Low dose computed tomography (LDCT) is the current gold-standard for lung cancer diagnosis. However, accuracy of diagnosis is limited by the radiologist's ability to discern cancerous from non-cancerous nodules. To assist with diagnoses, a 4D-CT lung elastography method is proposed to distinguish nodules based on tissue stiffness properties. The technique relies on a patient-specific inverse finite element (FE) model of the lung solved using an optimization algorithm. The FE model incorporates hyperelastic material properties for tumor and healthy regions and was deformed according to respiration physiology. The tumor hyperelastic parameters and trans-pulmonary pressure were estimated using an optimization algorithm that maximizes similarity between the actual and simulated tumor and lung image data. The proposed technique was evaluated using an in-silico study where the lung tumor elastic properties were assumed. Following that evaluation, the technique was applied to clinical 4D-CT data of two lung cancer patients. Results from the evaluation study show that the elastography technique recovered known tumor parameters with only 6% error. Tumor hyperelastic properties from the clinical data are also reported. Results from this proof of concept study demonstrate the ability to perform lung elastography with 4D-CT data alone. Advancements in the technique could lead to improved diagnoses and timely treatment of lung cancer.