EPITOME-1015-I: a Study to Investigate the Safety and Tolerability of MDG1015 in Patients with Epithelial Ovarian Cancer, Gastroesophageal Adenocarcinoma, Round Cell Liposarcoma And/or Synovial Sarcoma

Inclusion Criteria: 1. Adult, ≥ 18 years of age and weigh ≥ 40 kg for Dose levels 1-3 and ≥ 50 kg for Dose level 4 2. Subject must have a confirmed diagnosis of either High grade serous or endometrioid ovarian, primary peritoneal or fallopian tube cancer Gastric or esophageal (junction) adenocarcinoma Myxoid (round cell) liposarcoma Synovial sarcoma 3. Subject's must have tested positive for HLA-A\*02:01 genotype by a Sponsor designated central laboratory 4. Subject's tumor must have tested positive for NY-ESO-1 and/or LAGE-1a mRNA expression by a Sponsor designated central laboratory Both ≤1 year old archival tissue or fresh biopsy are allowed 5. Subjects diagnosed with an eligible indication must have exhausted treatment options with proven survival benefit 6. Subjects must have 1. measurable disease 2. Life expectancy ≥ 3 months per Investigator's opinion 8\. Eastern Cooperative Oncology Group (ECOG) performance status 0-1 9. Adequate vital organ function 10. Adequate bone marrow function 11. Adequate coagulation profile 12. Toxicities from prior/ongoing therapies must have recovered to ≤ Grade 2 according to the CTCAE v5.0 or Subject's baseline excluding alopecia 14. Prior toxicities related to surgical procedures should have recovered to Grade ≤ 1 15. Women of childbearing potential (WCBP) or men who can father children must be willing and able to use adequate (e.g. barrier or licensed hormonal methods) Exclusion Criteria: 1. Any uncontrolled medical or psychiatric disorder that would preclude participation as outlined 2. HLA-A\*02:02 or HLA-A\*02:03 genotype 3. Pregnant or lactating women 4. Viral serology: 1. Known infection with HIV-1/2, CMV (CMV required only for U.S. sites) or HTLV-1/2, 2. Active infection with HBV or HCV 3. Positive test for Mycoplasma or Treponema Pallidum 5. Uncontrolled infection(s) requiring intravenous anti-bacterial, anti-viral or anti-fungal treatment within 14 days prior to the first dose of LDC (patients receiving prophylactic antibiotics are eligible) 6. Inadequate venous access for or contraindications to leukapheresis 7. Contraindications or life-threatening allergies, hypersensitivity, or intolerance to MDG1015 excipients, LDC agents, rasburicase, methylprednisolone or tocilizumab. 8. Untreated CNS metastases or active CNS metastases (progressing or requiring corticosteroids for symptoms control) and leptomeningeal disease 9. Unstable/active ulcer, varices, or digestive tract bleeding or recent digestive surgery that may have increased risk of bleeding 10. History of another primary malignancy that requires intervention beyond surveillance or that has not been in remission for at least 1 year. The following are exempt from the 1-year limit: 1. non-melanoma skin cancer 2. curatively treated localized prostate cancer 3. carcinoma in situ (e.g. cervix, bladder, breast) 11. NYHA Class ≥ II, heart failure, unstable angina, a history of recent (≤ 6 months) arrythmias, myocardial infarction or sustained (\> 30 seconds) ventricular tachyarrhythmias 12. Subjects who are dependent on dialysis 13. Subjects with a history of pulmonary embolism or deep vein thrombosis that cannot safely withhold anti-coagulant therapy from leukapheresis until 7 days after administration of MDG1015 as determined by the Investigator 14. Active autoimmune disease requiring systemic therapy except for adequately controlled Type 1 diabetes mellitus, autoimmune hypothyroidism or Grave's disease 15. Previous allogeneic hematopoietic stem cell transplant within the last 5 years or solid organ transplant Specific to GAC/GEJ Subjects: 16. Positive history of esophageal or gastric resection that the Investigator considers is at increased risk of bleeding or perforation