BATTLE-2 Program: A Biomarker-Integrated Targeted Therapy Study in Previously Treated Patients With Advanced Non-Small Cell Lung Cancer

The Study Drugs: Erlotinib hydrochloride, MK-2206, AZD6244, and Sorafenib are targeted therapies. Targeted therapy is a type of drug that blocks the growth of cancer cells by interfering with specific targeted molecules needed for tumor growth, rather than by simply interfering with rapidly dividing cells (for example with traditional chemotherapy). Study Groups: If you are found to be eligible to take part in this study, you will be assigned to 1 of 4 groups: * If you are in Group 1, you will take erlotinib alone. * If you are in Group 2, you will take erlotinib and MK-2206. * If you are in Group 3, you will take AZD6244 and MK-2206. * If you are in Group 4, you will take sorafenib alone. The group you are in will be assigned by a computer. The results of your screening tests will be used to determine (and may limit) which of the 4 groups you may be eligible to be assigned to. Your likelihood of being in any 1 group over another may also be affected by how well that treatment has done in earlier study participants. Your study doctor will tell you which group you have been assigned. Study Drug Administration: Each cycle is 28 days. * If you are in Group 1, you will take erlotinib hydrochloride by mouth 1 time every day. * If you are in Group 2, you will take MK-2206 by mouth 1 time every week. You will take erlotinib hydrochloride by mouth 1 time every day. * If you are in Group 3, you will take MK-2206 by mouth 1 time per week. You will take AZD6244 by mouth 1 time every day. * If you are in Group 4, you will take sorafenib by mouth 2 times every day. If you will take erlotinib hydrochloride (Groups 1 and 2), you should take the tablets in the morning. You should take the drug 1 hour before or 2 hours after a meal, with no more than 7 ounces of water. If you forget to take a dose, the last missed dose should be taken as soon as you remember, as long as it is at least 12 hours before the next dose. The next day, you should take the scheduled dose at the usual time. You should try not to vomit for at least 30 minutes after taking the drug. If you feel nauseated before or after taking the erlotinib, you should take an anti-nausea drug. You may take an extra dose of erlotinib if you vomit within 30 minutes after taking the tablet. If you are in Group 2, you should take MK-2206 at about the same time each week with about 1 cup (8 ounces) of water. You must take MK-2206 at least 2 hours before or at least 2 hours after any food or a meal. You should not make up any missed doses. If you are in Group 3, you should take MK-2206 at about the same time each week with about 1 cup (8 ounces) of water. You must take MK-2206 at least 2 hours before or at least 2 hours after eating. You should not make up any missed doses. If you will take AZD6244 (Group 3), you should take the study drug in the morning before you eat or drink anything. You can eat breakfast 1 hour after you take the dose. If you will take sorafenib, you must take your doses 12 hours apart. You will take 2 tablets each morning, and again each evening. Sorafenib should be taken with about 1 cup of water on an empty stomach (either 1 hour before a meal or 2 hours after a meal). If you feel nauseated before or after taking the medication, anti-nausea medications should be used. If you miss a dose, you should skip it and take the next scheduled dose at the right time. Your medication should be stored at room temperature. Study Visits: The study visit schedule is described below. In certain cases, with the permission of your doctor, the study visits may occur up to 7 days earlier or later than described below. On Day 1 of each cycle: * You will have a complete physical exam, including measurement of your weight and vital signs. * You will be asked about any drugs you may be taking and any side effects you may be having. * Your performance status will be recorded. * Blood (about 3 teaspoons) will be drawn for routine tests. If your doctor thinks it is needed, you may have to have these blood tests more often. On Day 1 of every odd-numbered cycle (Cycles 3, 5, 7, and so on): * You will have a CT scan and/or MRI of the chest (and abdomen if the doctor thinks it is needed) to check the status of the disease. * You will have a chest x-ray. * MRI of the brain (if the doctor thinks it is needed) Additional Tests for Group 2: * During screening, on Day 1 of Cycle 2, and at the end-of study visit, you will have an eye exam. * On Day 1 of Cycle 1 you will have an ECG before taking your study medication and then again about 4 hours after you take your study medication. Additional Tests for Group 3: * During screening, on Day 1 of Cycle 2, and at the end-of study visit, you will have an eye exam. * During Cycle 2 (Week 6 of being on study drug), on Day 1 of Cycle 4, and then every 3 months, you will have a multigated acquisition (MUGA) scan and/or echocardiogram to check your heart function. You will have these tests more often if the doctor thinks it is needed. * On Day 1 of Cycle 1 you will have an ECG before taking your study medication and then again about 4 hours after you take your study medication. Length of Study: You may continue to take the study drug(s) for as long as you are benefitting. You will be taken off study if the disease gets worse or if you have intolerable side effects. End-of-Study Visit: When you go off study for any reason, you will have an end-of-study. This visit may occur up to 7 days earlier or later. The following tests and procedures will be performed: * Your medical history will be recorded. * You will have a complete physical exam, including measurement of your weight and vital signs. * You will be asked about any drugs you may be taking and any side effects you may be having. * Your performance status will be recorded. * Blood (about 3 teaspoons) and urine will be collected for routine tests. * You will have a CT scan and/or MRI of the chest (and abdomen if the doctor thinks it is needed) to check the status of the disease. * You will have a chest x-ray. * You will have an ECG (Group 3). Follow-Up: You will have a follow-up evaluation performed 4 weeks ± 7 days after therapy is discontinued. This evaluation may be a visit or contact by phone by the research personnel. If you are in Group 3, you will have a MUGA scan and/or echocardiogram to check your heart function. You will be called every 3 months for up to 3 years and asked about any cancer treatments you may be receiving. This phone call will take about 10 minutes. This is an investigational study. Erlotinib is FDA approved and commercially available for treatment of NSCLC that has gotten worse. Sorafenib is approved in renal cell cancer and hepatocellular carcinoma. Sorafenib has also has also been evaluated in unselected advanced patients with NSCLC both alone or with chemotherapy in the first-time treatment of patients . MK-2206, and AZD6244 are not FDA-approved or commercially available. At this time, they are only being used in research. Up to 450 patients will take part in this multicenter study. Approximately 350 will be enrolled at MD Anderson.