Stability and Controllability Issues in Mathematical ...

Stability and Controllability Issues in Mathematical Modeling of the Intensive Treatment of Leukemia L. Berezansky, S. BunimovichMendrazitsky & B. Shklyar

Journal of Optimization Theory and Applications ISSN 0022-3239 Volume 167 Number 1 J Optim Theory Appl (2015) 167:326-341 DOI 10.1007/s10957-015-0717-9

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Author's personal copy J Optim Theory Appl (2015) 167:326–341 DOI 10.1007/s10957-015-0717-9

Stability and Controllability Issues in Mathematical Modeling of the Intensive Treatment of Leukemia L. Berezansky · S. Bunimovich-Mendrazitsky · B. Shklyar

Received: 29 July 2014 / Accepted: 14 February 2015 / Published online: 27 February 2015 © Springer Science+Business Media New York 2015

Abstract We present a mathematical model of dynamic changes in clinical parameters following drug therapy for chronic myeloid leukemia (CML) using a system of ordinary differential equations (ODE), describing the interactions between effector T cells and leukemic cancer cells. The model successfully predicts clinical response to two separate drug therapies: targeted therapy with the tyrosine kinase inhibitor imatinib and immunotherapy with interferon alfa-2. Development of this model enables the identification of the treatment regimen for a determined time period, in order to reach an admissible concentration of cancer cells. To mathematically model the dynamics of CML progression, both without and with treatment, we have obtained the local and global stability and the local relative controllability conditions for this ODE system. Keywords Mathematical model of leukemia · Local and global stability · Local and global controllability Mathematics Subject Classification

93C95 · 92C50 · 34D20 · 34D23 · 93B05

L. Berezansky Ben Gurion University of the Negev, Beersheba, Israel S. Bunimovich-Mendrazitsky (B) Ariel University, Ariel, Israel e-mail: [email protected] B. Shklyar Holon Institute of Technology, Holon, Israel

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1 Introduction Leukemia is one of the twelve most common cancers [1], with approximately 400,000 new cases diagnosed per year globally. A total of 250,000 patients succumb to leukemia annually, most of whom in the developed countries of Europe and North America [2]. In this work, we undertook mathematical modeling of chronic myelogenous leukemia (CML). CML is definitively diagnosed by the presence of a reciprocal translocation between chromosome 9 and chromosome 22, known as the Philadelphia chromosome [3]. The clinical picture of myelogenous leukemia is characterized by progression through four distinct phases: the initial latent phase of disease (3–5 years); the chronic phase during which CML is most often diagnosed (3–6 years); acceleration phase to which the chronic disease progresses in the absence of treatment (period of months); and blastic phase, which is the fatal stage of the disease [3,4]. Until the year 2000, therapies for CML included administration of cytotoxic drugs such as busulfan and hydroxyurea, and immunotherapy as interferon-α and allogeneic stem cell transplantation. Treatment and control of CML underwent a dramatic change with the introduction of imatinib, which has proven to be an effective treatment for nearly all CML patients, especially in chronic phase [5]. However, imatinib is much less effective in the accelerated or the blastic phase of the disease. Moreover, a number of patients do not respond to imatinib due to drug resistance [6,7]. This clinical challenge is being partially met by a new generation of targeted drugs, such as dasatinib or nilotinib [8]. The results obtained with targeted drugs are indeed very promising; however, these drugs cannot definitively cure the disease [9]. An interruption of therapy is inevitably followed by a disease relapse. There are a number of reasons for CML to be a good candidate for immunotherapy. First, the slow development of CML provides a relatively long clinical window of opportunity for immunotherapeutic intervention. Second, the circulation of leukemic cells in the blood and the lymphatic system makes them easily accessible to the immune system. Third, CML cells carry a well-defined malignancy-specific antigen, such that the immunotherapy may be effectively targeted to diseased cells [10]. Interferon alfa2α (IFN-α) immunotherapy was the first treatment to produce the cytotoxic response in the CML patients. IFN-α is physiologically produced by a variety of immune system cells in response to viral infection. Recent data show that IFN-α activation of host immune cells, including T, B, natural killer cells and tumor antigen-presenting dendritic cells, contribute to apoptosis of leukemic cells [11]. Moreover, it was proven that IFN-α, which has long been considered as the standard maintenance therapy in CML, may exert its life-prolonging effect independently by activating immunological effector functions [12]. An impressive body of mathematical modeling research has accumulated on the basis of cancer-immune strategy, which proved the effectiveness of immunotherapy [13–20]. In addition to the experimental approach to the optimization of CML treatment, its mathematical modeling has been a subject of rapid development in recent years. The first mathematical models of CML were developed in 1969 [21], after which an innovative modeling of the hematopoietic system was presented by Fokas et al. [22]. More recently, Moore and Li [23] proposed an ODE-based model of CML treatment,

