Romosozumab as an Adjunct to Physiologic Estrogen Replacement in Functional Hypothalamic Amenorrhea

Inclusion Criteria: For FHA and controls: * Female, age 14-30 years, skeletally mature with bone age ≥ 12 years * For women of reproductive age, agree to use one of the following for the study duration: 1. an effective non-hormonal contraceptive method 2. a progestin releasing intrauterine device (no evidence of systemic skeletal effects) 3. a progestin releasing implant 4. does not have a history of sexual activity that could lead to pregnancy (i.e., same-sex partners only or total abstinence has been and is their preferred lifestyle) * Biochemical criteria: * Negative βHCG (pregnancy test) * TSH within twice the upper limit of normal; potassium, magnesium within the normal range; prolactin \<10 ng/mL above the upper limit of normal; FSH not elevated. * Serum ALT ≤ 3 times upper limit of normal, LDL ≤ 190 mg/dl * eGFR ≥ 30ml/minute * If the diagnosis of FHA is unclear, we may check additional labs (e.g., testosterone and sex hormone binding globulin if there is a suspicion of PCOS based on clinical hyperandrogenism). Additional inclusion criteria for FHA: * Less than 3 menses in the preceding 6 months * BMD Z-score ≤ -1.0 at ≥ 1 skeletal site (for subjects \<18 years old, we will use the height Z-score-adjusted BMD Z-score using the pediatric bone density calculator developed by the National Institutes of Health and currently maintained by the Children's Hospital of Philadelphia) * Dental check-up within the past year * If the menstrual status of the subject is unclear due to the presence of a progestin-releasing IUD, serum estradiol levels will be checked twice, at least one week apart. Both estradiol levels must be \< 50 pg/mL.At the discretion of the study clinician, subjects with a diagnosis of PCOS may undergo one to two serum estradiol levels, at least one week apart. The estradiol level(s) must be \< 50 pg/mL Exclusion Criteria: For FHA and controls * Disease other than FHA known to affect bone, including untreated thyroid dysfunction, Cushing's disease, renal failure, diabetes mellitus * Use of bisphosphonates in the past year * Use of other medications known to affect bone metabolism within 3 months of the study (other than calcium and vitamin D supplementation). * Current use of systemic corticosteroids * Migraine with aura as classified in IHS Classification ICHD-3. * Personal history of or first-degree relative with unprovoked thromboembolism (unless the subject has been tested and ruled out for a hypercoagulable state). * Active substance use disorder; currently smokes or vapes * History of malignancy or Paget disease of bone * Pregnant, planning to become pregnant within 12 months after the end of treatment and/or breastfeeding Additional exclusion criteria for FHA * Cardiovascular: History of myocardial infarction or stroke; history of hypertension or use of anti-hypertensive medications * Immunodeficiency or taking immunosuppressive therapy * Other conditions that can cause oligo-amenorrhea such as PCOS, primary ovarian insufficiency. A participant with PCOS may be included if a study clinician determines that missed menstrual periods are more likely a consequence of FHA. * Dental: Osteonecrosis of the jaw (ONJ) or risk factor for ONJ, such as invasive dental procedures (tooth extraction, dental implants, oral surgery in the past 3 months), poor oral hygiene, periodontal and/or pre-existing dental disease * Planned invasive dental procedure or other planned major surgery for 18 months after the baseline visit * Known sensitivity or absolute contraindication to any of the products or components of the medications to be administered (romosozumab, zoledronic acid, transdermal estradiol, micronized progesterone, calcium or vitamin D supplements) * Concerning EKG findings for ischemia Additional exclusion criteria for normal-weight healthy controls * BMD Z-score \<-2.5 (who we will refer for evaluation)