Patient description: A 16-year-old girl came to the clinic complaining of weight gain and abnormal menstrual cycles. Her BMI was 24, but she has gained 5 kg in the past few weeks. She gets tired more frequently and does not have energy to go dancing with her friends. Her lab results were remarkable for high TSH levels (15 mU/L) and low free T4 levels (0.18 ng/dl). Her anti-TPO levels were extremely high (136 IU/ml). She was diagnosed with Hashimoto disease. She does not smoke and she is not sexually active.

Clinical trial eligibility criteria: eligible ages (years): 2.0-25.0,  Tumors Solid Malignant Tumors Osteosarcoma Differentiated Thyroid Cancer (DTC) Histologically or cytologically confirmed diagnosis of solid malignant tumor. 1. Cohort 1: Any solid malignant tumor. 2. Cohort 2A: Differentiated Thyroid Cancer (DTC) with one of the following histological subtypes: i) Papillary thyroid cancer (PTC). i.a) Follicular variant. i.b) Other variants (including but not limited to tall cell, columnar cell, cribriform-morular, solid, oxyphil, Warthin's-like, trabecular, tumor with nodular fasciitis-like stroma, Hurthle cell variant of papillary carcinoma, or poorly differentiated carcinomas). ii) Follicular thyroid cancer (FTC). ii.a) Hurthle cell. ii.b) Clear cell. ii.c) Insular. c) Cohort 2B, 3A, and 3B: Relapsed or refractory osteosarcoma. 2. Relapsed or refractory solid tumor malignancy that has progressed on standard anticancer therapy with no available curative options. (Note: Osteosarcoma participants must be in first or subsequent relapse [greater than or equal to first relapse]). Only the osteosarcoma participants enrolled to Cohorts 3A and 3B must be deemed candidates for ifosfamide and etoposide chemotherapy). 3. Evaluable or measurable disease that meets the following 1. Participants must have evaluable or measurable disease based on 1.1 using computed tomography (CT)/magnetic resonance imaging (MRI). 2. Lesions that have had external beam radiotherapy (EBRT) or locoregional therapies such as radiofrequency (RF) ablation must have subsequently grown unequivocally to be deemed a target lesion. 4. DTC participants must be 131 iodine-refractory/ relapsed as defined by at least one of the following: 1. One or more evaluable or measurable lesions that do not demonstrate iodine uptake on any radioiodine scan; or 2. One or more evaluable or measurable lesions that have progressed based on 1.1, within 12 months of 131 iodine therapy, despite demonstration of radioiodine avidity at the time of that treatment by pre or post-treatment scanning. These participants must not be eligible for possible curative surgery; or 3. Cumulative activity of 131 iodine greater than 400 millicuries (mCi) or 14.8 gigabecquerels (GBq), with the last dose administered at least 6 months prior to study entry. 5. Participants with DTC must be receiving thyroxine suppression therapy and levels of thyroid stimulating hormone (TSH) should not be elevated (TSH should be less than or equal to 5.50 milliunits per liter (mU/L)). When tolerated by the participant, thyroxine dose should be changed to achieve TSH suppression (TSH less than 0.50 mU/L). 6. Participants with known central nervous system (CNS) primary tumors or metastases who have completed brain therapy (such as radiotherapy, stereotactic radiosurgery, or surgical resection) and have remained clinically stable, asymptomatic, and off of steroids for 2 weeks prior to Cycle 1 Day 1 will be eligible. 7. Male or female participants age 2 years to less than18 years and less than or equal to 25 years for osteosarcoma subjects at the time of informed consent. 8. Lansky play score greater than or equal to 50% or Karnofsky Performance Status score greater than or equal to 50%. Use Karnofsky for participants greater than or equal to 16 years of age and Lansky for participants less than 16 years of age. 9. Life expectancy greater than or equal to 3 months. 10. Adequate bone marrow function as evidenced by: 1. absolute neutrophil count (ANC) greater than or equal to 1.0 x 10^9/L (for Cohorts 3A and 3B leucocyte count greater than or equal to 2 x 10^9/L; participants with bone marrow involvement should have ANC greater than or equal to 0.8 x 10^9/L and leucocyte count greater than or equal to 1 x 10^9/L). 2. hemoglobin greater than or equal to 8.0 grams/deciliter (g/dL) (a hemoglobin less than 8.0 g/dL is acceptable if it is corrected by growth factor or transfusion before starting lenvatinib). 3. platelet count greater than or equal to 75 x 10^9/L. 11. Adequate liver function as evidenced by: 1. bilirubin less than or equal 1.5 times the upper limit of normal (ULN). 