Numerical computation of free boundary signal transduction during the formation of invadopodia

Abstract

Mathematical simulation is one of the methods that has been beneficial to the knowledge of cancer cell invasion, especially in listing the important components that are the main causes of the invasion. In fact, at subcellular view, invadopodia which are actin based protrusions formed on the cancer cell membrane degrade the extracellular proteins and lead the invasion. In this study, a mathematical model of invadopodia formation associated with signal transduction is investigated. The main objective of this study is to obtain the numerical solution of invadopodia formation of signal transduction in one and two dimensions. The signal equation is represented by heat-like equation with initial and boundary conditions. The free boundary of plasma membrane is considered and is moved by the velocity of the cancer cell which is equal to the gradient of the intracellular signal. The governing partial differential equation for a one-dimensional case with initial and boundary conditions is transformed into an approximation model by using a signal transformation variable. Then, the integrated penalty technique is applied to solve the model numerically. The results are validated with the exact solution obtained using separable variable approach with suitable similarity variable. From this study, it is discovered that both results have shown the same outcome with the constant boundary condition restriction. Simulation results demonstrated that the interface position increases from its initial location as time increases. This implies that the plasma membrane has moved in tandem with the increase in time. Meanwhile, in a two-dimensional case, free boundary of plasma membrane is represented by a zero level set function. The partial differential equations are discretized by using the level set method which combines the features of ghost points and extrapolation methods in order to solve the model numerically. The stability condition of the solution is enforced by using Gershgorin circle theorem. It is observed that some protrusions are developed on the membrane surface due to the presence of the signal density inside the cell with types of cosine and exponential boundary conditions. The highest concentration of signal is identified on the interface due to the stimulation of signal through contact between the ligand and the membrane associated receptor on the membrane. All numerical schemes obtained in this thesis by numerical methods (integrated penalty method in one dimension and level set method in two dimensions) and an analytical method (separable variable method in one dimension) are useful to solve the free boundary problem of an invadopodia formation model particularly on the signal transduction factor.