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modeling the dynamic interaction between cancer and cytotoxic T cells (naive and matured), and Komarova and Wodarz [24] used stochastic networks methods to study resistance to drugs, including imatinib. Mathematical models for CML treatment with imatinib were continuously advanced by Michor et al. [25], Kim et al. [26], and Paquin et al. [27]. These studies concluded that imatinib does not completely eliminate the leukemia cell population, and recommend the combination of imatinib with an additional form of treatment. Techniques for optimal control of CML treatment were discussed by Nanda et al. [28] and Ainseba and Benosman [29]. In these works, optimal disease control was achieved mathematically to obtain preferred dose delivery schedules for individual patients. Mathematical modeling of the combined treatment of CML with imatinib and IFNα was examined by Berezansky et al. [30], who designed a system of two nonlinear delay differential equations. This novel system was successfully utilized to obtain the explicit local and global stability conditions of the systems solutions and to quantify the inhibitory effect of IFN-α on CML cells. However, this technique has yet to be harnessed for identification of the optimal treatment regimen. In this study, we further develop the mathematical model of combined CML treatment utilizing the two-dimensional control ODE system. We analyze the dynamics of CML population changes, both without treatment and under combined therapy. In addition, we investigate the global and local stability and controllability of this ODE system. 2 Model Description Based upon the model of Moore and Li [23], we omitted from their ODE system one equation describing naive effector T cells behavior. In its place were added two additional terms, describing the influence of imatinib upon cancer cells and that of IFNα on the effector T cells, which were added to each one of the remaining equations. The obtained nonlinear two-dimensional system characterizes the dynamics of the interaction between these two biological components for the case of a time-varying scheduling drug application: y1 (t) presents the population of CML cells; y2 (t) presents the number of effector T cells cytotoxic to CML (CTLs). K − γ1 y1 (t)y2 (t) − γ3 y1 (t)ω (t) , y1 (t) β2 γ4 y1 (t)y2 (t) − γ2 y1 (t)y2 (t) + y 2 (t)in α (t) − μy2 (t), y˙2 (t) = η1 + y1 (t) η2 + y2 (t) 2 (1) y˙1 (t) = β1 y1 (t) ln

where K > 0, β j > 0, γ j > 0, j = 1, 2, μ > 0. The first term on the right-hand side of the first equation of the system (1) describes the growth of the CML cell population, according to the Gompertz law with the growth rate β1 .The Gompertz curve provides a significantly better fit for leukemic cancer data than do logistic, exponential or polynomial curves [31]. The constant K in the first term estimates the maximum carrying capacity of CML cells, taking into account the birth and death rate of cancer cells [32], such that, if the number of CML cells

