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Advances in Difference Equations

, 2012:217 | Cite as

A mathematical model with time-varying delays in the combined treatment of chronic myeloid leukemia

  • Leonid Berezansky
  • Svetlana Bunimovich-MendrazitskyEmail author
  • Alexander Domoshnitsky
Open Access
Research
Part of the following topical collections:
  1. Progress in Functional Differential and Difference Equations

Abstract

In this paper, we propose and analyze a mathematical model for the treatment of chronic myelogenous (myeloid) leukemia (CML), a cancer of the blood. Our main focus is on the combined treatment of CML based on imatinib therapy and immunotherapy. Treatment with imatinib is a molecular targeted therapy that inhibits the cells involved in the chronic CML pathogenesis. Immunotherapy based on interferon alfa-2a (IFN-α) increases cancer cell mortality and leads to improvement of outcomes of the combined therapy. Interaction between CML cancer cells and effector cells of the immune system is modeled by a system of non-linear differential equations, where we introduced biologically motivated time-varying delays in the treatment terms. The analysis of the described system shows the existence of a unique global positive solution and a unique non-trivial equilibrium. We also derive explicit local and global stability conditions for the non-trivial equilibrium.

Keywords

mathematical model time delay global stability CML treatment 

1 Introduction and biological motivation

Chronic myelogenous leukaemia (CML) is a progressive, malignant disease characterized by a large number of abnormal blood cells in the bone marrow and peripheral blood [1, 2]. The schematic progression in time of myelogenous leukemia is shown in Figure 1.
Figure 1

A scheme of myelogenous leukemia progression in time. Myelogenous leukemia progresses through four distinct phases. After an initial period of 3-5 years the abnormal cell counts rise to a relatively steady state, called the chronic phase. At this stage the disease can be diagnosed. Several years of the chronic phase where the abnormal cell count oscillates with a period of the order of months are followed by a phase characterized by oscillatory instability (the acceleration phase). Ultimately, this leads to the usually fatal acute phase with sharp increase in the abnormal cell count. This is known also as the blast crisis [15].

The benefit of the treatment of CML with imatinib (Gleevec (Novartis International AG, Basel, Switzerland)) was first indicated in 2000 [1, 3]. However, the loss of drug effectiveness was observed due to mutations of the target cancer cells, rendering the cells resistant to the drug [4, 5]. About 10 to 20% of patients do not respond to imatinib after prolonged therapy [6, 7].

One of the early mathematical models of CML was a model created by Fokas et al. (1991) [8] which includes hematopoietic process focused on the maturation and proliferation of the T cell precursors in the bone marrow. One of the models of anticancer CML therapy is Moore and Lee’s work [9] based on immune cell interactions with CML. Several recent mathematical models have been developed to study the dynamics of CML under imatinib treatment, including Komarova and Wodarz (2005) [10], Michor et al. (2005) [11], Nanda et al. (2007) [12], Kim et al. (2008) [13], and Paquin et al. (2011) [14]. In all of these studies, the authors conclude that imatinib does not completely eliminate the leukemia cell population and propose that imatinib therapy should be combined with an additional form of treatment.

In this paper, we present a model that combines imatinib with the interferon alfa-2α (IFN-α) immunotherapy. Including IFN-α treatment, we strengthen the immune response. We show that the immune response may play an important role in determining the length of time during treatment with imatinib for CML patients and would allow to keep the patient in the chronic stage for a longer period of time.

Recent data shows that IFN-α activation of immune cells (T killer cells, natural killer cells and others) contributes to the killing of cancer leukemic cells [16, 17]. Moreover, it is already proven that IFN-α may extend imatinib effect by activating immunological effector functions [18]. Therefore, we assume that the combination treatment of imatinib and IFN-α immunotherapy can be a good candidate to improve the current CML therapy.

The mathematical description of the above model is described in Section 2. In Section 3 we discuss the existence of a positive and unique equilibrium and prove local and global stability of this equilibrium. Section 4 validates the model results using published data taken from in vitro, mouse and human studies.

