br The outline of the paper
The outline of the paper is as follows. In Section 2, we give a brief overview of the modeling of the cancer L-NAME hydrochloride invasion into the extracellular matrix. In Section 3, we describe the adaptive moving mesh finite difference method. Section 4 provides the numerical details of the adaptive mesh algorithm and the solution procedure for the solution of the cancer cell invasion model and the adaptive mesh. Section 5 presents several numerical experiments to demonstrate the performance and efficiency of the proposed adaptive mesh method for solving the cancer cells invasion model. In Section 6, a discussion of the results and some concluding remarks are given.
2. The cancer cell invasion model
In this section, we give an overview of the mathematical model of the cancer cell invasion of tissue first introduced by Chaplain and Lolas  and subsequently described by Andasari et al. . The main part in the process of the cancer cell invasion into the surrounding tissue or extracellular matrix (ECM) is the over-expression of proteolytic enzymes, such as the urokinase-type plasminogen activator (uPA) and matrix metalloproteinases (MMPs), by cancer cells to breakdown ECM proteins which enables the cancer cells to migrate through the tissue. As in [4,5], the enzymatic system considered here is the urokinase plasminogen activation system (uPA system) which consists of: (1) uPA, urokinase plasminogen activator; (2) uPAR, urokinase plasminogen activator receptor; (3) plasmin, the degrading enzyme; (4) VN, the ECM protein vitronectin; and (5) PAI-1, the plasminogen activator inhibitor type-1. The process of the cancer cell migration and invasion into the surrounding ECM can be described by a dynamical system consisting of the five concentrations uPA, PAI-1, plasmin, uPAR and VN. The mechanism of the main interactions of the system is illustrated by the schematic diagram shown in Fig. 1.
The mathematical model describing the interactions between the five concentrations (uPAR, uPA, PAI-1, plasmin, and ECM component VN) is a nonlinear system consisting of five reaction–diffusion-taxis partial differential equations. The model, in dimensionless units, is given as follows (see [4,5] for more details):
where c = c(x, t) is the cancer cells density, v = v(x, t) denotes the VN concentration, u = u(x, t) represents the uPA concentration, p = p(x, t) is the PAI-1 concentration, m = m(x, t) denotes the plasmin concentration, and ∇ and ∆ are the spatial gradient and Laplacian operators. The system (1a)–(1e) is defined on a physical domain Ω representing the region of the ECM. The solution of the system (1a)–(1e) is determined uniquely using appropriate initial and boundary conditions.
In (1a), the parameter Dc denotes the random motility or diffusion rate of the cancer cells, where χu and χp are chemotaxis rates of the cancer cells due to the presence of uPA and PAI-1 concentrations, respectively. Haptotaxis rate χv of cancer cells is due to VN concentration, and µ1 is the cancer cells’ proliferation rate. In (1b), the parameter δ indicates the degradation rate of VN by plasmin that is balanced by PAI-1 inhibitor at a rate φ22, where the VN growth rate φ21 is indirectly affected by the PAI-1 and uPA interaction, and the proliferation rate of VN is given by the parameter µ2. In (1c), Du denotes the diffusion rate of uPA, φ31 is the interaction rate between uPA and PAI-1, φ33 gives the rate of interaction of uPA with its receptor uPAR, and α31 is the uPA’s production rate by cancer cell. In (1d), Dp denotes the PAI-1’s diffusion rate, φ41 and φ42 denote the interaction rates of PAI-1 with uPA and VN concentration, respectively, α41 is the production rate of PAI-1 due to plasmin. In (1e), Dm is the diffusion rate of plasmin; φ52 and φ53 are the binding rates of PAI-1 to VN and uPA to cancer cells, respectively; and φ54 is the degradation rate of plasmin.