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Discrete Dynamics in Nature and Society
Volume 2013 (2013), Article ID 841764, 8 pages
Modeling a Tumor Growth with Piecewise Constant Arguments
Department of Mathematics, Faculty of Education, Erciyes University, 38039 Kayseri, Turkey
Received 22 February 2013; Accepted 18 April 2013
Academic Editor: Qingdu Li
Copyright © 2013 F. Bozkurt. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
This study is based on an early brain tumor growth that is modeled as a hybrid system such as (A): , where the parameters , and denote positive numbers, and are negative numbers and is the integer part of . Equation (A) explains a brain tumor growth, where is embedded to show the drug effect on the tumor and is a rate that causes a negative effect by the immune system on the tumor population. Using (A), we have constructed two models of a tumor growth: one is (A) and the other one is a population model at low density by incorporating an Allee function to (A) at time . To consider the global behavior of (A), we investigate the discrete solutions of (A). Examination of the characterization of the stability shows that increase of the population growth rate decreases the local stability of the positive equilibrium point of (A). The simulations give a detailed description of the behavior of solutions of (A) with and without Allee effect.
Cancer biology has been revolutionized over the past several decades. Genetic alterations that lead to malignant phenotypes have been identified [1, 2], and mechanisms necessary to sustain a solid tumor  and that contribute to tumor-cell invasion [4, 5]. Mathematical modeling of both tumor growth and angiogenesis has been active areas of research. Such models can be classified into one of two categories: those that analyze the remodeling of the vasculature while ignoring changes in the tumor mass and those that predict tumor expansion in the presence of a nonevolving vasculature.
The works in [6, 7] are very important, since they developed a two-dimensional hybrid cellular automaton model of brain tumor growth. Showing a simple model for a single species, the well-known model is constructed by May  and May and Oster  such as who obtained that the asymptotic behavior of difference solutions can be complex and “chaotic” for certain parameter values of . In recent years, several studies have been conducted to investigate the difference solutions of specific logistic differential equations with respect to the parameters [10–19]. When in (A),  considered the logistic equation where , the parameters , and denote positive numbers, and denotes the integer part of . Here, the local asymptotic stability of the positive equilibrium point of (2) was proven by using the Linearized Stability Theorem and the global asymptotic stability by using a suitable Lyapunov function.
An investigation of (A) can be shown in , where it was obtained that under the condition and the positive equilibrium point of (A) is locally asymptotically stable if and only if . Furthermore, under specific conditions the global asymptotic stability, the semicycle, and oscillation results of the solutions of (A) were also studied.
An important research for population models was conducted in 1931, where Allee  demonstrated that “Allee effect” occurs when population growth rate is reduced at low population size. The logistic model assumes that per-capita growth rate declines monotonically when the density increases; it is shown, however, that for population subject to an Allee effect, per-capita growth rate gives a humped curve increasing at low density, up to a maximum intermediate density and then declines again. Many theoretical and laboratory studies have demonstrated the importance of the Allee effect in dynamics of small populations; see, for example, [21–28]. From this reasoning, biological facts lead us to assume the Allee function as follows:(a) if , then ; that is, there is no reproduction without partners, (b) for ; that is, Allee effect decreases as density increases,(c); that is, Allee effect vanishes at high density .
In this paper, a single species population (here especially about an early brain tumor growth) is modeled such as where , the parameters , and denote positive numbers, , negative numbers, and denotes the integer part of . The parameter is the population growth rate of this tumor and , and are rates for the delayed tumor volume and basis of a logistic population model. is embedded to show the drug effect on the tumor and is a rate that causes a negative effect by the immune system on the tumor population. In Section 2 we investigate the local and global behaviors of the nonlinear difference solutions of (3) basin under specific conditions. Additionally, a characterization of the stability when the population growth rate increases was also investigated. Section 3 gives results of the local and global asymptotic behaviors of the nonlinear difference solutions of (3) with Allee effect. The discrepancy of the stability behavior with and without Allee effect of (3) will give interesting results, which is discussed in Section 4.
2. Local and Global Asymptotic Stability Analysis
An integration of (A) on an interval of the form leads to In (4) if , then . Let ; it is clear that . This implies that we have positive solutions of (A) for positive initial conditions.
In addition, on an interval of the form one can write (A) as It is well known that (5) is a Bernoulli differential equation, and so for its solutions arewhere . The solution of (6) does not give any information about the global behavior of the differential equation. Hence, we can continue to investigate more about (6), since (6) is the difference equation of second order. Let and , where and are positive numbers. It is important to take into account that the drug therapy has more destroying effect on the tumor than the immune system. That is why . Considering (6) again, we obtain, for , where hereafter Computations reveal that the equilibrium points of (7) are The fundamental study contains the stability analysis of the positive equilibrium point . For this reason we show only the characteristic equation of (7) by linearizing (7) about . Computation gives a quadratic equation such as where .