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reaches K , then the patient dies. The second term of the first equation of the system (1) represents the loss of CML cancer cells due to CTL cells’ influence, according to the rate γ1 . The third term represents the cancer cell response to targeted therapy by imatinib, where the action of imatinib reduces the leukemia cell level (ω (t) is a dose of imatinib administered daily, similar as defined in papers [33,34]). The coefficient γ3 is the factor proportional to the imatinib dose. The second equation describes the dynamic balance between the stimulatory and inhibitory effects of CTL cells. The first term represents the growth of the CTL cell population y2 (t), accounting for the influence of the CML antigen in the lymph nodes, where β2 is the rate of this population growth and η1 is the standard half-saturation concentration in Michaelis–Menten term, taking into account the saturation levels of CML cells in the lymph nodes. The second term describes the loss of CTL following their interaction with CML cancer cells, according to the rate γ2 . The third term corresponds to the stimulatory augmentation of CTL cells due to IFN-α immunotherapy, where in α (t) represents the dose of IFN-α. IFN-α leads to increased expression of other cytokines, such as interferon-γ , that also contribute to the pro-inflammatory environment. The coefficient γ4 is the factor proportional to the IFN-α dose. The standard half-saturation concentration of CTL immune cells in a Michaelis–Menten term is η2 . The last term in the second equation describes the loss of CML cells due natural cell death, where μ is their death rate. This treatment strategy, in a finite time interval, is meant to reduce the amount of CML cancer cells to an acceptable level (denoted by s) through appropriate combination of the two drugs (imatinib and IFN-α).

3 Problem Statement We assume that the dose of imatinib ω and that of IFN-α in α depend upon t and can be considered as controls. Denote: ω (t) :=u 1 (t) , in α (t) :=u 2 (t) , t ≥ 0. Here, for any t ≥ 0, the controls u 1 (t) and u 2 (t) are not dimensional scalar functions; γ3 u 1 (t) is a dose of imatinib and γ4 u 2 (t) is a dose of IFN-α at the time point t ≥ 0. In the sequel, the model under consideration will be described by two-dimensional system of ordinary differential equations: K − γ1 y1 y2 − γ3 y1 u 1 , y1 γ4 y22 y1 y2 − γ2 y1 y2 − μy2 + u2, y˙2 = η1 + y1 η2 + y2 y1 (0) = y10 , y2 (0) = y20 . y˙1 = β1 y1 ln

(2) (3) (4)

The equilibrium points of the homogeneous system y˙1 = β1 y1 ln y˙2 =

K − γ1 y1 y2 , y1

y1 y2 − γ2 y1 y2 − μy2 η1 + y1

(5) (6)

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are: 1. y1 = K , y2 = 0, 2. y1 = γ11 1 − η1 γ2 − μ ∓ (1 − η1 γ2 − μ)2 − 4η1 γ2 μ , y2 =

β1 γ1

ln

K y1 ,

provided that (1 − η1 γ2 − μ)2 ≥ 4η1 γ2 μ, y1 < K . The pair (y1 , y2 ) = 0 is not an equilibrium, but it may be an attractor since the right-hand side of (5) tends to zero as y1 → 0, y2 = 0. We will show that it is impossible. As a corollary of Theorem 1 from [30], we obtain the following result. Theorem 3.1 For any initial condition (y1 (0), y2 (0)), where y1 (0) > 0, y2 (0) > 0, there exists a unique global solution of (5)–(6) such that y1 (t) > 0, y2 (t) > 0, for all t ≥ 0. Theorem 3.2 The pair (y1 , y2 ) = 0 is not an attractor for system (5)–(6). Proof Suppose (5)–(6) has an attractor y1 = 0, y2 = 0. After the substitution y1 = K e−x1 , y2 = x2 , the system (5)–(6) is transformed to x˙1 = −β1 x1 + γ1 x2 , K e−x1 −x1 x2 − μx2 . x˙2 = − γ2 K e η1 + K e−x1

(7)

By our assumption, forany solution of (7) we have limt→+∞ x1 (t) = +∞. Hence, e−x1 − γ2 K e−x1 < μ, then from the second equation for t sufficiently large η K+K e−x1 1 of system (7) we have limt→+∞ x2 (t) = 0. However, now the first equation of (7) implies that limt→+∞ x1 (t) = 0. We have a contradiction, so the theorem is proved. 4 Stability Problems First, we present the examination of the disease progression without treatment. Mathematically, this is possible to do by the well-known stability problem. The similar problems in the absence of therapy have been considered in [16,17,35], where the author considers the family of two-dimensional models for tumor-immune system interaction x˙ = x( f (x) − φ(x)π(y)), y˙ = β(x)y − μ(x)y + σ q(x) + θ (t),

(8)

which includes many known models and investigates in details stability problems for system (8). The model being investigated in our paper differs from system (8). Then, the results on stability obtained below are different and independent on results of [16,17,35].