2 Model description

The dynamics of the interaction between the immune system (effector T cells) and CML cancer cells in the body can be described by the following system of two ordinary non-linear differential equations:
x ˙ ( t ) = β 1 x ( t ) ln K x ( t ) γ 1 x ( t ) y ( t ) ω γ 3 x ( h ( t ) ) , y ˙ ( t ) = β 2 x ( t ) η 1 + x ( t ) y ( t ) γ 2 x ( t ) y ( t ) + in α γ 4 y ( t ) η 2 + y ( t ) y ( t τ ) μ y y ( t ) , Open image in new window
(1)

where x ( t ) Open image in new window is the CML cells population, y ( t ) Open image in new window is the number of effector T (CTL) cells and K > 0 Open image in new window, τ > 0 Open image in new window, β i > 0 Open image in new window, γ i > 0 Open image in new window, ω > 0 Open image in new window, in α > 0 Open image in new window, h ( t ) t Open image in new window is a continuous function such that t h ( t ) θ Open image in new window, θ > 0 Open image in new window, lim t ( t h ( t ) ) = 0 Open image in new window.

We will analyze system (1) behavior under the following initial conditions:
x ( t ) = φ ( t ) , y ( t ) = ψ ( t ) , t 0 . Open image in new window
(2)

Here, we assume that φ ( t ) Open image in new window, where θ t 0 Open image in new window and ψ ( t ) Open image in new window, where τ t 0 Open image in new window are nonnegative continuous functions where x ( 0 ) = φ ( 0 ) > 0 Open image in new window, y ( 0 ) = ψ ( 0 ) > 0 Open image in new window, φ ( t ) φ ( 0 ) Open image in new window, ψ ( t ) ψ ( 0 ) Open image in new window.

The first term on the right-hand side of the first equation of system (1) describes the growth of CML cancer cells population in the form of the Gompertz law with the growth rate β 1 Open image in new window. The Gompertz curve provides a significantly better fit for leukemic cancer data than logistic, exponential or polynomial curves [9]. The constant K in the first term represents the maximum carrying capacity of CML cells compartment [19, 20]. The second term of the first equation of system (1) accounts for the loss of CML cancer cells due to their interaction with CTL cells. The third term of this equation describes inhibition of cancer cells by imatinib, where ω is a dose of imatinib given every day and h ( t ) Open image in new window is the time-varying function accounting for the delay of the impact of imatinib treatment on the number of cancer cells in the blood. The presence of such a time lag was shown by Volpe [6], who demonstrated that reduction in the number of cancer cells does not occur immediately after the start of imatinib administration but rather after a certain time period. The maximum delay time (θ) is about three weeks. The influence of drugs tends to zero over time, so a natural candidate for the delay function is h ( t ) = t θ e λ t Open image in new window. Hence, this function satisfies the condition that lim t ( t h ( t ) ) = 0 Open image in new window in the formulation of the model (1).

The second equation describes the dynamic balance between stimulatory and inhibitory effects of CTL cells. The first term represents the growth of the population of the effector CTL cells y ( t ) Open image in new window due to the influence of CML antigen in the lymph nodes, where β 2 Open image in new window is the rate of this growth and η 1 Open image in new window is the standard half-saturation concentration in the Michaelis-Menten kinetics. The second term describes the loss of CTL cells due to the interaction between CTL and CML cancer cells with a rate γ 2 Open image in new window. CTL cells survive many hits by the target CML cells until they are inactivated and die [21]. The third term describes the stimulatory augmentation of the CTL cells due to IFN-α immunotherapy, where in α Open image in new window is the dose of IFN-α. Interferon-α leads to increased expression of other cytokines, such as interferon-γ that creates the pro-inflammatory environment with delay τ of about seven days [22]. In this term, η 2 Open image in new window is the standard half-saturation concentration of CTL immune within the Michaelis-Menten kinetics. Finally, the last term in the second equation describes the loss of CML cells due their natural death at a rate μ y Open image in new window.

It is well known from the medical practice that the value of x ( t ) Open image in new window of the population of CML cells cannot be reduced to zero by treatment of any type, so the therapy can be called successful if the value of x ( t ) Open image in new window does not increase over time. Imatinib has a marked inhibitory effect on the value of x ( t ) Open image in new window, while IFN-α add-on has an additional effect due to the stimulation of CML cell production. From mathematical standpoint, the role of the variable y ( t ) Open image in new window is to decrease the value of x ( t ) Open image in new window.

