Criteria: * Patients with Burkitt's lymphoma, T-ALL, CML in lymphoid blast crisis and mixed phenotype acute leukemia (MPAL). * Patients with newly diagnosed B-ALL who received prior treatments, with the exception of corticosteroid, hydroxyurea, or one dose of vincristine, are ineligible. * Patients with Ph+ B-ALL by FISH or RT PCR. * ECOG performance status \>3. * Left ventricular ejection fraction (LVEF) \<50%. * History of sinusoidal obstruction syndrome (SOS), also known as veno-occlusive disease (VOD). * Prior treatment with inotuzumab * History of liver cirrhosis * Ongoing need for systemic T-cell suppressive therapy (e.g. corticosteroids, tacrolimus, cyclosporine, etc.) Patients need to be off calcineurin inhibitors for at least 4 weeks in order to be eligible for enrollment. * Active Grade 2-4 acute graft versus host disease (GVHD), graded with the modified Glucksberg criteria and/or GVHD requiring systemic steroids in excess of physiologic replacement * Moderate or severe chronic GVHD graded with the NIH 2014 criteria * Pregnant or lactating women. Women and men of childbearing age should use effective contraception while on this study and continue for the following time periods: female patients of reproductive potential should use effective contraception during treatment and for 8 months after last treatment dose. Males with female partners of reproductive potential should use effective contraception during treatment and for 5 months after the last dose. * Patients with HIV or active hepatitis B or hepatitis C infection are ineligible. Patients with a prior history of hepatitis B or hepatitis C who have negative HBV/HCV PCR respectively at the time of screening are eligible * Patients with concurrent active malignancies as defined by malignancies requiring any therapy other than expectant observation or hormonal therapy, with the exception of squamous and basal cell carcinoma of the skin, in situ cervical cancer, adequately treated stage I/II cancer from which the patient is current in complete remission, or any other cancer from which the patient has been disease free for five years * Patients with history or presence of clinically significant neurological disorders such as epilepsy, generalized seizure disorder, or severe brain injuries. * Any other issue which, in the opinion of the treating physician, would make the patient ineligible for the study.