Study of Lenvatinib in Combination With Everolimus in Recurrent and Refractory Pediatric Solid Tumors, Including Central Nervous System Tumors

Inclusion Criteria * ≥2 years and \<18 years of age for enrolment in Phase 1 or ≥2 years and ≤21 years of age for enrolment in Phase 2. * Recurrent or refractory solid tumors * Phase 1: All solid tumors (measurable or evaluable disease), including primary central nervous system (CNS) tumors; exclusion of hepatoblastoma and lymphomas. Participants with diffuse intrinsic pontine glioma, optic pathway glioma, or pineal tumors with elevated tumor markers (alpha-fetoprotein \[AFP\] and beta-human chorionic gonadotropin \[ß-hCG\]\[or human chorionic gonadotropin \[hCG\])do not require histological or cytological confirmation of diagnosis * Phase 2: Ewing sarcoma, Rhabdomyosarcoma, High Grade Glioma (HGG) (all must have measurable disease); exclusion of Diffuse Intrinsic Pontine Glioma * Histologically or cytologically confirmed diagnosis * Measurable disease that meets the following criteria (Phase 2): 1. RECIST 1.1 (for all tumor types except HGG): At least 1 lesion of ≥1.0 cm in the longest diameter for a non lymph node or ≥1.5 cm in the short-axis diameter for a lymph node which is serially measurable according to RECIST 1.1 using computed tomography /magnetic resonance imaging (CT/MRI) 2. Response Assessment in Neuro-Oncology (RANO) for high grade glioma (HGG): At least one lesion must be measurable as defined as a bi dimensionally contrast enhancing lesion with clearly defined margins by CT or MRI scan, with a minimal diameter of 1 cm, and visible on 2 axial slices which are preferably at most 5 mm apart with 0 mm skip Lesions that have had external beam radiotherapy (EBRT) or locoregional therapies such as radiofrequency (RF) ablation must show evidence of progressive disease based on RECIST 1.1 to be deemed a target lesion * Karnofsky performance score ≥50 for participants\>16 year of age and Lansky play score ≥50 for participants ≤16 years of age. Neurologic deficits in participants with CNS tumors must have been relatively stable for at least 7 days prior to study enrollment. Participants who are unable to walk because of paralysis, but who are up in a wheelchair, will be considered ambulatory for the purpose of assessing the performance score * Prior Therapy * Participants must have fully recovered from the acute toxic effects of all prior anti-cancer therapy * Cytotoxic chemotherapy or other chemotherapy known to be myelosuppressive: ≥21 days after the last dose of cytotoxic or myelosuppressive chemotherapy (42 days if prior nitrosourea) * Anti-cancer agents not known to be myelosuppressive (eg, not associated with reduced platelet or absolute neutrophil counts): ≥7 days after the last dose of agent * Monoclonal antibodies: ≥21 days or 3 half-lives (whichever is shorter) of the antibody must have elapsed after the last dose of a monoclonal antibody (including checkpoint inhibitors). Toxicity related to prior antibody therapy must be recovered to Grade ≤1 * Corticosteroids: If used to modify immune adverse events related to prior therapy, ≥14 days must have elapsed since last dose of corticosteroid. Participants receiving corticosteroids, who have not been on a stable or decreasing dose of corticosteroid for at least 7 days prior to enrollment, are not eligible * Hematopoietic growth factors: ≥14 days after the last dose of a long-acting growth factor or 7 days for short-acting growth factor. For agents that have known adverse events occurring beyond 7 days after administration, this period must be extended beyond the time during which adverse events are known to occur * Interleukins, interferons, and cytokines (other than hematopoietic growth factors): ≥21 days after the completion of interleukins, interferons or cytokines (other than hematopoietic growth factors) * Stem cell infusions (with or without total body irradiation): Allogeneic (non-autologous) bone marrow or stem cell transplant, or any stem cell infusion including donor leukocytes infusion or boost infusion: ≥84 days after infusion and no evidence of graft versus host disease; Autologous stem cell infusion including boost infusion: ≥42 days * Cellular Therapy: ≥42 days after the completion of any type of cellular therapy (eg, modified T cells, natural killer cells, dendritic cells, etc) * Radiotherapy (XRT)/External