We used the model developed by Moore and Li [9] to explore treatment combinations. The difference between our model and the model by Moore and Li [9] is that we simplified the model describing the dynamics of their system by removing the third equation accounting for the behavior of the naive effector cells. Also, we added in the first equation the term accounting for the reduction in the number of cancer cells due to drug administration, while in the second equation we added the term describing the rate of increase in the number of immune cells that results from the treatment.

In our mathematical analysis of the above system, we aspire to stabilize the value of x ( t ) Open image in new window in such a way that x ( t ) Open image in new window does not exceed the limit of cancer CML cells in blood, which is characteristic of a chronic phase. The summary of values of the parameters used in the model is presented in Table 1.
Table 1

Parameter values

Param

Physical interpretation

Estimated value (units)

Reference

τ

delay for development of CTL cells

7 [days]

[9]

θ

maximal period to react to imatinib

20 [days]

[6]

μ x Open image in new window

death rate of cancer cells

0.2 [0, 0.8] [days−1]

[23]

μ y Open image in new window

death rate of effector T cells

0.06 [0, 0.5] [days−1]

[24]

η 1 Open image in new window

saturation effect of CML cells in the lymph nodes

100 [cells][ml−1]

[9]

η 2 Open image in new window

saturation effect of immune cell recruitment by cancer cells

2 × 107 [cells][ml−1]

[24]

β 1 Open image in new window

growth rate of CML cancer cells in the form of the Gompertz law

0.03 [0; 0.5] [days−1]

[9]

β 2 Open image in new window

change in the effector T cell (y(t)) population due to encounters with CML antigen

0.41 × 0.001 [days−1]

[25], [26]

γ 1 Open image in new window

loss of CML cancer cells due to encounters with the effector T cells

0.005 [days 1 [ cells ml ] 1 ] Open image in new window

[9]

γ 2 Open image in new window

loss of CTL cells due to these encounters between CTL and CML cancer cells

0.005 [days 1 [ cells ml ] 1 ] Open image in new window

[26]

γ 3 Open image in new window

factor using imatinib treatment

0.00014 [mg]−1

Estimated

γ 4 Open image in new window

factor using IFN-a treatment

0.005 [mg]−1

Estimated

ω

once-daily dose of imatinib

400 - 800 [mg/day]

[1]

i n α Open image in new window

IFN-a dose

13 [mg/days] (90 [mg] weekly)

[27]

K

constant, the maximum possible concentration of CML

[1.5 × 105;4 × 105] [cells/ml]

[9]

3 The local and global stability of combined model for tumor immunotherapy

3.1 Positivity and uniqueness of solution system (1)-(2)

In this section we discuss the positivity and uniqueness of the solution ( x ( t ) , y ( t ) ) Open image in new window of system (1)-(2).

Lemma 1 [28]

Consider the initial value problem for a scalar linear delay equation
u ˙ ( t ) = a ( t ) u ( h ( t ) ) , t 0 , u ( t ) = φ ( t ) , t < 0 , u ( 0 ) = u 0 . Open image in new window
If
a ( t ) 0 , t h ( t ) δ , δ sup a ( t ) 1 e , 0 φ ( t ) u 0 , u 0 > 0 , Open image in new window
then u ( t ) > 0 Open image in new window, t > 0 Open image in new window and for the solution of the inequality
v ˙ ( t ) a ( t ) v ( h ( t ) ) , t 0 , v ( t ) = u ( t ) , t 0 , Open image in new window

one has v ( t ) u ( t ) Open image in new window.

Theorem 1 Suppose that x ( 0 ) K Open image in new window, ω γ 3 θ 1 e Open image in new window. Then system (1)-(2) has the unique global solution ( x , y ) Open image in new window such that
0 < x ( t ) K , 0 < y ( t ) y ( 0 ) e ( | β 2 μ y | + in α γ 4 ) t , t > 0 . Open image in new window
(3)
Proof System (1) can be presented in the following form:
x ˙ ( t ) = f ( x ( t ) , y ( t ) , x ( h ( t ) ) ) , y ˙ ( t ) = g ( x ( t ) , y ( t ) , y ( t τ ) ) , Open image in new window

where f ( u , v , w ) Open image in new window, g ( u , v , w ) Open image in new window are continuous Lipschitz functions on the domain a u u b u Open image in new window, a v v b v Open image in new window, a w w b w Open image in new window for any positive constants a u Open image in new window, b u Open image in new window, a v Open image in new window, b v Open image in new window, a w Open image in new window, b w Open image in new window. Using Theorem 2.2.1 from [29], we can state that there is a unique local solution of system (1)-(2). Since x ( 0 ) > 0 Open image in new window, y ( 0 ) > 0 Open image in new window, this local solution is positive.

