A Study of Pomalidomide Monotherapy for Children and Young Adults With Recurrent or Progressive Primary Brain Tumors

Inclusion Criteria: 1. Subject is 1 to \< 21 years of age at the time of signing the Informed Consent Form/Informed Assent Form (ICF/IAF). 2. Subject (when applicable, parental/legal representative) must understand and voluntarily sign an ICF/IAF prior to any study-related assessments/procedures being conducted. 3. Subject has received at least one prior standard therapy (or generally accepted upfront therapy if no standard exists) and have no known curative therapy. 4. Subject has a diagnosis of high-grade glioma, medulloblastoma, ependymoma or diffuse intrinsic pontine glioma (DIPG) that is recurrent or progressive. Subjects with neurofibromatosis type 1 (NF-1) associated tumors are eligible if they meet all other eligibility criteria. 5. Subject has histological verification of tumor either at the time of diagnosis or recurrence. Subjects with DIPG are exempt from histologic verification if they have typical magnetic resonance imaging (MRI) findings of DIPG 6. Subject has measurable disease defined as a tumor that is measurable in 2 perpendicular diameters on MRI. For a lesion to be considered measurable, it must be at least twice the slice thickness on MRI (ie, visible on 2 or more axial slices) 7. To document the degree of tumor at study baseline, the following scan(s) must be obtained: * A brain MRI with and without contrast (ie, gadolinium) and a spine MRI with contrast within 21 days prior to first dose of study treatment. For subjects on steroids, baseline MRI scans must be performed while on stable or decreasing dose of steroids for at least 5 days. 8. Subject has Karnofsky (age ≥ 16 years) or Lansky (age \< 16 years) performance status score ≥ 50 at screening 9. Subject has adequate bone marrow function defined as: * Peripheral absolute neutrophil count (ANC) ≥ 1000/mm³ * Platelet count ≥ 100,000/mm³ (transfusion independent defined as no platelet transfusion within 7 days and recovery from nadir) * Hemoglobin ≥ 8 g/dL (red blood cell \[RBC\] transfusion is allowed) 10. Subject has adequate renal function defined as: * Serum creatinine based on age/gender calculated using the Schwartz formula, or a 24-hour creatinine clearance or radioisotope glomerular filtration rate (GFR) (radioisotope or iothalamate) ≥ 70 mL/min/1.73 m². 11. Subject has adequate liver function defined as: * Total bilirubin ≤ 1.5 X upper limit of normal (ULN) for current age (≤ 3 X ULN if increase in bilirubin is attributable to Gilbert's Syndrome) * Alanine aminotransferase (ALT) (SPGT) is ≤ 3 X ULN for age * Serum albumin ≥ 3 g/dL 12. Subject has adequate pulmonary function defined as: * No evidence of dyspnea at rest * A pulse oximetry ≥ 93% 13. Subject has recovered from clinically significant acute treatment related toxicities from all prior therapies. Recovery is defined as a toxicity Grade ≤ 2 (common terminology criteria for adverse events \[CTCAE\] v. 4.03). 14. Subject has no significant worsening in clinical status for a minimum of 7 days prior to first dose of study drug. 15. Subject (and when applicable, with parental/legal representative) is willing and able to adhere to the study visit schedule and other protocol requirements. 16. Females of Childbearing Potential (FCBP) and male subjects who have reached puberty (and when applicable, with parental/legal representative) must agree to undergo physician-approved reproductive education and discuss the side effects of the study therapy on reproduction. 17. Females of childbearing potential must agree and meet the following conditions below: * Medically supervised (ie, performed in a clinic) pregnancy testing, including those who commit to true abstinence. Two pregnancy tests must be conducted prior to starting pomalidomide. The first pregnancy test must be performed 10 to 14 days prior to the start of pomalidomide and the second pregnancy test must be performed within 24 hours prior to starting pomalidomide. Females of childbearing potential with regular or no menstrual cycles must also agree to have pregnancy tests weekly for the first 28 days study participation, every 28 days while on study, at study treatment discontinuation, and at Day 28 following pomalidomide discontinuation. If menstrual cycles are irregular, the pregnancy testing must occur weekly for the first 28 days of study participation and then every 14 days while on study, at study treatment discontinuation visit, and at Days 14 and 28 following pomalidomide discontinuation. * Female subjects must, as appropriate to age and at the discretion of the study Investigator, either commit to true abstinence from heterosexual contact and/or agree to the use of two reliable forms of approved and effective contraceptive methods simultaneously. The two methods of reliable contraception must include one highly effective method (ie, oral, injectable, or implantable hormonal contraceptive; tubal ligation; intra-uterine device; vasectomized partner) and one additional effective barrier method (ie, male condom, diaphragm, cervical cap) 28 days prior to starting pomalidomide, throughout the entire duration of study treatment including dose interruptions and 28 days after discontinuation of pomalidomide. * All male and female subjects must follow all requirements defined in the pomalidomide Pregnancy Prevention Program. 