Beam Irradiation including Protons: ≥14 days after local XRT; ≥150 days after total body irradiation, craniospinal XRT or if radiation to ≥50% of the pelvis; ≥42 days if other substantial bone marrow radiation. * Radiopharmaceutical therapy: ≥42 days after systemically administered therapy. * Vascular endothelial growth factor (VEGF)/VEGF receptor (VEGFR)-targeted or mammalian target of rapamycin (mTOR)-targeted therapies: Must not have received prior exposure to lenvatinib; May have previously progressed on an mTOR inhibitor; No more than 2 prior VEGF/VEGFR-targeted therapies (For Phase 2 only); Must not have received prior VEGF/VEGFR-targeted therapy in combination with an mTOR inhibitor (For Phase 2 only) * Adequate bone marrow function for participants with solid tumors without known bone marrow involvement * Adequate bone marrow function for participants with known bone marrow metastatic disease * Adequate renal function * Adequate liver function * Adequate cardiac function * Adequate neurologic function * Adequate blood pressure (BP) control with or without antihypertensive medications * Adequate coagulation * Adequate pancreatic function * Adequate metabolic function * Adequate glycemic control * Participants must have a minimum body surface area (BSA) of 0.6 m\^2 at study entry. Exclusion Criteria * Participants who have had or are planning to have the following invasive procedures * Major surgical procedure, laparoscopic procedure, open biopsy or significant traumatic injury within 28 days prior to enrolment * Central line placement or subcutaneous port placement is not considered major surgery. External central lines must be placed at least 3 days prior to enrollment and subcutaneous ports must be placed at least 7 days prior to enrollment * Fine needle aspirate within 7 days prior to enrolment * Surgical or other wounds must be adequately healed prior to enrolment * For purposes of this study, bone marrow aspirate and biopsy are not considered surgical procedures and therefore are permitted within 14 days prior to start of protocol therapy * Participants who have non-healing wound, unhealed or incompletely healed fracture, or a compound (open) bone fracture at the time of enrolment * Participants having an active infection requiring systemic therapy. * Participants with a known history of active hepatitis B (defined as hepatitis B surface antigen reactive or hepatitis B virus- deoxyribonucleic \[DNA\] detected) or known active hepatitis C virus (HCV, defined as HCV- Ribonucleic acid \[RNA\] detected). Note: No testing for hepatitis B and hepatitis C is required unless mandated by the local health authority. * Known to be human immunodeficiency virus (HIV) positive. Note: HIV testing is required at screening only when mandated by the local health authority * Clinical evidence of nephrotic syndrome prior to enrolment * Gastrointestinal bleeding or active hemoptysis (bright red blood of at least half teaspoon) within 21 days prior to enrolment * Thrombotic/ thromboembolic event requiring systemic anticoagulation within 90 days prior to enrollment * Evidence of new intracranial hemorrhage of more than punctate size on MRI assessment obtained within 28 days prior to study enrollment for Participants with HGG * Diagnosis of lymphoma * Radiographic evidence of major blood vessel invasion/infiltration. * Evidence of untreated CNS metastases (exception: participants with primary CNS tumors and leptomeningeal disease) * Participants who are currently receiving enzyme-inducing anticonvulsants * Participants chronically receiving strong cytochrome P450 3A4 (CYP3A4)/P-glycoprotein (P-gp) inhibitors or inducers within 7 days prior to study enrollment * Females who are breastfeeding or pregnant. For females of childbearing potential, a negative screening pregnancy test must be obtained within 72 hours before the first dose of study drug * Males who have not had a successful vasectomy (confirmed azoospermia) or if they and their female partners do not meet the criteria above (that is, not of childbearing potential or practicing highly effective contraception throughout the study period and for 7 days after lenvatinib discontinuation or 4 weeks after discontinuation of everolimus). No sperm donation is allowed during the study period and for 7 days after lenvatinib discontinuation or 4 weeks after discontinuation of everolimus.