Denote by [ 0 , c ) Open image in new window the maximum existence interval of the system’s solution. For y ( 0 ) > 0 Open image in new window we have from the second equation of system (1)
y ( t ) = y ( 0 ) e 0 t ( β 2 x ( s ) η 1 + x ( s ) γ 2 x ( s ) + in α γ 4 1 η 2 + y ( s ) y ( s τ ) μ y ) d s . Open image in new window
(4)

Hence y ( t ) > 0 Open image in new window, t [ 0 , c ) Open image in new window.

Suppose x ( 0 ) < K Open image in new window, which means that 0 < x ( t ) < K Open image in new window, t [ 0 , c ) Open image in new window or there is a t 1 > 0 Open image in new window such that x ( t 1 ) = K Open image in new window. Then x ˙ ( t 1 ) < 0 Open image in new window so x ( t ) < K Open image in new window, t > t 1 Open image in new window or there is a t 2 > t 1 Open image in new window such that x ( t ) < K Open image in new window, t 1 < t < t 2 Open image in new window and x ( t 2 ) = K Open image in new window. It is clear that x ˙ ( t 2 ) < 0 Open image in new window, which allows us to obtain that x ( t ) K Open image in new window for t t 1 Open image in new window. Repeating the process of our reasoning, we prove that x ( t ) K Open image in new window, t t 1 Open image in new window.

The case x ( 0 ) = K Open image in new window is same as the case (2) if we replace the point t 1 = 0 Open image in new window.

Now, let us estimate y ( t ) Open image in new window behavior.

Having
y ˙ ( t ) | β 2 μ y | y ( t ) + in α γ 4 y ( t τ ) , Open image in new window
we can state that
y ( t ) y ( 0 ) + 0 t ( | β 2 μ y | y ( s ) + in α γ 4 y ( s τ ) ) d s y ( 0 ) + 0 t ( | β 2 μ y | + in α γ 4 ) max τ ξ s y ( ξ ) d s . Open image in new window
Denote z ( t ) = max τ ξ t y ( ξ ) Open image in new window. Then
z ( t ) y ( 0 ) + 0 t ( | β 2 μ y | + in α γ 4 ) z ( s ) d s . Open image in new window
From the well-known Gronwall-Bellman inequality, we have
0 < y ( t ) z ( t ) y ( 0 ) e ( | β 2 μ y | + in α γ 4 ) t . Open image in new window

Hence, the inequalities in (3) hold for t [ 0 , c ) Open image in new window.

Suppose that c < Open image in new window. Then there are only two possibilities:
  1. (a)

    lim t c x ( t ) = + Open image in new window or lim t c y ( t ) = + Open image in new window;

     
  2. (b)

    lim t c x ( t ) = 0 Open image in new window or lim t c y ( t ) = 0 Open image in new window.

     

By inequalities (3), x ( t ) Open image in new window and y ( t ) Open image in new window are bounded on any final interval. Then (a) is impossible.

By (4) relation lim t c y ( t ) = 0 Open image in new window is also impossible.

Suppose now that lim t c x ( t ) = 0 Open image in new window. There exists C > 0 Open image in new window such that 0 < y ( t ) < C Open image in new window, t [ 0 , c ) Open image in new window. Hence, the first equation in (1) implies
x ˙ ( t ) γ 1 C x ( t ) ω γ 3 x ( h ( t ) ) . Open image in new window
Consider the following initial value problem:
w ˙ ( t ) = γ 1 C w ( t ) ω γ 3 w ( h ( t ) ) , w ( t ) = x ( t ) , t 0 . Open image in new window
(5)
After substitution w ( t ) = e γ 1 C t u ( t ) Open image in new window, equation (5) has a form
u ˙ ( t ) = ω γ 3 e γ 1 C ( t h ( t ) ) u ( h ( t ) ) . Open image in new window
(6)
We have 0 t h ( t ) θ Open image in new window. Hence
ω γ 3 e γ 1 C ( t h ( t ) ) θ ω γ 3 θ 1 e , t [ 0 , c ] . Open image in new window

