Brief Summary: The subject is invited to take part in this research study because s/he has been diagnosed with Diffuse Large B-Cell Lymphoma (DLBCL), Primary Mediastinal B-cell Lymphoma (PMBCL), or Post-transplant Lymphoproliferative Disorder (PTLD). In an attempt to improve cure rates while reducing harmful effects from drugs, oncologists are developing new treatment protocols. One such protocol, entitled dose-adjusted EPOCH-R, utilizes two major new strategies. First, the treatment approach utilizes continuous infusion of chemotherapy over four days, instead of being administered over minutes or hours. Secondly, the doses of some medications involved are increased or decreased based on how the drugs affect the subject's ability to produce blood cells, which is used as a measure of how rapidly the body is processing drugs. Using this approach in adults, researchers have shown improved cure rates in these cancers. Additionally, the harmful effects experienced by patients has been mild, with mucositis, severe infections, and tumor lysis syndrome occurring rarely. However, this new dosing method has never been used in children, and the effectiveness and side effects of this new method are unknown in children. The purpose of this study is to look at the safety of dose-adjusted EPOCH-R in the treatment of children with mature B-cell cancers, and to see if we can maintain cure rates (as has been shown in adults). This study represents the first trial of dose-adjusted EPOCH-R in children.

Detailed Description: The subject will need to have a variety of tests, exams, or procedures to find out if s/he can be on the study. The subject will also require placement of a catheter that stays in the vein for safe administration of chemotherapy drugs. During the study... If all of the tests that have been done show that s/he can participate and s/he chooses to participate, treatment cycles will begin. A cycle equals three weeks. The subject will have a minimum of 6 cycles of treatment, possibly 8. The cancer drugs s/he will receive are etoposide, vincristine, doxorubicin, cyclophosphamide, prednisone, and rituximab with each cycle. The amount of the drugs will be determined by the subject's weight at first, and some of the drugs will be adjusted up or down for later cycles. Rituximab will be given on Day 1 prior to continuous infusion drugs. After 4 cycles, the subject will have imaging scans again to see how the cancer responded to treatment. If the cancer responds completely after 4 cycles, s/he will get 2 more cycles (6 cycles total). If the cancer partially responds, s/he will get 4 more cycles (8 total cycles). If the subject has cancer in the Central Nervous System, s/he will receive a drug called methotrexate. If s/he does not have cancer in the CNS, s/he will get methotrexate to try to prevent CNS cancer.