Theorem 1. Let and . The positive equilibrium point of (7) is locally asymptotically stable if and only if
Proof. By the Linearized Stability Theorem  we get that the positive equilibrium point of (7) is locally asymptotically stable if and only if
holds. We can write (12) such as(a),(b).
Since , from (b) we have
From (a) we get where . Considering both (13) and (14), we will have since . This completes our proof.
Proof. We consider a Lyapunov function defined by
The change along the solutions of (17) is
Considering (18), we get
Furthermore, from (18) we will have whereSince , if then This implies that , which gives the condition for the global asymptotic stability of the positive equilibrium point of (7).
Theorem 3. Let and be population growth rates of (7) such that and suppose that and are positive equilibrium points of (7) with respect to and that hold the conditions in Theorem 1, respectively. Furthermore, assume that where and . If then the local stability of is weaker than . That is, increase of the population growth rate decreases the local stability of the positive equilibrium point in (7).
Proof. Let us write from (7)where . In this case, the positive equilibrium points of (27) are
respectively. The characteristic equations of (27) are
where , respectively. Since , we can write
The inequality (32) can be also written as
Since the inequality
always holds, we get
The inequality (35) can be also written as
It is obvious that from (37), we get
then the inequality
holds. In view of (39), considering (31), (33), (38), and (40) together, we obtain
From the Linearized Stability Theorem we want to obtain the conditions that satisfy the inequality
Simplifications of (42) give us the inequalitySince and , we get
Considering both (44) and (45), we get (42).
Furthermore, from the Linearized Stability Theorem we must also show that the inequality holds. Simplifying (46), we can write If then The inequalities (49) imply that (46) holds. This result explains that increase of the population growth rate decreases the local stability of the positive equilibrium point in (7), which completes our proof.
3. Local and Global Asymptotic Stability Analysis with Allee Effect
In this section we use an Allee function of time . Let (3) be written as where and . Applying to (57) an Allee function where is an Allee constant, we get By defining and taking the derivative of g with respect to , we obtain By showing the sign of (61), we get that is an increasing function for and is a decreasing function for This also means that if the density is then a population model without an Allee function will not give realistic results. But if then it is not important to use a model with an Allee function as it is also explained in the introduction. Applying to (3) an Allee function such as where , we obtain It is clear that (67) is a Bernoulli differential equation on the interval . Solving (67) for and , we get for It can be shown that the equilibrium points of (68) are also the equilibrium points of (7).
Linearizing (68) about the positive equilibrium point, we obtain the characteristic equation as follows: where .
Theorem 5. Let and . The positive equilibrium point of (68) is locally asymptotically stable if
Proof. The proof is similar to that in Theorem 1 and will be omitted.
Proof. The proof is similar to that in Theorem 2 and will be omitted.
Example 7. The goal of this investigation is to examine the development of monoclonal tumors under the effects of treatment. In view of , the carrying capacity of a monoclonal tumor is ca. 38 mm. We select , , and . By dividing these values by the carrying capacity, we obtain , and . These values are also suitable for the hypotheses in Theorems 1 and 5. To have a compatible result, a relation between the model and the data is constructed by multiplying these values with 10. Thus, the parameters for (7) are in this case , , and . For a therapy of ca. 75 mg drug and under the assumption that the effect on the tumor is 1.6% we obtain . The effect on the immune is ca. 10% compared with the effect of the drug treatment, so we have . Figure 1 shows us the behavior of the solution of (7). Differently from this, we can see Figure 2, where we have used the Allee function for . Studies demonstrated that Allee effects play an important role in the stability analysis of equilibrium points of a population dynamics model. Generally, an Allee effect has a stabilizing effect on population dynamics. So, in (68) our expectation is that the chaos begins later as it can be also shown in Figure 2.
Section 2 was constructed to obtain specific conditions for local and global asymptotic stability of the positive equilibrium point of (7) without Allee effect by applying the Linearized Stability Theorem and the theory of the use of a Lyapunov function, respectively. Furthermore, we showed that increase of the population growth rate decreases the stability of the positive equilibrium point of (7), which is given in Theorem 3. Finally, in Section 2 we proved Theorem 4, which has given information about the bound of the solutions of (7). By using the data of  about the radius of tumor at 111 days, we take the radius as 4.96 mm for a population growth rate . By multiplying it with 10, we use the growth rate . The volume of such a tumor will be then mm3. Considering Theorem 4, we can see that the above mentioned value for is suitable. For , , , , and we consider Section 3 in view of , In this case, for the volume mm3 the radius of the tumor must be around 1.0609 mm, which the temporal development of a cross-central section of a tumor growing without angiogenesis show a tumor more than at day 80 and the temporal development of a cross-central section of a tumor growing with angiogenesis around 40 days of the tumor (see ). This means that during the 90 days of the tumor we shall use the model given in (68). For days more than 90, both models are suitable ((7) and (68)). Using the above mentioned values, it can be shown that the local stability of both theorems (Theorems 1 and 5) hold. However, Theorem 5 give us that the stable interval for the growth rate is wide as assumed in Theorem 1, which is important for the drug therapy.
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