Lemma 1 and the inequality u ( t ) < u ( 0 ) Open image in new window, t < 0 Open image in new window imply that u ( t ) > 0 Open image in new window, t [ 0 , c ] Open image in new window, then also w ( t ) > 0 Open image in new window, t [ 0 , c ] Open image in new window. Hence, there exists δ > 0 Open image in new window such that w ( t ) > δ Open image in new window, t [ 0 , c ] Open image in new window. By Lemma 1 we have x ( t ) w ( t ) δ > 0 Open image in new window, t [ 0 , c ) Open image in new window. This is a contradiction to the assumption lim t c x ( t ) = 0 Open image in new window.

Hence, c = Open image in new window and the theorem is proven. □

Theorem 2 If
μ y > β 2 + in α γ 4 , Open image in new window
(7)

there is the unique equilibrium ( X , 0 ) Open image in new window of system (1) where X = K e ω γ 3 β 1 Open image in new window.

Proof It is obvious that ( X , 0 ) Open image in new window is an equilibrium of system (1). We have to prove the uniqueness only of this equilibrium or, in another words, we have to prove there is no other equilibrium ( X 0 , Y 0 ) Open image in new window, where X 0 > 0 Open image in new window, Y 0 > 0 Open image in new window.

Suppose that ( X 0 , Y 0 ) Open image in new window, where X 0 > 0 Open image in new window, Y 0 > 0 Open image in new window is the equilibrium of system (1). Then, from the second equation in (1), we have
β 2 X 0 η 1 + X 0 γ 2 X 0 + in α γ 4 Y 0 η 2 + Y 0 μ y = 0 . Open image in new window
Hence,
Y 0 = η 2 [ μ y + γ 2 X 0 β 2 X 0 η 1 + X 0 ] in α γ 4 μ y γ 2 X 0 + β 2 X 0 η 1 + X 0 . Open image in new window
Thus, we have
μ y + γ 2 X 0 β 2 X 0 η 1 + X 0 μ y β 2 > 0 , in α γ 4 μ y γ 2 X 0 + β 2 X 0 η 1 + X 0 in α γ 4 μ y + β 2 < 0 . Open image in new window

Then Y 0 < 0 Open image in new window contradicts our assumption and, therefore, system (1) has only one equilibrium ( X , 0 ) Open image in new window. □

In the rest of the paper we assume that the conditions of Theorems 1 and 2 hold.

3.2 Local stability of the equilibrium ( X , 0 ) Open image in new window

To analyze local stability of the equilibrium ( X , 0 ) Open image in new window for system (1), we will use the lemma defined below. Consider the scalar linear equation
x ˙ ( t ) + k = 1 m a k x ( h k ( t ) ) = f ( t ) , Open image in new window
(8)

where a k > 0 Open image in new window, lim t h k ( t ) = Open image in new window.

Lemma 2 [30]

If lim sup t > 0 k = 1 m a k ( t h k ( t ) ) < 3 2 Open image in new window, lim t f ( t ) = 0 Open image in new window, then for any solution x of (8), lim t x ( t ) = 0 Open image in new window.

Theorem 3 If
ω γ 3 < β 1 , γ 2 X + μ y > β 2 X η 1 + X , Open image in new window
(9)

then the equilibrium ( X , 0 ) Open image in new window is locally asymptotically stable.

Proof After substitution of x ( t ) = u ( t ) + X Open image in new window, y ( t ) = v ( t ) Open image in new window, system (1) can be rewritten in the following form:
u ˙ ( t ) = β 1 ( u ( t ) + X ) ln K u ( t ) + X u ˙ ( t ) = γ 1 ( u ( t ) + X ) v ( t ) ω γ 3 ( u ( h ( t ) ) + X ) , v ˙ ( t ) = β 2 u ( t ) + X η 1 + u ( t ) + X v ( t ) γ 2 ( u ( t ) + X ) v ( t ) v ˙ ( t ) = + in α γ 4 v ( t ) η 2 + v ( t ) v ( t τ ) μ y v ( t ) . Open image in new window
(10)
The linearized system (10) has a form of
u ˙ ( t ) = ( β 1 ω γ 3 ) u ( t ) γ 1 X v ( t ) ω γ 3 u ( h ( t ) ) , v ˙ ( t ) = ( γ 2 X + μ y β 2 X η 1 + X ) v ( t ) . Open image in new window
(11)

The second inequality (9) implies that the second equation of (11) is exponentially stable. Hence lim t v ( t ) = 0 Open image in new window.

