The cancer cell invasion into other organs through the metastasis process can contribute to the high mortality rate among cancer patients. To flee from the primary tumor, the metastatic cells must penetrate several physical barriers, aided by actin-rich protrusions called invadopodia. Generally, invadopodia is the consequence of the movement of the plasma membrane (interface), which is precisely accounted for as a free boundary. Nevertheless, matrix metalloproteinases (MMPs) in degrading the extracellular matrix are the starting point for invadopodia formation. Hence, this work applies four different functions of MMPs to investigate the stability of invadopodia with the aid of a ligand and signal model. The model is discretized using the finite difference and ghost fluid methods. Meanwhile, the interface is detected using a zero-level set function approach. In the sense of the behavior of the protrusions, the result showed that using cosine and exponential functions is more stable than polynomial and sine functions.