Sirolimus and Pemetrexed to Treat Non-Small Cell Lung Cancer

* INCLUSION CRITERIA: 1. Histologically documented non small cell lung cancer (NSCLC) that is confirmed by the Laboratory of Pathology at the Clinical Center/National Institutes of Health (NIH) or the Laboratory of Pathology at National Naval Medical Center (NNMC). 2. Tumor biopsy will be requested from all study subjects unless the procedure poses too great a risk. If the subject declines, he or she may still participate in the study. We will ask subjects not undergoing biopsy to provide 6 unstained slides or a tissue block of archived tissue for immunohistochemistry (IHC) evaluation. Tumors from subjects enrolling in the phase II portion of the study will be analyzed retrospectively to demonstrate mammalian target of rapamycin (mTOR) activation as assessed by immunohistochemistry in a fresh biopsy. mTOR activation will be defined using distribution and intensity of staining for phosphorylation of mTOR, or its downstream substrates S6 kinase (S6K), and S6. Standard operating procedures (SOPs) describing the acquisition and handling of PBMCs and tissues are outlined in appendix 10.3 and 10.4. At a minimum, a total score (sum of intensity and distribution scores) of 2 for phospho-S6 or phospho-mTOR (S2448) mTOR will be required to determine that mTOR is active. Either measurement will be sufficient to ascertain that mTOR is active. Measurement of phosphorylation of Akt, factor 4E binding protein 1 (4E-BP1), and total levels of thymidylate synthase (TS) will also be measured, but are not part of the eligibility requirements. In the event of limited tissue availability, the stains will be prioritized as follows: S6, mTOR, S6K, Akt (S473), Akt (T308), and TS. Phosphorylation of S6 correlates most closely with mTOR activity, while phosphorylation of mTOR at S2448 best predicts response to sirolimus. 3. Tissue from the time of original diagnosis will be adequate for enrollment on study. Optional fresh tissue biopsy must be obtained AFTER their most recent chemotherapy (including small molecule or targeted therapy) or radiation therapy. Tumors that can be biopsied percutaneously (with or without computed tomography (CT)/ultrasound guidance) or via bronchoscopy will be considered accessible if there are no other competing risk factors such as coagulopathy, hypoxemia, unstable cardiovascular disease, uncontrolled pain, or inability to give informed consent. 4. Individuals with relapsed NSCLC who have received at least one standard chemotherapeutic regimen are eligible. Patients who received adjuvant chemotherapy and then relapse or recur less than or equal to 12 months after completion of chemotherapy will be eligible. Patients who received adjuvant chemotherapy and relapse greater than 12 months after completion of chemotherapy should receive frontline therapy for metastatic disease before enrollment, as should individuals who initially present with incurable disease that is chemotherapy naive. Individuals unwilling to receive standard front line therapy for metastatic lung cancer may enroll. 5. Patients must have not received any chemotherapy, biological, or radiation therapy in the 21 days prior to protocol enrollment. All previous chemo and radiation therapy induced toxicities must have resolved to grade 1 or less prior to enrollment. 6. Because sirolimus may affect the efficacy of hormonal birth control via cytochrome P450 3A4 (CYP 3A4), study subjects of child bearing potential must be willing to use barrier birth control while receiving sirolimus therapy and for 12 weeks after discontinuation of sirolimus. 7. Patients must have measurable disease for the phase II portion of the study. 8. Age greater than or equal to 18 years of age. 9. Eastern Cooperative Oncology Group (ECOG) performance score of 0-2. 10. An expected survival of at least 3 months. 11. Patients must have the capacity to provide informed consent and demonstrate willingness to comply with an oral regimen. 12. Patients must have normal organ and marrow function as defined below: * Absolute neutrophil count greater than or equal to 1,500/mL. * Platelets greater than or equal 100,000/mL. * Total bilirubin less than 1.5 times upper limit of institutional normal. * Aspartate aminotransferase (AST) serum glutamic oxaloacetic transaminase (SGOT) less than 2.5 times upper limit of institutional normal. * Alanine aminotransferase (ALT) serum glutamic pyruvic transaminase (SGPT) less than 2.5 times upper limit of institutional normal. * Creatinine Estimated creatinine clearance as calculated using the modification of diet in renal disease (MDRD) equation must be greater than or equal to 60ml/min/1.73m\^2. The formula to be used is MDRD: 186 times (Scr)(-1.154) times (Age)(0.203) times (0.742 if female) times (1.212 if African American) * Serum triglycerides less than or equal to 2.5 times upper limit of normal; serum cholesterol less than or equal 300 mg/dl (includes subjects with familial and acquired hyperlipidemia). 13\. Subjects on steroids must be on a stable or tapering dose of less than or equal 20 mg/day of prednisone (or equivalent dose of another glucocorticoid) for at least one week prior to study entry. EXCLUSION CRITERIA: 1. Human immunodeficiency virus (HIV) positive patients. 2. Pregnant or lactating women. 3. Patients who received pemetrexed previously for Phase 1 only. Patients with prior pemetrexed are eligible for Phase 2. 4. Patients who have had prior therapy with mTOR inhibitors such as sirolimus or its analogues within six months. 5. Any concurrent therapy with chemotherapeutic agents or biologic agents or radiation therapy. 6. Subjects with brain metastases may participate if the metastases are asymptomatic. Subjects are ineligible if brain metastases are symptomatic. 7. Patients who are on the following drugs that modulate CYP3A4 and cannot replace these medications with other equivalent medications for the period of the study: amprenavir, atazanavir, bromocriptine, cimetidine, clarithromycin, clotrimazole, cyclosporine, danazol, diltiazem, erythromycin, fluconazole, fosamprenavir, other HIV protease inhibitors, indinavir, itraconazole, ketoconazole, metoclopramide, nefazodone, nelfinavir, nicardipine, nifedipine, ritonavir, saquinavir, telithromycin, troleandomycin (TAO), verapamil, voriconazole, nevirapine, rifampicin, rifampin, rifabutin, rifapentine, phenytoin, carbamazepine, phenobarbital, and St. John's Wort. 8. Subjects taking non steroidal anti-inflammatory agents who are unable to stop or replace the agents for the 5 days prior to and the 2 days after pemetrexed will not be eligible. 9. Patients who have received live vaccines in the past 21 days.