The first equation can be rewritten in the form (8), where m = 2 Open image in new window, a 1 = β 1 ω γ 3 > 0 Open image in new window, a 2 = ω γ 3 > 0 Open image in new window, h 1 ( t ) = t Open image in new window, h 2 ( t ) = h ( t ) Open image in new window, f ( t ) = γ 1 X v ( t ) Open image in new window. Since t h 1 ( t ) = 0 Open image in new window and lim t ( t h 2 ( t ) ) = 0 Open image in new window, then
lim sup t > 0 k = 1 2 a k ( t h k ( t ) ) = 0 < 3 2 . Open image in new window

By Lemma 2 lim t u ( t ) = 0 Open image in new window. The theorem is proven. □

3.3 Global stability of the equilibrium ( X , 0 ) Open image in new window

To analyze the global stability of the equilibrium ( X , 0 ) Open image in new window for system (1), we will use the following lemmas.

Lemma 3 [29]

Consider the scalar linear delay differential equation
x ˙ ( t ) = a ( t ) x ( t ) + b ( t ) x ( h ( t ) ) , Open image in new window

where a, b are continuous bounded on [ 0 , ) Open image in new window functions, h, k = 1 , , m Open image in new window are continuous functions, lim t h ( t ) = Open image in new window.

If a ( t ) a 0 > 0 Open image in new window, | b ( t ) | q a ( t ) Open image in new window, 0 < q < 1 Open image in new window, then the solution of the above equation is asymptotically stable.

Lemma 4 [31]

Consider the following equation and inequalities:

where a, a k Open image in new window, k = 1 , , m Open image in new window are continuous bounded on [ 0 , ) Open image in new window functions, a k ( t ) 0 Open image in new window, h k Open image in new window, k = 1 , , m Open image in new window are continuous functions, lim t h k ( t ) = Open image in new window.

Denote by X ( t , s ) Open image in new window the fundamental function of equation (12). Then X ( t , s ) > 0 Open image in new window, 0 s t Open image in new window. Moreover, for any t 0 Open image in new window, the equality x ( t ) = y ( t ) = z ( t ) Open image in new window, t t 0 Open image in new window, implies y ( t ) x ( t ) z ( t ) Open image in new window, t > t 0 Open image in new window, where x, y, z are the solutions of (12), (13), (14) respectively.

Let us formulate one of the classical results by Chaplygin [32] (see also [33]).

Lemma 5 Consider the ODE and corresponding differential inequalities:
x ˙ ( t ) = f ( t , x ( t ) ) , t t 0 , y ˙ ( t ) f ( t , y ( t ) ) , t t 0 , z ˙ ( t ) f ( t , z ( t ) ) , t t 0 , Open image in new window

where f ( t , u ) Open image in new window is a continuous function. If y ( t 0 ) x ( t 0 ) z ( t 0 ) Open image in new window, then y ( t ) x ( t ) z ( t ) Open image in new window, t > t 0 Open image in new window.

Theorem 4 If condition (7) holds, then for any solution ( x , y ) Open image in new window of system (1), we have lim t x ( t ) = X Open image in new window, lim t y ( t ) = 0 Open image in new window, which means that ( X , 0 ) Open image in new window is a global attractor for all solutions of system (1)-(2).

Proof Suppose ( x , y ) Open image in new window is an arbitrary solution of (1). From the second equation (1) we have
y ˙ ( t ) ( μ y β 2 ) y ( t ) + in α γ 4 y ( t τ ) . Open image in new window
By Lemma 3 we have 0 < y ( t ) u ( t ) Open image in new window, where u is the solution of the equation
u ˙ ( t ) = ( μ y β 2 ) u ( t ) + in α γ 4 u ( t τ ) , Open image in new window

where u ( t ) = y ( t ) Open image in new window, t 0 Open image in new window. Lemma 4 implies that lim t u ( t ) = 0 Open image in new window. Hence lim t y ( t ) = 0 Open image in new window. The second part of the theorem is proven.

