Abstract
There are many works (i.e. [1]) aiming to find out numerically how positive feedback affects the formation of invadopodia and invasion of cancer cells; however, studies on the cancer cell invasion model with free boundary are fairly rare. In this paper, we study modified cancer cell invasion model with free boundary, where, free boundary stands for cancer cell membrane, so that we can more precisely describe the positive feedback affects. Firstly, we simplized the model by means of characteristic curve and semi-groups’ property, and obtained the Stefan-like problem by introducing Gaussian Kernel and Green function. Secondly, based on the classical Stefan problem, we derived the integral solution of simplified model, which could lead us a further step to find the solution of modified cancer cell invasion model.
Highlights
As well known, cancer disease is one of the leading causes of death worldwide
We study modified cancer cell invasion model with free boundary, where, free boundary stands for cancer cell membrane, so that we can more precisely describe the positive feedback affects
How actin filaments are driving cancer cell invasion has been discussed in [1]. They considered four particles, actins, ECMs, MMPs and ECM fragments, where actins and MMPs act in intracellular area, ECMs and ECM fragments exist in extracellular area
Summary
Cancer disease is one of the leading causes of death worldwide. Many natural and man-made factors (for example, smoking, car exhaust fumes, ultraviolet rays, air pollution and radiations, etc.) are the main risks for cancer disease. Signaling molecule VEGF (vascular endothelial growth factor) can be secreted by cancer cells and can bind the normal endothelial cells to form new blood vessels. Scientists found inhibitors, such as bevacizumab, to block VEGF [2]. In this paper, we study a modified cancer cell invasion model with free boundary problem. Invadopodia are the invasive feet (as shown in Figure 1) of cancer cells which can degrade the surrounding matrix (mainly ECMs), and cause metastasis [12]. ECM fragments (laminin γ2 chains) bind to cell membrane receptors and send signals to the actin reorganization, where signals have random motility. The detailed calculation from (48) to (49) and (50) is given in Appendix A, B, C
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