Abemaciclib and Letrozole in Patients With Estrogen Receptor-positive Rare Ovarian Cancer

Inclusion Criteria: 1. Voluntary written informed consent of the participant or their legally authorized representative has been obtained prior to any screening procedures. 2. Use of highly effective methods of birth control; defined as those that, alone or in combination, result in low failure rate (i.e., less than 1% per year) when used consistently and correctly; such as implants, injectables, combined oral contraceptives, some IUDs, true sexual abstinence (i.e. refraining from heterosexual intercourse during the entire period of risk associated with the Trial treatment(s)) or commitment to a vasectomised partner. 3. Histological confirmation of diagnosis of low-grade serous (original diagnosis of low-grade serous carcinoma or original diagnosis of serous borderline tumor with subsequent diagnosis of low-grade serous carcinoma )or low-grade endometrioid carcinoma of ovary, fallopian tube or peritoneum or granulosa-cell tumor of the adult type and ER positivity on immunohistochemistry. In order to prevent inclusion of patients with high-grade serous carcinoma, diagnosis of low-grade serous carcinoma will be verified as part of screening review by a gynecologic pathologist. Tissue for confirmation can be from primary tumor or recurrence. 4. For Stage 1: only patients where platinum is still an option are eligible with no limitations in prior chemotherapy regimens and a maximum of 2 prior endocrine therapy regimens. For Stage 2: a further 20 patients where platinum is still an option will be included, with no limitations in prior chemotherapy regimens and a maximum of 2 prior endocrine therapy regimens. Fifteen patients where platinum is not an option are allowed with no limitations in prior chemotherapy regimens and maximum of 2 prior endocrine therapy regimens. Patients cannot have received chemotherapy for platinum resistant or refractory disease. 5. Age \> 18 years at time of study entry. 6. Eastern Cooperative Oncology Group (ECOG) performance status (PS) 0-1. 7. Patient must have recurrent, measurable disease by RECIST v1.1. Measurable disease is defined as at least one lesion that can be accurately measured in at least 1 dimension (longest dimension to be recorded). Each lesion must be ≥10 mm when measured by computed tomography (CT), magnetic resonance imaging (MRI), or caliper measurement by clinical exam or must be ≥20 mm when measured by chest x-ray. Lymph nodes must be \>15 mm in short axis when measured by CT or MRI. 8. Pre- and post-treatment tissue biopsy and ct-DNA blood sample are mandatory for translational studies. Tissue from an archival tissue sample or fresh tissue obtained from a core or excisional biopsy of a tumor lesion. 9. Patients who were previously treated with letrozole or another aromatase inhibitor are allowed, but capped at 10 patients in each cohort. 10. Patients who received radiotherapy must have completed and fully recovered from the acute effects of radiotherapy. A washout period of at least 14 days is required between end of radiotherapy and randomization. 11. Patients must not have remaining ovarian function. In women who have at least one retained ovary, menopause must be confirmed with laboratory confirmation. Women who have ovarian function are eligible but must be placed on hormonal suppression after a negative serum or urine human chorionic gonadotropin (hCG) test. 12. Abnormal organ function is permitted. However, patients must have: 1. absolute neutrophil count ≥1500/mL 2. platelets ≥100.000/mL 3. hemoglobin ≥9 g/dL 4. estimated creatinine clearance ≥ 45 ml/min as calculated using the method standard for the institution 5. total serum bilirubin ≤1.5 X ULN 6. aspartate aminotransferase (AST/SGOT) and/or alanine aminotransferase (ALT/SGPT) ≤3 X ULN 7. alkaline phosphatase ≤2.5x ULN (or ≤5.0x ULN if liver or bone metastases) Exclusion Criteria: 1. For Stage 1: patients where platinum is not an option and platinum refractory patients are not allowed. For Stage 2: patients with platinum refractory disease are not allowed. Patients who received chemotherapy must have recovered (Common Terminology Criteria for Adverse Events \[CTCAE\] Grade ≤1) from the acute effects of chemotherapy except for residual alopecia or Grade 2 peripheral neuropathy prior to randomization. A washout period of at least 21 days is required between last chemotherapy dose and randomization (provided the patient did not receive radiotherapy). 2. The patient has serious preexisting medical condition(s) that would preclude participation in this study (for example, interstitial lung disease, severe dyspnea at rest or requiring oxygen therapy, severe renal impairment (e.g. estimated creatinine clearance \<30 mL/min), history of major surgical resection involving the stomach or small bowel, or preexisting Crohn's disease or ulcerative colitis or a preexisting chronic condition resulting in baseline Grade 2 or higher diarrhea). 3. Current use of food or drugs known to be potent CYP3A4 inhibitors, drugs known to be potent CYP3A4 inducers (for examples, see the Prohibited Concomitant Medications section). 4. Diagnosis of another malignancy within 3 years, except for adequately treated basal cell or squamous cell skin cancer, or carcinoma in situ of the cervix. 5. Patient cannot have previously received a prior cyclin dependent kinase inhibitor (CDKi). 6. Known Hepatitis B, Hepatitis C or human immunodeficiency virus (HIV) infection. 7. Inability or unwillingness to swallow pills. 8. Patient has had major surgery within 14 days prior to starting study drug or has not recovered from major side effects (tumor biopsy is not considered as major surgery). 9. Active infection requiring intravenous (IV) antibiotics or antifungals, or other uncontrolled recurrent illness requiring hospitalization. 10. History of any of the following: syncope of cardiovascular etiology, ventricular arrhythmia of pathological origin (including, but not limited to, ventricular ta