#### Abstract

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 cancer cells, the matrix degrading enzyme and the host tissue. The main objective of the present paper is to show, using numerical computations, the ability of a relatively simple model to produce complicated dynamics. We build some novel techniques and numerical algorithms based on the Generalized Finite Difference Method (GFDM) and test the method in several examples.

**Keywords:** Chemotaxis Haptotaxis; Meshless Method; Generalized Finite Difference
Method

#### 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 Ω ⊂ R^{2} is a bounded domain with a regular
boundary. The model is the following:

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

Tao and Winkler in [3] have demonstrated that whenever the
initial data (U_{0}, V_{0}, W_{0}) are regular fulfilling

U>_{o}0, W_{o} ≤ 1 and K >η^{2}/8, solution (U,V,W) converges asumptotically
to the constant stationary solution (1, 1, 0) in the L∞(Ω).
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
Section 3 we present numerical examples. In particular, the first example
shows the convergence of the solution to the constant steady
state (1, 1, 0), in accordance with the theory [3]. The second and
third examples show that for an appropriate initial data the solution
converges to (0, 0, ω), with ω arbitrary. Finally, we obtain
some conclusions in Section 4.

#### 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 Ω ⊂ R^{2}. Then,

Therefore, system (1) reads as equations (3), (4) and (5) together with the nonnegative initial data

Withand 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

Where

For j = 0; 1; 2. The time derivative is approximated by

Then, the GFD scheme for the system (3)-(5) is:

#### Numerical Computations

We consider an irregular cloud of points see (Figure 1) modelling
our *in vitro* domain. For all examples we denote by (u, v, w) the
approximate solution given by the GFDM. For all numerical examples
we use as time step Δt = 0.001 and, then, T (s) (total time) is
calculated by nΔt .

#### Example 1

In this first example we solve numerically system given by (3)-
(5). We consider η = ρ = 0.5 and k =1.5 .

We take into account the following initial data:

As we have mentioned in Section 1, since the assumption κ >
η^{2}/8 is fulfilled, we expect to find convergence of the solution in the
sense

In (Table 1)l∞ 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 η = ρ = 0.5 and K =1.5 . As initial data we consider

Notice that W_{0} ≤ 1 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].

#### Example 3

Let us choose now η = 0.5 , ρ = 0.6 and K =1.5 . As initial data we consider the following

Table 3 shows the values norm of the approximate solutions. In this third case, where w(X ) ≤1 does not hold, we obtain convergence to the steady state (0, 0, 2.8109). In general, this means that solution to (1) converges to

(0,0,wˆ)

where wˆ is an arbitrary function (which seems to depend on
∫_{Ω}W_{0}(X )dX). 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.

#### Acknowledgement

The authors acknowledge the support of the Escuela Technical Superior de Ingenious Industrials (UNED) of Spain, project 2019- IFC02, and of the University Polytechnical de Madrid (UPM) (Research groups 2019). This work is also partially support by the Project MTM2013-42907-P from MICINN (Spain).

#### References

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