A Trial of Lenvatinib (E7080) in Subjects With Iodine-131 Refractory Differentiated Thyroid Cancer to Evaluate Whether an Oral Starting Dose of 18 Milligram (mg) Daily Will Provide Comparable Efficacy to a 24 mg Starting Dose, But Have a Better Safety Profile

1. Participants must have histologically or cytologically confirmed diagnosis of one of the following differentiated thyroid cancer (DTC) subtypes:

1. Papillary thyroid cancer (PTC)

* Follicular variant

* Variants (including but not limited to tall cell, columnar cell, cribriform-morular, solid, oxyphil, Warthin's-like, trabecular, tumor with nodular fasciitis-like stroma, Hürthle cell variant of papillary carcinoma, poorly differentiated)

2. Follicular thyroid cancer (FTC)

* Hurthle cell

* Clear cell

* Insular

2. Measurable disease meeting the following criteria and confirmed by central radiographic review:

1. At least 1 lesion of greater than or equal to (\>=)1.0 centimeter (cm) in the longest diameter for a non-lymph node or \>=1.5 cm in the short-axis diameter for a lymph node that is serially measurable according to Response Evaluation Criteria in Solid Tumors (RECIST 1.1) using computed tomography/magnetic resonance imaging (CT/MRI). If there is only 1 target lesion and it is a non-lymph node, it should have a longest diameter of \>=1.5 cm.

2. 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.

3. Participants must show evidence of disease progression within 12 months (an additional month will be allowed to accommodate actual dates of performance of screening scans, that is, within less than or equal to \<=13 months) prior to signing informed consent, according to RECIST 1.1 assessed and confirmed by central radiographic review of CT and/or MRI scans.

4. Participants must be Iodine-131 refractory/resistant as defined by at least one of the following:

1. One or more measurable lesions that do not demonstrate iodine uptake on any radioiodine scan.

2. One or more measurable lesions that have progressed according to RECIST 1.1 within 12 months (an additional month will be allowed to accommodate actual dates of performance of screening scans, ie, within ≤13 months) after Iodine-131 therapy, despite demonstration of radioiodine avidity at the time of that treatment by pre- or posttreatment scanning. These participants must not be eligible for possible curative surgery.

3. Cumulative activity of Iodine-131 of greater than (\>) 600 millicurie (mCi) or 22 gigabecquerels (GBq), with the last dose administered at least 6 months prior to study entry.

6. Participants must be receiving thyroxine suppression therapy and thyroid stimulating hormone (TSH) should not be elevated (TSH should be \<=5.50 micro-international units per liter \[mcIU/ML\]). When tolerated by the participant, thyroxine dose should be changed to achieve TSH suppression (TSH \<0.50 mcIU/ML) and this dose may be changed concurrently upon starting study drug treatment.

7. All chemotherapy or radiation-related toxicities must have resolved to Grade \<2 severity per Common Terminology Criteria for Adverse Events (CTCAE v4.03), except alopecia and infertility.

8. Participants must have an Eastern Cooperative Oncology Group (ECOG) Performance Status of 0, 1, or 2.

9. Adequately controlled blood pressure (BP) with or without antihypertensive medications, defined as BP ≤150/90 mmHg at Screening and no change in antihypertensive medications within 1 week prior to Cycle 1/Day 1.

10. Adequate renal function defined as calculated creatinine clearance \>=30 mL/min per the Cockcroft and Gault formula.

11. Adequate bone marrow function:

1. Absolute neutrophil count (ANC) \>=1500/mm\^3 (\>=1.5\*10\^3/uL)

2. Platelets \>=100,000/mm\^3 (\>=100\*10\^9/L)

3. Hemoglobin \>=9.0 g/dL

12. Adequate blood coagulation function as evidenced by an International Normalized Ratio (INR) \<=1.5.

13. Adequate liver function:

1. Bilirubin \<=1.5\*upper limit of normal (ULN) except for unconjugated hyperbilirubinemia or Gilbert's syndrome.

2. Alkaline phosphatase, alanine aminotransferase (ALT), and aspartate aminotransferase (AST) \<=3\*ULN (\<=5\*ULN if participant has liver metastases). If alkaline phosphatase is \>3\*ULN (in absence of liver metastases) or \>5\*ULN (in presence of liver metastases) AND the participant also is known to have bone metastases, the liver-specific alkaline phosphatase must be separated from the total and used to assess the liver function instead of total alkaline phosphatase.

14. Males or females age \>=18 years at the time of informed consent.

15. Females must not be lactating or pregnant at Screening or Baseline (as documented by a negative beta-human chorionic gonadotropin \[B-hCG\] test with a minimum sensitivity of 25 IU/L or equivalent units of B-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.

16. All females will be considered to be of childbearing potential unless they are postmenopausal (amenorrheic for at least 12 consecutive months, in the appropriate age group and without other known or suspected cause) or have been sterilized surgically (ie, bilateral tubal ligation, total hysterectomy or bilateral oophorectomy, all with surgery at least 1 month before dosing).

17. Females of childbearing potential should avoid becoming pregnant and use highly effective contraception while on treatment with lenvatinib and for at least 1 month after finishing treatment. Females of childbearing potential must not have had unprotected sexual intercourse within 30 days before study entry and must agree to use a highly effective method of contraception (example, total abstinence, an intrauterine device, a contraceptive implant, an oral contraceptive, or have a vasectomized partner with confirmed azoospermia) throughout the entire study period and for 30 days after study drug discontinuation. Females who are using hormonal contraceptives must have been on a stable dose of the same hormonal contraceptive product for at least 4 weeks before dosing and must continue to use the same contraceptive during the study and for 30 days after study drug discontinuation. Women using oral hormonal contraceptives should add a barrier method.

18. Participants must voluntarily agree to provide written informed consent.

19. Participants must be willing and able to comply with all aspects of the protocol.

What this study is about