18. Male subjects must, as appropriate to age and the discretion of the study physician: * Practice true abstinence or agree to use a condom during sexual contact with a pregnant female or a female of child bearing potential while participating in the study, during dose interruptions and for at least 28 days following pomalidomide discontinuation, even if he has undergone a successful vasectomy or practices complete abstinence. Exclusion Criteria: 1. Subject has a history of non-central line related thrombosis (arterial or venous), more than one prior central-line related thrombosis or known coagulopathy. 2. Subject has first degree family member with a known hereditary coagulopathy. 3. Subject is actively on anticoagulation therapy. 4. Subject has had major (per Investigator discretion) surgery, with the exception of tumor resection, within 21 days from first dose of study drug. 5. Subject has previously received (presence of any of the following will exclude a subject from enrollment): * Any prior treatment with pomalidomide. Subjects who have prior treatment with other immunomodulatory compounds (thalidomide, lenalidomide) are eligible if they meet all other eligibility criteria and did not have allergic reactions or other "significant toxicity" per Investigator discretion associated with lenalidomide or thalidomide use. * Myelosuppressive chemotherapy, immunotherapy, or any investigational agent: ≤ 21 days (≤ 42 days if a nitrosourea) prior to screening. * Biological (anti-neoplastic) therapy: ≤ 7 days prior to screening. * Immunomodulatory therapy: ≤ 28 days prior to screening. * Monoclonal antibody treatment and agents with known prolonged half-lives: \< 3 halflives have elapsed or ≤ 28 days prior to screening, whichever is longer. * Prior radiation: * Cranial irradiation, total body irradiation (TBI), or ≥ 50% radiation of pelvis ≤ 3 months prior to screening. * Focal irradiation: ≤ 3 weeks prior to screening if radiation field involved a nontarget lesion; ≤ 6 weeks prior to screening if radiation field involved a target lesion. Note: True disease progression following prior irradiation therapy must be confirmed by Investigator prior to screening. * Bone marrow transplant: * Presence of graft versus host disease (GVHD). * \< 6 months since allogeneic bone marrow transplant prior to screening. * \< 3 months since autologous bone marrow/stem cell transplant prior to screening. * \< 3 months since stem cell transplant (SCT) or Rescue without TBI with no evidence of GVHD prior to screening. * Radioisotopes: fluorothymidine (18FLT) ≤ 72 hours prior to first dose of study drug 6. Subject has received therapy with a known moderate to potent CYP1A2 inhibitor within 14 days or 5 half-lives of first dose of study treatment (whichever is longer). 7. Subject has received colony-stimulating growth factor(s) within 7 days prior to screening (or within 14 days if subject received polyethylene glycol formulations). 8. Subject is pregnant, breast-feeding or lactating. 9. Subject has an untreated or uncontrolled infection defined as ongoing signs/symptoms related to the infection without improvement despite appropriate antibiotics, antiviral therapy and/or other treatment. 10. Subject has active infectious hepatitis, type A, B, or C, or chronic carriers of hepatitis C. 11. Subject has any prior history of malignancies, other than high-grade glioma, medulloblastoma, ependymoma or DIPG (Note: radiation-associated gliomas are excluded from enrollment) 12. Subject who, in the opinion of the Investigator, has any significant medical condition, laboratory abnormality, or psychiatric illness that would prevent the subject from participating in the study. 13. Subject has any condition including the presence of laboratory abnormalities which, in the opinion of the Investigator, places the subject at unacceptable risk if he/she were to participate in the study. 14. Subject has any condition that confounds the ability to interpret data from the study. 15. Subject has symptomatic cardiac disorders (CTCAE v. 4.03 Grade 3 and 4).