In order to prove that lim t x ( t ) = X Open image in new window, at first we will show that x ˙ Open image in new window is a bounded function. From the first equation (1), we have
x ˙ ( t ) β 1 x ( t ) ln K x ( t ) . Open image in new window
Since
lim x 0 x ln K x = 0 , lim x + x ln K x = , Open image in new window

then for some M 1 > 0 Open image in new window, x ˙ ( t ) M 1 Open image in new window.

Since lim t y ( t ) = 0 Open image in new window, then 0 < y ( t ) < M y Open image in new window for some M y > 0 Open image in new window. Thus
x ˙ ( t ) ( β 1 ln K max { x ( 0 ) , K } γ 1 M y ω γ 3 ) max { x ( 0 ) , K } = M 2 Open image in new window

and | x ˙ ( t ) | C Open image in new window for some C > 0 Open image in new window.

Using all the above, we can rewrite the following:
x ˙ ( t ) β 1 x ( t ) ln K x ( t ) ω γ 3 x ( h ( t ) ) β 1 x ( t ) ln K x ( t ) ω γ 3 x ( t ) + ω γ 3 h ( t ) t | x ˙ ( s ) | d s β 1 x ( t ) ln K x ( t ) ω γ 3 x ( t ) + ω γ 3 C ( t h ( t ) ) . Open image in new window

Since lim t ( t h ( t ) ) = 0 Open image in new window, then for any ϵ > 0 Open image in new window, there exists t 1 Open image in new window such that ω γ 3 C ( t h ( t ) ) < ϵ Open image in new window, t t 1 Open image in new window.

And therefore,
x ˙ ( t ) β 1 x ( t ) ln K x ( t ) ω γ 3 x ( t ) + ϵ , t t 1 . Open image in new window
Consider now an ODE associated with the previous differential inequality
u ˙ ( t ) = β 1 u ( t ) ln K u ( t ) ω γ 3 u ( t ) + ϵ , t t 1 , Open image in new window
(15)
where u ( t 1 ) = x ( t 1 ) Open image in new window, and a functional equation F ( p , ϵ ) = 0 Open image in new window, where
F ( p , ϵ ) = β 1 p ln K p ω γ 3 p + ϵ . Open image in new window

We have F ( X , 0 ) = 0 Open image in new window, F p ( X , 0 ) = β 1 0 Open image in new window. By the implicit function theorem, for small ϵ > 0 Open image in new window, there exists the unique solution p = p ( ϵ ) Open image in new window of the equation F ( p , ϵ ) = 0 Open image in new window such that lim ϵ 0 p ( ϵ ) = X Open image in new window. It is obvious that p ( ϵ ) Open image in new window is a positive equilibrium of ODE (15).

For the solution of equation (15), we have u ˙ ( t ) > 0 Open image in new window, t > t 1 Open image in new window if u ( t 1 ) < p ( ϵ ) Open image in new window and u ˙ ( t ) < 0 Open image in new window, t > t 1 Open image in new window if u ( t 1 ) > p ( ϵ ) Open image in new window. Hence, for any solution of this equation, we have lim t u ( t ) = p ( ϵ ) Open image in new window. By Lemma 5 we have x ( t ) u ( t ) Open image in new window, t t 1 Open image in new window.

We also have
x ˙ ( t ) β 1 x ( t ) ln K x ( t ) γ 1 x ( t ) y ( t ) ω γ 3 x ( t ) ω γ 3 C ( t h ( t ) ) . Open image in new window
Since
lim t y ( t ) = 0 , lim t ( t h ( t ) ) = 0 , Open image in new window
then for any ϵ > 0 Open image in new window, there exists t 2 Open image in new window such that
x ˙ ( t ) β 1 x ( t ) ln K x ( t ) ϵ x ( t ) ω γ 3 x ( t ) ϵ , t t 2 . Open image in new window
Consider an ODE
v ˙ ( t ) = β 1 v ( t ) ln K v ( t ) ϵ v ( t ) ω γ 3 v ( t ) ϵ , t t 2 . Open image in new window
(16)

Performing the same calculations as for equation (15), we can show that for small ϵ > 0 Open image in new window, equation (16) has the unique equilibrium q ( ϵ ) Open image in new window such that lim ϵ 0 q ( ϵ ) = X Open image in new window. Hence, for any solution of equation (16), we have lim t v ( t ) = q ( ϵ ) Open image in new window. By Lemma 5 we obtain x ( t ) v ( t ) Open image in new window, t t 2 Open image in new window.

