On strategies on a mathematical model for leukemia therapy

doi:10.1016/j.nonrwa.2011.02.027

Nonlinear Analysis: Real World Applications

Volume 13, Issue 3, June 2012, Pages 1044-1059

https://doi.org/10.1016/j.nonrwa.2011.02.027 Get rights and content

Abstract

A mathematical model for leukemia therapy based on the Gompertzian law of cell growth is studied. It is assumed that the chemotherapeutic agents kill leukemic as well as normal cells.

Effectiveness of the medicine is described in terms of a therapy function. Two types of therapy functions are considered: monotonic and non-monotonic. In the former case the level of the effect of the chemotherapy directly depends on the quantity of the chemotherapeutic agent. In the latter case the therapy function achieves its peak at a threshold value and then the effect of the therapy decreases. At any given moment the amount of the applied chemotherapeutic is regulated by a control function with a bounded maximum. Additionally, the total quantity of chemotherapeutic agent which can be used during the treatment process is bounded too.

The problem is to find an optimal strategy of treatment to minimize the number of leukemic cells while at the same time retaining as many normal cells as possible.

With the help of Pontryagin’s Maximum Principle it was proved that the optimal control function has at most one switch point in both monotonic and non-monotonic cases for most relevant parameter values.

A control strategy called alternative is suggested. This strategy involves increasing the amount of the chemotherapeutical medicine up to a certain value within the shortest possible period of time, and holding this level until the end of the treatment.

The comparison of the results from the numerical calculation using the Pontryagin’s Maximum Principle with the alternative control strategy shows that the difference between the values of cost functions is negligibly small.

Section snippets

Statement of the problem

A number of mathematical models describing growth of the cancer cells have been proposed [1], [2]. We shall consider a mathematical model which describes the dynamics of leukemia based on Gompertzian law [3], [4], [5], [6].

Let $m = m (t)$ be the number of the leukemic cells at the time $t$ . We describe the dynamics of cancer cells growth (the external influences are not taken into account) by the following equation $\frac{d m}{d t} = r m \cdot ln \frac{m_{a}}{m}, r, m_{a} \in R_{+} (constants) .$

The solution of Eq. (1.1) is given by $m (t) = exp {ln m_{a} - ($

Optimal control strategy

The most important results of this section are the following: in the case of the monotonic therapy functions of the form $f (h) = \frac{λ h}{1 + h}, λ \in R_{+}$ and of the non-monotonic therapy functions of the form $f (h) = λ h exp (- b h) λ, b \in R_{+}$ , the control function $u$ is a piecewise constant function with at most one switch point for the most relevant parameters. More precisely, it depends on the difference $r_{n} - r_{l}$ (see Theorem 1, Theorem 2).

The Hamiltonian of the system (1.11) has the following form [28]: $H = ψ_{1} (- r_{l} l + γ_{l} + f_{l}) + ψ_{2} (-$

Alternative control strategy

Let $f (h)$ be a monotonic function. We shall investigate the behavior of system (1.11) assuming now that $u$ is a constant parameter such that $0 \leq u \leq \bar{R}, \bar{R} \in R_{+} .$

The unique critical point of (1.11) is $l = \frac{γ_{l} + f_{l} (h)}{r_{l}}, n = \frac{γ_{n} + c_{a} e^{- l} + f_{n} (h)}{r_{n}}, h = \frac{u}{γ_{h} + ε (L_{a} e^{- l} + N_{a} e^{- n})} .$

It is easy to prove for $ε = 0$ or a sufficiently small $ε > 0$ that all eigenvalues of the Jacobian matrix of system (1.11) at the critical point are negative. Therefore, the steady-state of the system is stable for small values of $ε$ .

It follows from the implicit

Numerical results

In this section we present some numerical results concerning optimal and alternative strategies for the leukemia therapy. All results are obtained for the same model parameters and constraints: $r_{l} = 0.25, r_{n} = 0.25, γ_{h} = 0.01, c_{a} = 3.7 \cdot 1 0^{- 5}$ , $λ_{l} = 4.0, λ_{n} = 1.8, a_{1} = 4.0$ , $a_{2} = 2.0$ , $b_{1} = 0.01$ , $b_{2} = 0.01 R = 1, Q = 250$ , $L_{a} = 10^{10}$ , $L_{n} = 10^{10}, ε = 0$ . The initial values $N (0) = 10^{8}, L (0) = 1.8 \cdot 10^{8}$ and $\hat{N} = 2.7 \cdot 10^{7}$ are chosen for all numerical calculations (Fig. 1, Fig. 2, Fig. 3, Fig. 4, Fig. 5).

We use the iterated (successive) approximation

Conclusion

Mathematical modeling of leukemia therapy is a complex optimal control problem. The number of cells and the interaction between cells and chemotherapeutic agent are determined by certain nonlinear laws. The cumulative amount of chemotherapeutic agent which can be applied during the therapy, as well as the intensity of this application, is restricted by some prescribed values.

The chemotherapeutic agent destroys not only leukemic but normal cells too, and this is reflected in the asymmetric form

Acknowledgment

The authors thank A.S. Novozhilov for his valuable and constructive comments.

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On strategies on a mathematical model for leukemia therapy

Abstract

Section snippets

Statement of the problem

Optimal control strategy

Alternative control strategy

Numerical results

Conclusion

Acknowledgment

Math. Biosci.

J. Theoret. Biol.

Bull. Math. Biol.

Bull. Math. Biol.

Math. Biosci.

Math. Biosci.

J. Theoret. Biol.

Nonlinear Anal. RWA

Math. Biosci.

Nonlinear Anal. RWA

A history of the study of solid tumour growth: the contribution of mathematical modeling

Bull. Math. Biol.

Mathematical Models of Cancer and their Relevant Insights

A cell kinetics justification for Gompertz’s equation

SIAM J. Appl. Math.

Quiescence as an explanation of Gompertzian tumor growth

Growth Dev. Aging