Clinical Study of Vorinostat in Combination With Etoposide in Pediatric Patients < 21 Years at Diagnosis With Refractory Solid Tumors

Inclusion Criteria: * Phase I Component: Histologic confirmation of relapsed/refractory solid tumors, including tumors of the central nervous system that have failed to respond to standard therapy, progressed despite standard therapy, or for which standard therapy does not exist. Patients with diffuse pontine glioma are not required to have histologic confirmation of disease, and are eligible with radiologic confirmation. Phase II Component: the population will be restricted to relapsed/refractory sarcomas * Patient must be between 4-21 years of age at the time of study enrollment. Efforts will be made to enroll patients \<13 years of age so that adequate information about the biologic effects of this agent in younger patients can be obtained. * Patient must have Karnofsky \> or = to 60% for patients \>10 years of age; Lansky Play Scale \> or = to to 60 for children \< or = to 10 years of age * Patient must have a life expectancy of \> 8 weeks. * There is no limit to the number of prior treatment regimens provided that performance status and life expectancy meet the criteria above. * Absolute neutrophil count (ANC) ≥ 1000 / mcL * Platelets ≥100,000 / mcL (transfusion not permitted) * Hemoglobin ≥ 9 g/dL qualifications (transfusion permitted) * Coagulation Prothrombin Time or INR ≤ 1.5x upper limit of normal (ULN) * Serum creatinine ≤ 1.5x upper limit of normal (ULN) OR calculated creatinine clearance ≥ 60 mL/min for patients with creatinine levels \> 1.5x institutional ULN. or calculated creatinine clearance Creatinine clearance should be calculated per institutional standard. * Serum total bilirubin ≤ 1.5 x ULN Patient's who don't meet this criteria must have a Direct bilirubin ≤ 1.5 x ULN * AST (SGOT) and ALT (SGPT) Alkaline Phosphatase (liver fraction) ≤ 2.5 x ULN. If AST or ALT is \> 2.5 x ULN, then the liver fraction of Alkaline Phosphatase should be ≤ 2.5 x ULN * Phase I component: Patients may have measurable or non-measurable disease. Phase II component: Patients may only have measurable disease. * Patient must have no persistent toxicities from prior therapy \> or = to Grade 2 with the exception of hematologic indices (i.e. hemoglobin, WBC, ANC, ALC). * For females of childbearing potential, a negative serum pregnancy test must be documented within 72 hours of receiving the first dose of vorinostat. * Patient, or the patient's legal representative, has voluntarily agreed to participate by giving written informed consent. * Female patients of childbearing potential must be willing to use 2 adequate barrier methods of contraception to prevent pregnancy or agree to abstain from heterosexual activity throughout the study, starting with visit 1. * Male patients must agree to use an adequate method of contraception for the duration of the study. * Prior Therapy: Patients must have fully recovered from the acute toxic effects of all prior chemotherapy, immunotherapy, or radiotherapy prior to entering this study. * Myelosuppressive chemotherapy: At least 2 weeks must have elapsed since the administration of previous therapy. Six weeks must have elapsed since administration of nitrosoureas or mitomycin C. Seven days must have elapsed since the administration of G-CSF and/or GM-CSF. * Biologic agents: At least 14 days must have elapsed since the completion of therapy with a biologic agent such as a monoclonal antibody. Seven days must have elapsed since the last dose of retinoids. * Radiation therapy (XRT): \> or = to 2 weeks must have elapsed for local XRT (small port); \> or = to 6 months must have elapsed if prior radiation to \> or = to 50% of the pelvis or if other substantial bone marrow irradiation, including total body irradiation. * Patient must be able to swallow capsules. * Patient must have an available archival/pre-treatment block or fresh tumor biopsy for molecular profiling to be performed. Exclusion Criteria: * A patient meeting any of the following criteria is not eligible to participate in this study: * Patients currently participating or has participated in a study with an investigational compound or device within 4 weeks of initial dosing with study drugs. * Patients with a prior history of treatment with HDAC inhibitors ( e.g. SNDX-275/entinostat, LAQ-824, LBH589, PXD-101/belinostat, etc). Patients who have received Valproic acid will be excluded from this study. * Patients with non CNS primary tumors who have known brain metastases or symptomatic CNS disease (e.g. cranial nerve abnormalities) without cytologic abnormality in the CSF should be excluded from this clinical trial because of their poor prognosis and known propensity for the development of progressive neurologic dysfunction that would confound the evaluation of neurologic and other adverse events. Patients with metastatic CNS tumors will not be excluded from enrollment on this study in the phase I component only. * Patients who have undergone prior autologous stem cell transplantation or allogeneic transplantation. * Uncontrolled intercurrent illness or circumstances that could limit compliance with the study requirements including, but not limited to: ongoing or active bacterial or fungal infection, acute or chronic graft versus host disease, symptomatic congestive heart failure, cardiac arrhythmia, or psychiatric illness/social situations. * Patients who are pregnant or breastfeeding, or expecting to conceive within the projected duration of the study. Because there is an unknown but potential risk for adverse events in nursing infants secondary to treatment of the mother with vorinostat, lactating patients will be excluded from this study. * Patients known to be Human Immunodeficiency Virus (HIV)-positive. * Patients with known hypersensitivity to the components of the study drugs or their analogs. * Patients with symptomatic ascites or pleural effusion. A patient who is clinically stable following treatment for these conditions is eligible. * Patients who are at the time of signing informed consent, a regular user of any illicit drugs, substance abuser or who have a recent history of drug or alcohol abuse. * Patients with a known history of Hepatitis B or C. * Patients who have a history of gastrointestinal surgery or other procedures that might in the opinion of the investigator, interfere with the absorption or swallowing of the study drug. * Patients who are unable to take or tolerate oral medications on a continuous basis. * Patients with a history of a prior malignancy who have undergone potentially curative therapy with no evidence of that disease for five years or patients, who are deemed low risk for recurrence by his/her treating physician are permitted to enroll.