Thus, for t 0 = max { t 1 , t 2 } Open image in new window, we have v ( t ) x ( t ) u ( t ) Open image in new window, t t 0 Open image in new window. Hence
q ( ϵ ) lim t x ( t ) p ( ϵ ) . Open image in new window
But
lim ϵ 0 p ( ϵ ) = lim ϵ 0 q ( ϵ ) = X . Open image in new window
Hence
lim t x ( t ) = X . Open image in new window

The theorem is proven. □

By definition, local stability and global attractivity imply global stability. Hence, we have the following result.

Theorem 5 Suppose conditions (7) and (9) hold. Then the equilibrium ( X , 0 ) Open image in new window is globally asymptotically stable.

4 Simulation results

In order to verify the mathematical model of the biological system, it is widely accepted to estimate the conditions defined in the presented theorems based on the parameters set taken from the biological and medical literature (these parameters are taken from Table 1).

System (1) has an equilibrium ( X , 0 ) Open image in new window where X = K e ω γ 3 β 1 Open image in new window. Substituting the relevant parameters from Table 1, we get that X = 1.5 × 10 5 exp ( 400 × 0.00014 0.03 ) = 23 , 196 Open image in new window, which means that the cancer cells number will not exceed the 23,196 cells at a constant daily dose of 400 mg imatinib.

With increasing the daily dose of imatinib to 600 mg, the amount of cancer cells decreases to almost 10,000. These numerical results are consistent with the data given in [12].

In order to check the local and global stability conditions, we substitute parameters to the expression in α × γ 4 + β 2 < μ y Open image in new window. The following numerical inequality is received: 13 × 0.005 + 4.1 × 10 4 < 0.05 Open image in new window. The obtained result supports that the model (1)-(2) has a positive stable local and global equilibrium if condition (4) is satisfied.

5 Conclusion

In this work we have proposed a mathematical model for the combination treatment of chronic myelogenous leukemia with imatinib and IFN-α to overcome immune suppressive side effects of imatinib, prolonging the chronic phase of the disease. Our model consists of a system of two non-linear delay differential equations with logarithmic and rational nonlinearities.

In order to describe the influence of two types of the treatment component (imatinib and IFN-α) on the model compartments, we introduced delays and explained a biological motivation for it. The proposed two-compartment model allows to evaluate directly the expected steady states of the system.

In our work we investigate the contribution of IFN-α (immunotherapy) in eliciting strong killer cells (CTL) responses against cancer (CML) cells in addition to imatinib therapy by examination of CTL and CML cells populations only. The dose of IFN-α has an inhibitory effect on the value of x ( t ) Open image in new window, as seen from the formula (4). As a result of calculations shown in Section 4, we observe that y ( t ) Open image in new window inhibits x ( t ) Open image in new window, destroying itself to 0 in spite of constant addition of IFN-α.

The existence of unique global solutions for CML model was defined. Explicit local and global stability conditions for the unique non-trivial equilibrium were obtained by applying the method of delays in differential inequalities and linear stability theory of non-linear delay differential equations. The numerical results show that our model replicates the averaged behavior of the combined treatment.

In the future, in order to avoid resistance to imatinib, it looks reasonable to change the IFN-α to another type of the treatment in combination with imatinib [34]. The best form of combination therapy that leads to improved survival in patients remains to be seen in future by mathematical models and clinical trials.

Notes

Acknowledgements

We thank professor Dina Ben-Yehuda from Hadassah University Hospital (Division of Hematology) for helpful comments and suggestions. The authors are very grateful to the reviewers whose valuable comments greatly contributed to the paper presentation.

Supplementary material

13662_2012_326_MOESM1_ESM.eps (2.1 mb)
Authors’ original file for figure 1

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Copyright information

© Berezansky et al.; licensee Springer 2012

This article is published under license to BioMed Central Ltd. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Authors and Affiliations

  • Leonid Berezansky
    • 1
  • Svetlana Bunimovich-Mendrazitsky
    • 2
    Email author
  • Alexander Domoshnitsky
    • 2
  1. 1.Department of MathematicsBen-Gurion University of NegevBeer-ShevaIsrael
  2. 2.Department of Computer Science and MathematicsAriel University Center of SamariaArielIsrael

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