Hyperbaric Oxygen Therapy and Allogeneic Peripheral Blood Stem Cell (PBSC) Transplant

Inclusion Criteria: * Voluntary written informed consent * Men or women, age ≥ 18 years of age, with upper limit of 75 years old. * Subjects with acute myeloid leukemia (AML) or Myelodysplastic Syndrome (MDS) for cohort 1. * Subjects with chronic myelomonocytic leukemia (CMML), atypical chronic myeloid leukemia (aCML), CML, chronic neutrophilic leukemia (CNL), myelofibrosis, and myelodysplastic/myeloproliferative (MDS/MPN) overlap syndrome for cohort 2. * Karnofsky performance status (KPS) of ≥ 70% * Patients should have New York Heart Association (NYHA) Functional Classification, Class I (ordinary physical activity does not cause undue fatigue, palpitation, dyspnea, or anginal pain) or Class II (ordinary physical activity results in fatigue, palpitation, dyspnea, or anginal pain). * Adequate hepatic, renal, cardiac and pulmonary function to be eligible for transplant. Minimum criteria include: Hepatic: ALT, AST \< 4x IULN and serum total bilirubin ≤ 2.0 mg/dL; Renal: serum creatinine: ≤ 2.0 mg/dL; Left ventricular ejection fraction ≥ 45% measured by 2D-ECHO or MUGA scan; EKG with no clinically significant arrhythmia; FEV1, FVC and DLCO ≥ 50% of predicted value (corrected to serum hemoglobin) * Women of child-bearing potential and men with partners of child-bearing potential must agree to use adequate contraception (hormonal or barrier method of birth control; abstinence) prior to study entry, for the duration of study participation, and for 30 days following completion of therapy. Should a woman or partner become pregnant or suspect she is pregnant while participating in this study, she should inform her treating physician and the investigator immediately. * A woman of child-bearing potential is any female (regardless of sexual orientation, having undergone a tubal ligation, or remaining celibate by choice) who meets the following criteria: Has not undergone a hysterectomy or bilateral oophorectomy; or Has not been naturally postmenopausal for at least 12 consecutive months (i.e., has had menses at any time in the preceding 12 consecutive months) * Women of child-bearing potential should have a negative urine or serum pregnancy test within 4 weeks of starting preparative regimen Exclusion Criteria: * Pregnant or breastfeeding * Severe chronic obstructive pulmonary disease requiring oxygen supplementation * History of spontaneous pneumothorax, prior chest surgery requiring thoracotomy or direct chest irradiation to the lungs * Evidence of pneumothorax or significant pulmonary fibrosis on chest imaging within 60 days of transplant. * Active malignancy excluding AML, MDS, CMML, aCML CML, CNL, MF and MDS/MPN overlap syndrome. * Active ear/sinus infection. Patients with chronic sinusitis or sinus headaches are excluded unless cleared by ear, nose, and throat specialist. * Recent sinus surgery (within the last 5 years). * Ear surgery excluding myringotomy or ear tubes * Subjects must agree to refrain from active tobacco or e-cigarette use 72 hours prior to transplant until complete transplant recovery. Nicotine replacement therapy is allowed. * Claustrophobia * History of recurrent seizures within 5 years of study enrollment. * Uncontrolled asthma * Uncontrolled viral or bacterial infection at the time of study enrollment * Active or recent (prior 6 months) invasive fungal infection without interdisciplinary (ID) consult and approval * Patients who had intrathecal chemotherapy within 2 weeks of starting preparative regimen or cranial irradiation within 4 weeks of starting preparative regimen