Mathematical modeling of quasi-static signal and ligand during invadopodia formation with velocity jump

Cancer cell invasion in the metastasis process contributes to the high death cases among cancer patients. The spread of a tumor from one location to the other part of the body is as a result of the existence of finger-like protrusions or invadopodia on the plasma membrane. On the biological view, the formation of invadopodia involves several processes that include the activity of matrix metalloproteinase (MMP) in degrading the extracellular matrix (ECM), the creation of ligand, stimulation of signal from the binding of a ligand with epidermal growth factor receptor (EGFR), up-regulation of MMP, and actin polymerization. The purpose of this study is to investigate the appearance of invadopodia on the plasma membrane through the mathematical modeling of quasi-static signal and ligand. The method of level set is emphasized to detect the movement of the moving interface (plasma membrane) by setting the interface as the zero level set function. Also, the regular and neighboring points are solved using second-order centered difference and ghost fluid with linear extrapolation method, respectively. The velocity is solved using the difference of gradient between the intracellular signal and extracellular ligand that consequently moves the interface. Results showed that the protrusions or invadopodia are formed on the plasma membrane with the aid of the above-mentioned methods