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A periodic mathematical model of cancer treatment by radiotherapy is presented and studied in this paper. Conditions on the coexistence of the healthy and cancer cells are obtained. Furthermore, sufficient conditions on the existence and globally asymptotic stability of the positive periodic solution, the cancer eradication periodic solution, and the cancer win periodic solution are established. Some numerical examples are shown to verify the validity of the results. A discussion is presented for further study.

Cancer is a well-known killer of humans worldwide, and its treatments are varied and sporadically successful. There are four main types of cancer treatments, which are surgery, chemotherapy, radiotherapy, and immunotherapy. In this paper, we only consider cancer treatment by radiotherapy.

Radiotherapy, as a primary treatment strategy, has been proven to be an effective tool in combating with cancer [

It is an important and effective way to deeply understand the real-world problems by establishing mathematical models and analyzing their dynamical behaviors (see [

This paper is organized as follows. In Section

To simplify the model, we assume that the concentrations of cancer and healthy cells exist in the same region of the organism; the administration of radiation removes a large amount of cancer cells and a small amount of healthy cells from the system. Here, the terms “large” and “small” are used as a relation to the appropriate cell population at a particular location in the organism. Radiotherapy is in fact a control mechanism on the rates of change of the concentrations of cancer and healthy cells by harvesting them.

In a given tissue, let

In the absence of radiation, cancer (i.e.,

According to biological interpretation, we only consider the nonnegative solutions. Hence, we suppose that

(i) Nonnegative quadrant of

Let

(i) If

(ii) From the first equation of system (

In this section, we will investigate the coexistence of the healthy and cancer cells. We will find that when the radiation dosage

To understand the model more clearly, we rewrite system (

Before giving the main result of this section, we firstly consider the following two-specie Lotka-Volterra competitive system:

System (

System (

On the coexistence of the healthy cells and cancer cells, we have the following theorem.

Assume that the following conditions

From system (

Condition (

(a) Consider

(b) Consider

(c) Consider

Condition (

Under conditions of Theorem

The existence of a positive

Firstly, let us investigate the existence of cancer eradication periodic solution of the system. Consider the following subsystem of system (

System (

The existence of the cancer win periodic solution of system (

System (

Up to now, we have completed the studies on the existences of the positive periodic solution, the cancer eradication periodic solution, and the cancer win periodic solution of system (

Let

The positive periodic solution

On the uniqueness and global stabilities of the cancer eradication periodic solution and cancer win periodic solution, we have the following results.

Assume that condition

The existence of the cancer eradication periodic solution has been established by Theorem

Let

However, when

By a simple calculation and from the last inequality of condition (

Assume that condition

Let

In the discussion of the existence and global stability of the positive periodic solution, we need the conditions that

In this section, we give three groups of numerical examples to verify the validity of the three cases of periodic solutions, respectively. We consider system (

The dynamics of system (

The dynamics of system (

The dynamics of system (

From Figures

To show the existence and global stability of the cancer eradication periodic solution, we illustrate Theorem

(a) The time series for the cancer eradication periodic solution

The dynamics of system (

The dynamics of system (

It can be seen from Figures

As we all know, larger dosage radiation can kill cancer cells more effectively, but it also may increase the rate to the healthy cells from the radiation. We now investigate effects of the variance of the parameter

If we take

(a) The time series for the cancer eradication periodic solution

The dynamics of system (

The dynamics of system (

It is also a helpful suggestion for doctors that under what situation the cancer will win the competition. Theorem

(a) The time series for the cancer win periodic solution

The dynamics of system (

The dynamics of system (

Throughout Figures

In this paper, we employed a pair of ordinary differential equations to model the dynamics between the healthy cells and cancer cells for the cancer treatment by radiotherapy. We separated the treatment into two stages: treatment stage and recovery stage (no treatment stage). During the treatment stage, the radiation harvesting amount is

The cancer treatment model discussed in this paper is only based on one treatment measure. It may be more effective for the cancer treatment if we add a medication during the recovery stage, which is still an open problem and we will carry out the research in the further work.

The authors declare that there is no conflict of interests regarding the publication of this paper.

This research was supported by the National Natural Science Foundation of China (11401060) and Zhejiang Provincial Natural Science Foundation of China (LQ13A010023).