Supercharged cord blood cells take on resistant blood cancers

Copied word for word from the study's registry entry, so the wording is the study team's rather than ours.

Inclusion Criteria: 1. 18-75 years of age. 2. Diagnosis of relapsed B-NHL, HL, T-NHL, or B-ALL in refractory relapse, defined as: * B-NHL: Failure of \>/= 1 salvage line and failure of or ineligibility for CAR-T. * HL and T-NHL: Failure of \>/= 1 salvage line or prior SCT. * ALL: Active disease (\>5% of blasts or positive MRD at a level of \>0.1% measured by multiparameter flow cytometry) after ≥ 2 two lines of therapy. Patients with B-ALL must have either failed of or be ineligible for CAR-T cell therapy. Patients who have mutations for which there are FDA approved targeted therapies (i.e., BCR-ABL) must also have received at least one of such agents. 3. Expression of CD70 in the pre-enrollment sample \>/= 20% measured by immunohistochemistry or flow cytometry. 4. Measurable disease, defined by \>/= 1 histologically confirmed hypermetabolic lesion on PET/CT scan. 5. ECOG PS ≤ 2 (Karnofsky ≥60%). 6. Adequate blood counts (WBC \>/= 2K, HGB \>/= 8 g/dL, platelets \>/= 50K). 7. Creatinine clearance ≥ 30 ml/min. 8. ALT and/or AST ≤ 3 x ULN, and bilirubin and ALP ≤ 2 x ULN. 9. FEV1, FVC and DLCOc ≥ 50%. 10. LVEF ≥ 40%, without active arrythmias. 11. If female of child-bearing potential, she must not be pregnant or breastfeeding and required to have a negative urine or serum pregnancy test prior to enrollment. 12. For patients with evidence of chronic hepatitis B virus (HBV) infection, the HBV viral load must be undetectable on suppressive therapy, if indicated. 13. Patients with a history of hepatitis C virus (HCV) infection must have been treated and cured. For patients with HCV infection still on treatment, they are eligible if they have an undetectable HCV viral load. 14. Patients with treated brain metastases are eligible if follow-up brain imaging after central nervous system (CNS)-directed therapy shows no evidence of progression. 15. Patients with a prior or concurrent malignancy whose natural history or treatment does not interfere with the safety or efficacy assessment of the investigational regimen are eligible for this trial. 16. Patients with known history or current symptoms of cardiac disease, or history of treatment with cardiotoxic agents, should have a clinical risk assessment of cardiac function using the New York Heart Association Functional Classification. To be eligible for this trial, patients should be class 2B or better. 17. The effects of CAR-NK cells on the developing human fetus are unknown. For this reason and because fludarabine and cyclophosphamide, as well as other therapeutic agents, used in this trial are known to be teratogenic, women of child-bearing potential and men must agree to use adequate contraception (hormonal or barrier method of birth control; abstinence, see Appendix 1) prior to study entry and for the duration of study participation. (Refer to Pregnancy Assessment Policy MD Anderson Institutional Policy # CLN1114). * This includes all female patients, between the onset of menses (as early as 8 years of age) and 55 years unless the patient presents with an applicable exclusionary factor which may be one of the following: * Postmenopausal (no menses in greater than or equal to 12 consecutive months). * History of hysterectomy or bilateral salpingo-oophorectomy. * Ovarian failure (Follicle Stimulating Hormone and Estradiol in menopausal range, who have received whole pelvic radiation therapy). * History of bilateral tubal ligation or another surgical sterilization procedure. * Approved methods of birth control (see Appendix 1) are as follows: Hormonal contraception (i.e. birth control pills, injection, implant, transdermal patch, vaginal ring), intrauterine device (IUD), tubal ligation or hysterectomy, subject/partner post vasectomy, Implantable or injectable contraceptives, and condoms plus spermicide. Not engaging in sexual activity for the total duration of the trial and the drug washout period is an acceptable practice; however periodic abstinence, the rhythm method, and the withdrawal method are not acceptable methods of birth control. Should a woman become pregnant or suspect she is pregnant while she or her partner is participating in this study, she should inform her treating physician immediately. • Men treated or enrolled on this protocol must also agree to use adequate contraception prior to the study, for the duration of study participation, and 4 months after completion of CAR NK cell administration. 18. Ability to understand and willingness to sign a written informed document. 19. Agree to sign consent to the long-term follow-up protocol PA17-0483 to fulfill the institutional responsibilities to various regulatory agencies. Exclusion Criteria: 1. Lymphoma or ALL in CR with no measurable sites of disease. 2. Major surgery \<4 weeks prior to first dose of study drug. 3. Any other severe or uncontrolled disease or condition which mightincrease the risk associated with study participation. 4. Any other malignancy known to be active, with the exception of treated cervical intraepithelial neoplasia and non-melanoma skincancer. 5. Grade \>/= 3 non-hematologic toxicity from prior therapy that has notimproved to grade \</= 2. 6. Active hepatitis B, either active carrier (HBsAg +) or viremic (HBV DNA \>/=10,000 copies/mL, or \>/=2,000 IU/mL), or hepatitis C (detectable viral load by HCV RNA PCR). 7. Active infection requiring parenteral antibiotics. 8. HIV infection. 9. Treatment within prior 2 weeks with any anti-cancer agent,investigational or approved. 10. Active CNS involvement (untreatedparenchymal brain metastasis or positive cytology of cerebrospinal fluid). 11. Life expectancy \</= 6 months. 12. Active and uncontrolled neurological disorder. 13. Patients receiving systemic steroid therapy at time of enrollment (physiological replacement doses are allowed) or have received antithymocyte globulin or lymphocyte immune globulin within 14 days of enrollment or alemtuzumab within 28 days of enrollment. 14. Patients receiving immunosuppressive therapy. 15. Patients who are receiving any other investigational agents. 16. Patients with psychiatric illness/social situations that would limit compliance with study requirements.