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We will study now an equilibrium point x1 = K , x2 = 0. The investigation of other two equilibrium points is similar. Let us write the system (2)–(3) in a neighborhood of steady point (5)–(6) by changing of variables: y1 := x1 + K ,

y2 :=x2 .

(9)

System (2)–(3) in new variables is: K − γ1 (x1 + K ) x2 − γ3 (x1 + K ) u 1 x1 + K γ4 x22 (x1 + K ) x2 x˙2 = − γ2 (x1 + K ) x2 − μx2 + u2, η1 + K + x 1 η2 + x 2 x1 (0) = x10 , x2 (0) = x20 . x˙1 = β1 (x1 + K ) ln

(10) (11) (12)

Thus, the equilibrium points of the homogeneous system x˙1 = β1 (x1 + K ) ln

K − γ1 (x1 + K ) x2 , x1 + K

(13)

(x1 + K ) x2 − γ2 (x1 + K ) x2 − μx2 , (14) η1 + K + x 1 x1 (0) = x10 , x2 (0) = x20 , (15) are x1 = 0, x2 = 0 or x1 = γ11 1 − η1 γ2 − μ ∓ (1 − η1 γ2 − μ)2 − 4η1 γ2 μ − K , x˙2 =

K x2 = βγ11 ln x1 +K − K , provided that (1 − η1 γ2 − μ)2 ≥ 4η1 γ2 μ, x1 < 0. 1 In the following, the standard stability definitions will be used.

Definition 4.1 We will assume that the equilibrium solution (x1 , x2 ) = (0, 0) of system (13)–(14) is locally stable, iff for any ε > 0 there exists δ > 0 such that, for every initial x1 (0), x2 (0), the inequality max{|x1 (0)|, |x2 (0)|} < δ implies max{|x1 (t)|, |x2 (t)|} < ε for the solution (x1 (t) , x2 (t)). Furthermore, if for any solution with max{|x1 (0)|, |x2 (0)|} < δ lim x1 (t) = 0,

t→∞

lim x2 (t) = 0,

t→∞

(16)

then the equilibrium point (0, 0) is locally asymptotically stable (LAS). If system (13)–(14) is locally stable, and for any solution of this equation with arbitrary initial data the condition lim x1 (t) = 0,

t→∞

lim x2 (t) = 0,

t→∞

(17)

holds true, then the equilibrium point (0, 0) is globally asymptotically stable (GAS).

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If there exist two positive numbers M > 0, α > 0, such that for any solution of this system with arbitrary initial conditions max{|x1 (t)|, |x2 (t)|} ≤ M max{|x1 (0)|, |x2 (0)|}e−αt ,

(18)

then this system is globally exponentially stable (GES). The local asymptotic stability of the nonlinear system (13)–(14) can be established by asymptotic stability of its linearized system x˙1 = −β1 x1 − γ1 K x2 , K − γ2 K − μ x 2 . x˙2 = K + η1 The eigenvalues of this system are λ1 = −β1 , λ2 = stability condition of system (19)–(20) is:

K K +η1

(19) (20) − γ2 K − μ. Hence, the

K < γ2 K + μ. η1 + K

(21)

Using the stability condition by linear approximation, we thereby obtain the validity of the following assertion: Theorem 4.1 If K < γ2 K + μ, η1 + K

