The purpose of this research study is to determine whether taking either of two low dose drugs that would prevent new blood vessels from growing after stem cell transplant is feasible, and what the side effects of taking each of these drugs after autologous transplant might be. The reason the investigators are looking at these drugs is because one of the things that allows tumors to grow quickly is their ability to stimulate the growth of new blood vessels. By suppressing the growth of new blood vessels after stem cell transplant, the investigators hope to prevent the tumors from coming back or continuing to grow.

Inclusion Criteria:

• Patient must have an original diagnosis, benefited by autologous transplantation, confirmed by biopsy* of high-grade glial tumor, low-grade glial tumor, ependymoma, medulloblastoma, primitive neuro-ectodermal tumor (PNET), Wilms' tumor, rhabdomyosarcoma, Ewing's sarcoma, retinoblastoma, or miscellaneous poor-prognosis solid tumors. Lymphomas and other lymphoid malignancies will not be studied in this protocol.

  * Brain stem glioma patients who have progressed after radiation therapy do not require histologic confirmation. Brain stem gliomas are defined as intrinsic tumors of the pons causing diffuse enlargement. These patients are diagnosed on clinical and radiographic appearance of the lesion and the biopsy requirement will be waived for this group.

• Patient must be ≥ 6 months of age and ≤ 21 years of age at the time of study entry.

• Patient must have a Karnofsky performance status or Lansky* play status ≥ 50

  * For purpose of determining performance scores, wheelchair-bound patients will be considered ambulatory.

• Patient must have an adequate supply of stem cells for transplant harvested prior to study enrollment, with adequate supply defined as 3 x 10^6 CD34+ cells/kg for peripheral blood stem cells (PBSC). Cell mobilization method will be left up to the treating physician's discretion and may include mobilization growth factor alone or mobilization after chemotherapy. If patient is unable to mobilize the proper amount of peripheral stem cells, bone marrow may be harvested as the source of hematopoietic stem cells. In this instance, 3 x 10^8 mononuclear cells/kg will be considered adequate. If necessary, a combination of peripheral stem cells and bone marrow can be used.

• Prior radiation therapy and/or chemotherapy, including cyclophosphamide, are permitted.

• Prior anti-angiogenic therapy, including thalidomide and oral cyclophosphamide, is permitted.

• If on corticosteroids for mass effect and/or edema related to the tumor, patient must be on a stable or decreasing dose for at least 2 weeks prior to study entry.

• Patient must have a life expectancy > 3 months.

• Patient must have an adequate bone marrow reserve as defined by:

  • Hemoglobin ≥ 8.0 g/dl and
  • Peripheral absolute neutrophil count (ANC) ≥ 750/mm3

• Patient must have adequate cardiac function tested within 4 weeks of study enrollment as defined by:

  • Shortening fraction of ≥ 27% by echocardiogram or
  • Ejection fraction of ≥ 50% by radionuclide angiogram

• Patient must have adequate pulmonary function tested within 4 weeks of study enrollment as defined by:

  • Pulse oximetry > 94% on room air or O2 by nasal cannula and
  • No evidence of dyspnea at rest.

• Patient must have adequate hepatic function as defined by:

  • Total bilirubin ≤ 1.5x upper limit of normal (ULN) for age and
  • SGOT (AST) or SGPT (ALT) ≤ 2.5 x ULN (SGOT ≤ 4x ULN if on Zantac)

• Patient must have adequate renal function as defined by:

  • Serum creatinine < 1.5 mg/dl
  • Glomerular filtration rate (GFR), calculated via I-125 iothalamate clearance, 24-hour creatinine clearance, or Schwartz formula*, ≥ 70 mL/min and ≥ 50 mL/min/1.73 m2 done within 4 weeks of study entry
  • The Schwartz formula is an estimated glomerular filtration rate in children based upon serum creatinine and height. Height (Ht) should be measured in cm and serum creatinine (Cr) in mg/dL. Proportionality constant (k) is 0.55 for children and adolescent girls and 0.7 for adolescent boys aged 13-21. This constant is based upon a series of evaluations performed by Schwartz. Formula: GFR= (k x Ht)/Cr

• Enrollment in the Celgene THALOMID REMSTM Program:

  1. If enrolled in Arm III of this study, patient must be registered at the Celgene THALOMID REMSTM Program prior to day +30 post-ASCR.

  2. If enrolled in Arm III of this study, patient must be willing to practice birth control as outlined in the THALOMID REMSTM Program from the beginning of the thalidomide treatment until at least 4 weeks following discontinuation of thalidomide therapy. Two reliable forms of contraception must be used simultaneously unless continuous abstinence from heterosexual sexual contact is chosen. Contraceptive methods must include at least one highly effective method (e.g. oral contraceptive pills, injections, hormonal patches, IUD, or implants), AND one additional effective barrier method (e.g. latex condom, diaphragm, cervical cap).

     If hormonal or IUD contraception is medically contraindicated, another highly effective method or two barrier methods must be used at the same time.

  3. Pregnancy surveillance:

  i. Patient must have a negative in office pregnancy test sensitive to within 50 mIU/mL (serum or urine) within 24 hours prior to beginning thalidomide even if continuous abstinence is the preferred method of birth control. ii. A pregnancy test must be performed weekly during the first 4 weeks of therapy and repeated monthly for patients with regular menses or every 2 weeks for patients with irregular menses iii. Negative pregnancy tests are valid for only 7 days. iv. If irregular bleeding or skipped menses, pregnancy test should be performed and pregnancy counseling given. v. If pregnancy occurs during treatment, thalidomide must be immediately discontinued. Any suspected lethal exposure must be reported immediately to Celgene Customer Care Center at 1-888-423-5346, and the patient referred to an OB/GYN experienced in reproductive toxicity for further evaluation and counseling.

• Patient (or legally authorized representative) must be able to understand and willing to sign a written informed consent document.

Exclusion Criteria:

• Patient must not have any active, uncontrolled cardiac, hepatic, renal, or psychiatric disease defined as ≥ grade 3 based on NCI Common Terminology Criteria for Adverse Events v4.0 (CTCAE).
• Patient must not be receiving any other investigational agents.
• Patient must not have any active infection or concurrent illness obscuring toxicity or dangerously altering drug metabolism.
• Patient must not have any thromboembolic event (deep vein thrombosis or pulmonary embolism) less than 3 weeks prior to enrollment.
• Patient must not have a history of allergic reactions attributed to compounds of similar chemical or biologic composition to any of the agents used in the study.
• Patient must not be pregnant or breastfeeding.