Numerical Simulation of a Mathematical Model for Cancer Cell Invasion

In this paper we analyze a mathematical model of cancer cell invasion consisting of a system of partial differential equations which describes the interactions between...


Introduction
In this paper we focus on the discretization of a mathematical model describing the process of cells invasion in the surrounding extracellular matrix using a Generalized Finite Difference Method.
Chaplain and Lolas in [1,2] developed a mathematical model consisting of three partial differential equations describing the evolution in time and space of the system variables. It is assumed that the key physical variables are tumor cell density (denoted by U); protein density of the extracellular matrix (denoted by W) and the concentration of the chemical substance responsible for the chemotaxis (denoted by V) each of them considered at and time t > 0.
Throughout this paper 2 R Ω ⊂ is a bounded domain with a regular boundary. The model is the following: (1) where we consider the chemotactic and hepatotactic coefficients, η and ρ, respectively, to be constant and positive. The parameter k represents the net growth of the tumor cell density. It is natural to assume homogeneous Neumann boundary condition as we assume that invasion takes place within an isolated system (see [2]). We consider for our numerical simulation (2) Tao and Winkler in [3] have demonstrated that whenever the initial data (U 0 , V 0 , W 0 ) are regular fulfilling  The GFDM has been recently proved to obtain highly accurate approximations to the solutions of nonlinear PDEs (see for instance [6,5,4]. The paper is organized as follows: in Section 2 we present the explicit formula of the GFDM and obtain the explicit scheme. In

GFD Scheme
As stated in the introduction, our objective is to derive a discretization of system (1) using the GDF explicit formulae. To do so, let us consider a bounded domain 2 R Ω ⊂ . Then, Therefore, system (1) reads as equations (3), (4) and (5) together with the nonnegative initial data Ω and the boundary conditions The explicit formulae of the GFD explicit scheme can be seen in [4,5,6], although for the sake of completeness we reproduce below

Example 1
In this first example we solve numerically system given by (3)- We take into account the following initial data: (12) As we have mentioned in Section 1, since the assumption κ > 2 8 η is fulfilled, we expect to find convergence of the solution in the sense In (Table 1) we present the of the approximate solution for different times. Figure 2 shows the different solutions at such times.

Example 2
For this second case we also consider Notice that 0 1 W ≤ does not hold. Then, the assumptions of [3] are not fulfilled. Table 2 shows the maximum value of the approximate solutions for different times [4]. Figure 3 shows the approximate solutions at 1, 10 and 20 seconds. We obtain convergence to the steady state (0, 0, 1). To the best of our knowledge no analytical proof of this convergence is known [5,6].   ). Figure 4 shows the approximate solution of this third case. Note that we introduce the solutions at small times in order to capture the dynamical complexity of the model.

Conclusion
We have derived the discretization of the chemotaxis-hypotaxis system (1) using a GFD scheme. The discrete solution obtained inherits the complicated dynamical behavior of the analytical solution. The Generalized Finite Difference Method solves this strongly coupled highly nonlinear parabolic-elliptic system efficiently and with high accuracy.