IIT2022-04-Sasine-CAR-T: A Phase 1 Single-arm, Open-label Study to Evaluate the Feasibility and Safety of Collecting and Combining Autologous Hematopoietic Stem Cells With Chimeric Antigen Receptor (CAR) T-Cell Therapy in Subjects With Relapsed/Refractory Hematological Malignancies

Inclusion Criteria: * Age 18 - 85 years. * Histologically proven hematological malignancy according to the World Health Organization 2016 classification criteria for which a commercially available, FDA-approved CAR T product exists. * Relapsed or refractory disease, defined by the following: * Disease progression after last regimen, or * Refractory disease: failure to achieve a partial response (PR) or complete remission (CR) to the last regimen * At least 2 weeks or 5 half-lives, whichever is shorter, must have elapsed since any prior systemic therapy for the malignancy at the time the subject is planned for leukapheresis. * Toxicities due to prior therapy must be stable or recovered to ≤ Grade 1 with the exception of alopecia. * Subjects with an active uncontrolled infection should not start CAR T treatment until the infection has resolved. * Eastern cooperative oncology group (ECOG) performance status 0 - 2. * Adequate hematologic, hepatic, and cardiac function * Serum pregnancy test for women of childbearing potential (WOCBP) at Screening. * Willing to comply to research specimen collection as specified in the protocol. * Written informed consent obtained from subject and ability for subject to comply with the requirements of the study. Exclusion Criteria: * Autologous hematopoietic cell transplant intent or execution within 8 weeks of planned CAR T infusion. * History of allogeneic cell transplantation within 8 weeks of planned CAR T infusion. * Presence or suspicion of fungal, bacterial, viral, or other infection that is uncontrolled or requiring IV antimicrobials for management at time of screening. * History of myocardial infarction, cardiac angioplasty or stenting, unstable angina, or other clinically significant cardiac disease within 6 months of enrollment. * History of a seizure disorder, cerebrovascular ischemia/hemorrhage, dementia, or any autoimmune disease with CNS involvement. * Doses of corticosteroids of greater than or equal to 5 mg/day of prednisone or equivalent doses of other corticosteroids and other immunosuppressive drugs are not allowed prior to enrollment. A washout period of 10 days prior to leukapheresis and 10 days prior to anti-CD19 CAR T cell administration is required. * Any medical condition likely to interfere with assessment of feasibility or safety of study treatment. * Live vaccine ≤ 6 weeks prior to planned start of conditioning regimen. * History of severe immediate hypersensitivity reaction to any of the agents used in this study. * Current pregnancy or breastfeeding because of the potentially dangerous effects of the preparative chemotherapy on the fetus or infant. * Subjects of both sexes who are not willing to practice birth control from the time of consent through 6 months after the completion of conditioning chemotherapy. Females who have undergone surgical sterilization or who have been postmenopausal for at least 1 year 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. * Patients with obvious myeloid clonal hematopoiesis on the screening bone marrow biopsy will be excluded based on the risk of developing myeloid neoplasms with aHSC infusion.