Lung cancer is one of the leading causes of cancer-related deaths worldwide and is characterized by hijacking immune system for active growth and aggressive metastasis. Neutrophils, which in their original form should establish immune activities to the tumor as a first line of defense, are undermined by tumor cells to promote tumor invasion in several ways. In this study, we investigate the mutual interactions between the tumor cells and the neutrophils that facilitate tumor invasion by developing a mathematical model that involves taxis-reaction-diffusion equations for the critical components in the interaction. These include the densities of tumor and neutrophils, and the concentrations of signaling molecules and structure such as neutrophil extracellular traps (NETs). We apply the mathematical model to a Boyden invasion assay used in the experiments to demonstrate that the tumor-associated neutrophils can enhance tumor cell invasion by secreting the neutrophil elastase. We show that the model can both reproduce the major experimental observation on NET-mediated cancer invasion and make several important predictions to guide future experiments with the goal of the development of new anti-tumor strategies. Moreover, using this model, we investigate the fundamental mechanism of NET-mediated invasion of cancer cells and the impact of internal and external heterogeneity on the migration patterning of tumour cells and their response to different treatment schedules.