(22)

then the equilibrium point (0, 0) of system (13)–(14) is LAS. This theorem asserts that if the concentration of cancer cells is closed to the K and immune cells is small enough, it follows that the outcome will be fatal and treatment is needed. Below, we investigate the global stability (GAS) of system (13)–(14). Theorem 4.2 If μ > 1, then the equilibrium point (0, 0) of system (13)–(14) is GAS. Proof The local stability follows from Theorem 4.1. Thus, we need only to prove that any solution of the system (13)–(14) tends toward trivial equilibrium, by fixing a solution of system (13)–(14) and rewriting (14) in the form x1 (t) + K x˙2 (t) = − γ2 (x1 (t) + K ) − μ x2 (t). (23) η1 + x1 (t) + K Since x1 (t) + K > 0,

x1 (t) + K < 1, ∀t ≥ 0, η1 + x1 (t) + K

for some γ > 0 we have β(t) =

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x1 (t) + K − γ2 (x1 (t) + K ) − μ ≤ −γ < 0, ∀t ≥ 0. η1 + x1 (t) + K


333

Therefore, there exists a positive constant C, such that for this solution of (23), we can estimate |x2 (t)| < Ce−γ t . (24) After the substitution of x1 (t) = K e−y(t) − K in (13), this equation takes the form y˙ (t) = −β1 y(t) + γ1 x2 (t).

(25)

The solution of (25) is obtained by y (t) = e−β1 t y (0) + γ1 e−β1 t

t

eβ1 τ x2 (τ ) dτ.

(26)

0

Using estimation (24) in (26) we obtain y (t) < e

−β1 t

−β1 t

t

L + γ1 e

e(β1 −γ )τ dτ = e−β1 t L + γ1

0

e−γ t − e−β1 t . β1 − γ

(27)

From (27) it follows, that lim y(t) = 0, implying that also lim x1 (t) = 0, thus t→∞ t→∞ proving the theorem. One can easily prove that the inequality μ > 1 implies the inequality K < γ2 K + μ η1 + K of Theorem 4.1. This reflects the clear fact that the global asymptotical stability implies the local asymptotical stability. Unfortunately, the gap between these conditions is still open. Now, we will improve Theorem 4.2. Theorem 4.3 If μ > 1, then the equilibrium point (0, 0) of system (13)–(14) is GES. Proof By substituting x2 (t) = e−λt z(t) with small positive number λ in (14), this equation takes a form x1 (t) + K − γ2 (x1 (t) + K ) − μ + λ z(t). (28) z˙ (t) = η1 + x1 (t) + K If we choose λ > 0 such that μ − λ > 1, as in the proof of the previous theorem, we obtain that for some γ > 0 : x1 (t) + K − γ2 (x1 (t) + K ) − μ + λ ≤ −γ < 0. η1 + x1 (t) + K

(29)

Hence, for any solution of (28) we have |z(t)| ≤ e−γ t C, where C and γ are positive constants that do not depend on x2 (0) . Again, for any solution of (14), we have |x2 (t)| ≤ e−(γ +λ)t C. Similarly, we can obtain an exponential estimation for the function x1 (t) of (13), thus proving the theorem.

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Theorem 4.3, proven above, claims when μ > 1, in the absence of treatment, x1 (t) always tends to K exponentially. In this paper, we are considering the case μ < 1 (μ = 0.06, from Table 1 in [30]). When there is no evidence of any mortal outcome or recovery from the disease, treatment is considered to be necessary. The similar results can be obtained for two other equilibrium points. 5 Controllability Problem The goal of the treatment process is: For any initial values of the CML cancer cells and the effector T cells specific to CML (CTL), find a medical treatment released by a dose of imatinib ω (t) and a dose of IFN-α in α (t) such that the population of CML will be reduced to the safe level at the predefined time t1 . Again, consider the system (2)–(3) in the variables x1 and x2 , where x1 = y1 − K , x2 = y2 . For the reader’s convenience, we write it once more: K − γ1 (x1 + K ) x2 − γ3 (x1 + K ) u 1 , x1 + K γ4 x22 (x1 + K ) x2 x˙2 = − γ2 (x1 + K ) x2 − μx2 + u2, η1 + K + x 1 η2 + x 2 x1 (0) = x10 , x2 (0) = x20 . x˙1 = β1 (x1 + K ) ln

(30) (31)

Using the notations x:=col (x1 , x2 ) , u:=col (u 1 , u 2 ) , f (x, u) := col ( f 1 (x, u) , f 2 (x, u)) , C = (1, 0) , where K − γ1 (x1 + K ) x2 − γ3 (x1 + K ) u 1 , (32) x1 + K γ4 x22 (x1 + K ) x2 − γ2 (x1 + K ) x2 − μx2 + f 2 (x, u) : = u2, (33) η1 + (x1 + K ) η2 + x 2 f 1 (x, u) : = β1 (x1 + K ) ln

one can write above system by x˙ = f (x, u) , x (0) = 0,

y = C x.