2. alkaline phosphatase, alanine aminotransferase (ALT), and aspartate aminotransferase (AST) less than or equal to 3 times ULN. 12. Adequate renal function as evidenced by: a) Serum creatinine based on age/gender as below. If serum creatinine is greater than maximum serum creatinine for age/gender as shown in the table below, then creatinine clearance (or radioisotope glomerular filtration rate [GFR]) must be greater than 70 milliliter/minute/1.73 square meter (mL/min/1.73 m2). Maximum Serum Creatinine in milligrams/deciliter (mg/dL) for male: i. Age 2 to less than 6 years = 0.8 ii. Age 6 to less than 10 years = 1.0 iii. Age 10 to less than 13 years = 1.2 iv. Age 13 to less than 16 years = 1.5 v. Age greater than or equal to 16 years = 1.7 Maximum Serum Creatinine (mg/dL) for Female: vi. Age 2 to less than 6 years = 0.8 vii. Age 6 to less than 10 years = 1.0 viii. Age 10 to less than 13 years = 1.2 ix. Age 13 to less than 16 years = 1.4 x. Age greater than or equal to 16 years = 1.4 The threshold creatinine values in this Table were derived from the Schwartz formula for estimating glomerular filtration rate using child length and stature data published by the CDC. b) Urine dipstick less than 2+ for proteinuria. Participants who have greater than or equal to 2+ proteinuria on dipstick urinalysis should undergo a spot protein-creatinine (P/C) ratio that should be Grade less than 2. c) No clinical evidence of nephrotic syndrome. 13. Adequate cardiac function as evidenced by left ventricular ejection fraction (LVEF) greater than or equal to 50%) at baseline as determined by echocardiography. 14. Adequately controlled blood pressure (BP) with or without antihypertensive medications, defined as: BP less than 95th percentile for sex, age, and height/length at screening (as per National Heart Lung and Blood Institute guidelines) and no change in antihypertensive medications within 1-week prior to Cycle 1/Day 1. Osteosarcoma subjects 18 to 25 years should have BP ≤150/90 mm Hg at screening and no change in antihypertensive therapy within 1 week prior to Cycle 1/Day 1. 15. Washout of 3 weeks in case of prior chemotherapy, 6 weeks if treatment included nitrosoureas; 4 weeks for definitive radiotherapy, and 2 weeks for palliative radiotherapy; 3 months from high-dose chemotherapy and stem cell rescue; 3 weeks from major surgery. Participants must have recovered from the acute toxic effects of all prior anticancer therapy before enrollment into the study. 16. Written and signed informed consent from the parent(s) or legal representative (guardian) and assent from the minor participant. Written informed consent from subjects ≥18 years. 17. Willing and able to comply with the protocol, scheduled follow-up, and management of toxicity as judged by the Investigator. Cohort 3B (Combination Expansion): Osteosarcoma subjects who progressed in Cohorts 1 or 2B and opt to receive combination therapy. 18. Osteosarcoma participants receiving combination therapy of lenvatinib with ifosfamide and etoposide should meet only Numbers 6 through 17 (after progression in Cohort 2B)  Any active infection or infectious illness unless fully recovered prior to dosing. 2. Any medical or other condition that in the opinion of the investigator(s) would preclude the participant's participation in a clinical study. 3. Other organ toxicity due to prior anticancer therapy (investigational agent, chemotherapy, or radiation therapy) except alopecia, and ototoxicity due to cisplatin not already covered in the inclusion/  which has not recovered to Grade less than 2 per Common Terminology for Adverse Events (CTCAE) v4.0. 4. Known hypersensitivity to any component of the product (lenvatinib or ingredients). 5. Concurrent administration of any other antitumor therapy. 6. Previous treatment with lenvatinib (except for participants previously enrolled into Cohorts 1 or 2B of this study). 7. Two or more prior vascular endothelial growth factor/vascular endothelial growth factor receptor (VEGF/VEGFR) targeted therapies. 8. Currently receiving any investigational drug or device in another clinical trial or within 30 days preceding informed consent. 9. A clinically significant ECG abnormality, including a marked baseline prolonged QT or QTc interval (eg, a repeated demonstration of a QTc interval greater than 480 msec). 10. Gastrointestinal malabsorption or any other condition that in the opinion of the investigator might affect the absorption of lenvatinib. 11. Gastrointestinal bleeding or active hemoptysis (bright red blood of at least half teaspoon) within 3 weeks prior to the first dose of study drug. 12. Active second malignancy within 2 years prior to enrollment ([in addition to the primary tumor types specified by cohort in Criterion Number 1], but not including definitively treated superficial melanoma, in-situ, basal or squamous cell carcinoma of the skin). 13. Previous treatment with ifosfamide and Grade greater than or equal to 3 nephrotoxicity or encephalopathy (Cohorts 3A and 3B). 14. Females who are breastfeeding or pregnant at Screening or Baseline (as documented by a positive beta-human chorionic gonadotropin [ß-hCG] (or human chorionic gonadotropin [hCG]) test with a minimum sensitivity of 25 IU/L or equivalent units of ß-hCG [or hCG]). A separate baseline assessment is required if a negative screening pregnancy test was obtained more than 72 hours before the first dose of study drug. Cohort 3B (Combination Expansion): Osteosarcoma participants who progressed in Cohorts 1 or 2B and opt to receive combination therapy. 15. Osteosarcoma participants receiving combination therapy of lenvatinib with ifosfamide and etoposide should meet all the  with the exception of Criterion Number 6