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Immunotherapy is one of the future treatments applicable in most cases of cancer including malignant cancer. Malignant cancer usually prevents some genes, e.g., p53 and pRb, from controlling the activation of the cell division and the cell apoptosis. In this paper, we consider the interactions among the cancer cell population, the effector cell population that is a part of the immune system, and cytokines that can be used to stimulate the effector cells called the IL-2 compounds. These interactions depend on both time and spatial position of the cells in the tissue. Mathematically, the spatial movement of the cells is represented by the diffusion terms. We provide an analytical study for the constant equilibria of the reaction-diffusion system describing the above interactions, which show the initial behaviour of the tissue, and we conduct numerical simulation that shows the dynamics along the tissue that represent the immunotherapy effects. In this case, we also consider the steady-state conditions of the system that show the long-time behaviour of these interactions.

Cancer is one of the malignant diseases triggered by gene mutations on the cells in the tissues. The gene mutations affect the cell cycle, DNA damages, and some anomalies on the cells. Some anomalies’ behaviour of the enzymes in the cells due to the DNA damage caused by cancer has been shown in [

The mathematical model that shows the dynamics of cancer on the tissues has been given in [

In the subcellular level, the cancer infections are mainly caused by the gene mutation represented by the shifting concentration of the enzymes. There are some enzymes that can be used as the indicators of the mutation, e.g., p53, pRb, and EBNA1; see [

The immunotherapy model of cancer that involves the interaction between the cancer cells, the effector cells, and the IL-2 compounds was motivated by Kirschner et al. [

The other immunotherapy method for cancer is done by using oncolytic virus; see [

In the real situation, the growth direction of the cancer cells depends on the weakest parts of the tissue. The evolution profiles of the infection in a tissue are interesting to study. Therefore, following the results in [

Our model is a three-dimensional system of partial differential equations, and the analysis is focused on the study of the steady state and the cycle of infection of the disease that is represented by the limit cycle of the system. We apply the Runge–Kutta method to determine the solutions of the system numerically. It is important to understand the behaviour of the system.

We separate the cell populations into three parts, which are the cancer cells (

In our model, we use constant parameters that have the following meanings. The antigenicity for the cancer cells that measures the ability of the immune system to recognize cancer via the non-self-protein antigens and the average of the natural lifespan of the effector cells are represented by parameters

The cancer growth is assumed to follow the logistic model where the constant birth rate is

Our model is formulated as a three-dimensional system of PDE as follows:

We apply the nondimensional transformation for the variables and parameters of system (

The initial condition of system (

The analysis of the cancer-free equilibrium is important to determine the conditions of the patients to be cured from cancer. For the numerical simulation, we focus on the dynamics of the solutions near the cancer-free equilibrium.

System (

If

If

If

If

The proof of the theorem is given by using the linearization of system (

In the beginning of cancer invasions, the cancer cells enter the tissue via the epithelial layer. In this case, the effector cells and the IL-2 compounds, which are parts of the immune system, will respond to attack the cancer cells before they enter the body via the outer epithelial layer.

The numerical data that we used in this paper are based on the clinical data from [

In Figure

The initial position of the effector cell, the cancer cell, and the IL-2 compound concentrations. (a) Concentration of effector cells,

In the following sections, we perform some numerical simulations to study the dynamics of system (

We consider the dynamics of system (

According to Theorem

Evolution profiles of the effector cell (blue), the cancer cell (green), and IL-2 (red) concentrations for the cancer cell random motility coefficient

In Figure

In Figure

Evolution profiles of the effector cell (blue), the cancer cell (green), and IL-2 (red) concentrations for the cancer cell random motility coefficient

By the results in Figures

The trajectory and the limit cycle of system (

In this section, we consider the case that

The evolution profile of the effector cell, cancer cell, and the IL-2 compound concentrations is shown in Figure

Evolution profiles of the effector cell (blue), the cancer cell (green), and IL-2 (red) concentrations for the cancer cell random motility coefficient

The evolution profile of the cancer cells’ concentrations for the nontreatment case is shown in Figures

The spatial-temporal evolutions of the cancer cells’ diffusion coefficients for

The spatial-temporal evolutions of the cancer cells’ diffusion coefficients for

From the results in Figure

Different situations occur in system (

From Figures

The spread of the cancer cells on the tissue does not only depend on the time but also on the position of the cancer cells in the tissue. There are two important parameters in our system that represent the antigenicity for cancer and the motility of the cancer cells. The higher motility of the cancer cells implies that the spread of the cancer cells in the tissue becomes faster. In this case, the lower antigenicity causes higher concentration of the cancer cells to spread faster in the tissue. For the higher antigenicity, although the cancer cells cover most of the tissue, the cells’ concentration is on the intermediate level.

One of the important phenomena in our system is the appearance of the stable limit cycle when we choose a certain value of the diffusion parameter of the cancer cells. The limit cycle behaviour is usually caused by the Hopf bifurcation when the parameter value is varied. This bifurcation is one of the entry points to the possibility for the system to have chaotic behaviour that represents the metastases of the cancer cells. The analysis of this case is one of the open problems for our system.

The other open problem is that, in the real situation, the cancer cells can grow and spread to other tissues and organs via blood vessels. In this case, we can approach the model using the moving boundary condition that determines the spread of the cancer cells in the body.

The authors do not use the data from any repositories to provide the results of the research. For the numerical simulations, the value of parameters are adopted from the reference [

The authors declare that they have no conflicts of interest.

This work was partially supported by the Deputy for Research and Strengthening Development of the Ministry of Research and Technology, Indonesia (the National Research and Innovation Agency), which has provided research funding through Penelitian Dasar Unggulan Perguruan Tinggi (PDUPT) with contract agreement numbers 6/AMD/E1/KP.PTNBH/2020 and 2773/UN1.DITLIT/DIT-LIT/PT/2020. The authors would also like to thank some colleagues in UGM and the Cancer Modelling Team, UGM, for the discussions during the research.