Sirolimus and Pemetrexed to Treat Non-Small Cell Lung Cancer

Background: * Lung cancer is the most deadly cancer due to late stage of diagnosis and intrinsic resistance to chemotherapy. * Pemetrexed is a well tolerated Food and Drug Administration (FDA)-approved second line chemotherapeutic agent with a 9% response rate. * Increasing the efficacy of pemetrexed could provide clinical benefit for patients with refractory NSCLC. * Inhibition of the phosphoinositide 3-kinase (PI3K)/protein kinase B (Akt)/mTOR pathway may increase response to chemotherapy. * Combining sirolimus, an mTOR inhibitor, with pemetrexed could improve patient outcomes. Objectives: * Determine the safety, tolerability, pharmacokinetics (PKs), and maximum tolerated dose (MTD) of the combination of sirolimus with pemetrexed in subjects with NSCLC subjects with activation of the Akt/mTOR pathway. * Determine the clinical response rate at the MTD of sirolimus plus pemetrexed in NSCLC subjects. * Determine effects of sirolimus on activation of the PI3K/Akt/mTOR pathway in peripheral blood mononuclear cells (PBMCs) and/or tumor tissues, to determine metabolic changes using PET scans, and measure PKs. Eligibility: \- Adults with refractory or relapsed NSCLC regardless of mTOR pathway activation are permitted to enroll in the trial. Design: * Phase I followed by Phase II study * For phase I/II subjects, documentation of mTOR pathway activation is not mandatory. If accessible, tissue will be obtained at baseline and following two cycles of therapy or at time of progression, whichever occurs first. Tumor tissue will be obtained at baseline and after two cycles of therapy or at time of progression, whichever occurs first. All subjects will have pathway analysis using PBMCs at baseline, day 8 and every two cycles of therapy or at time of progression, whichever occurs first. Cycle 1 is 28 days in length and all others 21 days. * Each dose level incorporates a lead-in period of sirolimus alone that will allow for correlations of dose level, pharmacokinetics, and biologic effects. * The phase I portion of the study has 5 dose cohorts beginning below the FDA approved doses for both agents. There are 3 dose escalations for sirolimus and 2 for pemetrexed. Up to 30 subjects may enroll in the phase I study. * The Phase II portion will utilize the MTD from the Phase I and enroll up to 60 subjects. * Sirolimus will be administered by mouth daily, and pemetrexed will be administered intravenously every 21 days until unacceptable toxicity or disease progression. * Clinical imaging (CT or MRI) and a PET CT will be obtained at baseline and after two cycles of treatment. Clinical imaging will be performed every two cycles until disease progression.