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We provide a family of ordinary and delay differential equations to model the dynamics of tumor-growth and immunotherapy interactions. We explore the effects of adoptive cellular immunotherapy on the model and describe under what circumstances the tumor can be eliminated. The possibility of clearing the tumor, with a strategy, is based on two parameters in the model: the rate of influx of the effector cells and the rate of influx of IL-2. The critical tumor-growth rate, below which endemic tumor does not exist, has been found. One can use the model to make predictions about tumor dormancy.

Cancer is one of the most difficult diseases to be treated clinically, and one of the main causes of death. It is the second fatal disease after the cardiovascular diseases. The World Health Organization estimates that the annual cancer-induced mortality number exceeds six million people. Accordingly, the fight against cancer is of major public health interest. For this and other economy-related reasons, a great research effort is being devoted to understand the dynamics of cancer and to predict the impact of any changes on the system reactors. Hence, mathematical models are required to help design therapeutic strategies.

In cancer modeling, we have to care about the scaling problem, where the class of equations, used to describe the model, are to be determined. Indeed, there are three natural scales, which are connected to the different stages of the disease and have to be identified. The first is the subcellular (or molecular) scale, where we focus on studying the alterations in the genetic expressions of the genes contained in the nucleus of a cell, as a result of some special signals, which are received by the receptors on the cell surface and transmitted to the cell nucleus. The second is the cellular scale, which is an intermediate level between the molecular and the macroscopic scale. The third is the macroscopic scale, where we deal with heterogeneous tissues. In the heterogeneous tissues, some of the layers (the external proliferating layer, the intermediate layer, and the inner zone with necrotic cells) constituting the tumor may occur as islands, leading to a tumor comprised of multiple regions of necrosis, engulfed by tumor cells in a quiescent or proliferative state [

A great research effort is being devoted to understand the interaction between the tumor cells and the immune system. Mathematical models, using ordinary, partial, and delay differential equations [

Many mathematical models have been proposed to model the interactions of cytotoxic T lymphocyte (CTL) response and the growth of an immunogenic tumor (see, e.g., [

The treatment of cancer is then one of the most challenging problems of modern medicine. The treatment should satisfy two basic conditions: first, it should destroy cancer cells in the entire body. Second, it should distinguish between cancerous and healthy cells. Other treatments such as surgery and/or chemoand radiotherapies have played key roles in treatment [

Numerous research papers have been made to explore the effects of the immune system in eliminating the tumor cells in the host, by stimulating the host's own immune response to kill cancer cells [

Immunotherapy models and their predictions have been extensively studied in [

In this paper, we investigate mathematical models for the dynamics between tumor cells, immune-effector cells, and the cytokine interleukin-2 (IL-2). It is worth stressing that we operate at a supermacroscopic scale, namely, by ordinary differential equations. However, the link to lower cellular scale is represented by the delay. The delay differential equations have long been used in modeling cancer phenomena [

The organization of this paper is as follows: in Section

The model of Kuznetsov et al. [

The idea in this paper is to simplify the above model and reduce it into a two- or three-equation model to describe the interactions of three types of cell populations: the activated immune-system cells,

System (

Yafia [

The nondimensionalization parameters of bilinear model (

Solution of the DDEs (

Solution of the DDEs (

Solution of the DDEs (

The solutions of practical interest should have nonnegative population

To ease the analysis, we start with the 2-population model (

In case of the three-equation model (

Consider again the two-equation model (

If the endemic equilibrium

Since

Similarly, it is easy to prove the following proportion.

If the point

We extend the above analysis to the case of the three-equation model (

The

In this subsection, we then analyze the bifurcation so that the tumor growth rate

The parameter

If we solve

shows the bifurcation diagrams for the bilinear model (

No Treatment Case,

Interleukin-2 Case

Adoptive cellular immunotherapy case,

Small interval of

Immunotherapy with both ACI and IL-2

Small interval of

Given the threshold point

In addition to the tumor-free equilibrium, (

Equation (

Since the bifurcation direction at the point

Now, to find the conditions, on the model parameters, being required for the existence of the persistent equilibria, we make the use of both Descant's rule of signs and the Sturm sequence [

The number of positive steady states (SS) is determined by the signs of the coefficients (

SS | ||||
---|---|---|---|---|

0 | + | − | − | |

1 | − | − | ||

3 | − | + | − | + |

We may note that the tumor-persistent equilibria do not exist for

If we consider the general case of immunotherapy with both ACI and IL-2 treatments, then according to the conditions given in Table

The two-dimensional transition structure as a function of

The phase spaces at different equilibria where stable steady states [•], unstable steady states [○], stable manifold [- - -], unstable manifold [

Tumor-free equilibrium (

One persistent-tumor equilibrium with a stable limit cycle (

One persistent-tumor equilibrium at little

One persistent-tumor equilibrium (

Three persistent-tumor equilibria with a stable limit cycle (

Three persistent-tumor equilibria (

Let us introduce the following definition to facilitate the analysis.

The threshold parameter

The parameter

The critical

In the tumor-clearance problem, we have the following three cases:

if

if

if

In this paper, we introduced a family of differential models (ODEs and DDEs) to describe the dynamics of cancer. The ODEs models model cancer at supermacroscopic, in the sense that they describe the interaction between the tumor cells and the normal (immune) cells [

The numerical simulations (have been obtained by semi-implicit RK methods [

The authors would like to thank the referees and Professor Pedro Serranho for their valuable comments on the paper. The first author should thank Emirates Foundation for partially funding this research.