| Summary: | The degradation of the extracellular matrix (ECM) is driven by actin-rich membrane protrusions called invadopodia, which leading to the cancer cell invasion across the surrounding tissue barriers. Signaling pathways through ligand and membrane associated receptor bindation are vital point in order to enhance the actin polymerization activities that push the membrane of migrating cells. The results presented by Saitou et. al, [36] are contradict to this fact since actins are not only pushed, but also diffused beyond the cell membrane into the ECM region. Hence, in this study, we considered mathematical modeling for an individual cancer cell. We investigated one-dimensional Stefan-like problem of the signal process, (CM-I-B) and treated the cell membrane as a free boundary surface to separate any activity happen in intra- and extracellular regions. An approximation problem, (CM-I-C) is introduced by transforming the Stefan-like problem into an initial-boundary value problem for the signal equation with penalty term. The velocity concerning the movement of the free boundary is then calculated by the integration of the penalty term. The auxiliary problem is solved numerically using finite-difference scheme, (CM-I-C0) for the above integrated penalty method, [21]. Two convergences of CM-I-C and CM-I-C0 into CM- I-B are investigated by taking E and ox goes to 0, respectively. Our results showed a good agreement with the other known fixed domain method for the free boundary positions and the signal distributions.
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