A Study in Non Small Cell Lung Cancer

Inclusion Criteria:

* Phase 1 portion:

  * Participants with a cytologic or histologic diagnosis of nonsquamous NSCLC that is classified as Stage IV according to the 7th edition of the American Joint Committee on Cancer (AJCC) classification and for whom the combination of pemetrexed and cisplatin is deemed to be appropriate
  * Participants with histologic or cytologic diagnosis of malignant mesothelioma that is unresectable
  * Participants with histologic or cytologic diagnoses of advanced or metastatic solid tumors who are not candidates for any standard therapy and for whom the combination with pemetrexed and cisplatin is deemed to be appropriate
* Phase 2 portion:

  * Have a histological diagnosis of NSCLC other than predominantly squamous cell histology that is classified as Stage IV according to the 7th edition of the AJCC classification
  * Eligible for a first line of palliative treatment with a platinum doublet
  * Have archived or fresh tumor tissue (not cytology)
* Phase 1 participants can have measurable or nonmeasurable disease. Phase 2 participants must have at least 1 measurable lesion according to Investigational New Drug (Response Evaluation Criteria in Solid Tumors \[RECIST\], v1.1) definitions. Tumor lesions located in a previously irradiated area can be considered measurable if they are new or if have shown unequivocal progression.
* Have a performance status ≤1 on the Eastern Cooperative Oncology Group (ECOG) scale
* Have adequate hematologic, hepatic, and renal organ function
* Prior radiation therapy for treatment of cancer is allowed to \<25% of the bone marrow, and participants must have recovered from the acute toxic effects of their treatment prior to study enrollment. Prior radiation to the whole pelvis is not allowed. Prior radiotherapy must be completed at least 4 weeks before study entry
* For women: Must be surgically sterile, postmenopausal, or compliant with a highly reliable contraceptive method (failure rate \<1%) during and for 6 months after the treatment period; must have a negative serum or urine pregnancy test within 7 days before study enrollment and must not be breast-feeding. For men: Must be surgically sterile or compliant with a contraceptive regimen during and for 6 months after the treatment period

Exclusion Criteria:

* Have serious preexisting medical conditions or serious concomitant systemic disorders that would compromise the safety of the participant or his/her ability to complete the study, at the discretion of the investigator (for example, unstable angina pectoris or uncontrolled diabetes mellitus). Special attention should be paid to kidney and heart conditions that may be worsened with cisplatin treatment or hydration
* Have central nervous system (CNS) metastases (unless the participant has completed successful local therapy for CNS metastases and has been off corticosteroids for at least 4 weeks before starting study therapy). A screening computed tomography scan or magnetic resonance imaging before enrollment in the absence of a clinical suspicion of brain metastases is not required.
* Have current active infection that would, in the opinion of the investigator, compromise the participant's ability to tolerate therapy
* Have known allergy to pemetrexed, cisplatin, LY2603618, or any ingredient of pemetrexed, cisplatin, or LY2603618
* Have clinically significant (by physical exam) third-space fluid collections; for example, ascites or pleural effusions that cannot be controlled by drainage or other procedures prior to study entry
* Participants taking non-steroidal anti-inflammatory drugs who cannot interrupt the treatment appropriately according to the guidelines
* Have received a recent yellow-fever vaccination (within 28 days of enrollment) or are receiving concurrent yellow-fever vaccination
* Phase 1 portion:

  * Have received more than 2 previous lines of chemotherapy for the advanced/metastatic disease
  * Have received more than 6 cycles of therapy containing an alkylating agent