For the sake of generality, consider n-dimensional system (34), where x ∈ Rn , u ∈ Rr ,

y ∈ Rm ,

C is a constant m × n-matrix. Let x 1 be a predefined final state of (34). Denote y ∗ :=C x 1 .

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(34)


335

Definition 5.1 System (34) is said to be locally relatively y ∗ -controllable on [0, t1 ] iff there exists a number δ > 0, such that, for any x0 , C x0 − C x 1 < δ, there exists a square-integrable control u (·) ∈ L r2 [0, t1 ] , such that the corresponding solution x (t) , x (0) = x0 satisfies the final condition C x (t1 ) = y ∗ . Let f x 1 , 0 = 0. Together with nonlinear system (34), consider the linear system w˙ = Aw + Bu, w (0) = 0,

y = Cw,

(35)

where A:= f x x 1 , 0 , B:= f u x 1 , 0 are constant n × n and n × r -matrices, and w (t) := x (t) − x 1 , w(0) = x (0) − x 1 . Using the well-known relative controllability results proven in [36] for linear systems, one can easily obtain the following Theorem 5.1 If

rank C B, C AB, . . . , C An−1 B = rank {C}

(36)

then system (35) is relatively y ∗ -controllable on [0, t1 ] for any y ∗ ∈ Rm . Theorem 5.2 If f x 1 , 0 = 0 and condition (36) holds, then system (34) is locally relatively y ∗ -controllable on [0, t1 ] . Proof Denote: τ :=t1 −t, z (τ ) := x (t1 − τ )−x (t1 ) , v (τ ) :=u (t1 − τ ) , where x (t) is a solution of nonlinear system (34). Obviously nonlinear system (34) is transformed to the system z = −g (z, v) , z (0) = 0, z (t1 ) = −x 1 ,

y = C z,

(37)

and system (35) is transformed to z = −Az − Bv, z (0) = 0,

y = C z,

(38)

where g (z, v) := f (z + x (t1 ) , v) , gz (0, 0) = f x x 1 , 0 = A, gv (0, 0) = f u (0, 0) = B. Let us prove that the set K t1 , x 1 of final points of all outputs y (t1 ) = C z (t1 ) of trajectories of nonlinear system (37) with initial condition z (0) = 0 contains an open neighborhood of the origin. If so, then, returning to the time t, and to the state x (t) = z (t1 − t) + x 1 , we will prove, that one can transfer every initial point x0 such that C x0 is from the neighborhood of the point C x 1 , to the final point x 1 such that C x 1 = y ∗ by system (34). This will prove the theorem. We have, for linearized system (38)

τ z (τ ) = −

e 0

−A(τ −θ)

τ Bv (θ ) dθ, y (τ ) = C z (τ ) = −

Ce−A(τ −θ) Bv (θ ) dθ.

0

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Since condition (36) holds, linearized system (38) is relative y ∗ -controllable on [0, t1 ], so for any m linear independent points y 1 , . . . , y m ∈ Rm , there exist controls v 1 (τ ) , . . . , v m (τ ) ∈ L r2 [0, t1 ] such that

t1 y =− j

Ce−A(t1 −θ) Bv j (θ ) dθ,

j = 1, . . . , m.

(39)

0

For any ξ = (ξ1 , . . . , ξm ) ∈ Rm denote: v (τ, ξ ) := mj=1 ξ j v j (τ ) , and let z (τ, ξ ) be a solution of linearized system (38) with initial condition z (0, ξ ) = 0, generated by control v (t, ξ ), and let z j (t) be a solution of linear system ( 38) with initial condition z (0) = 0, generated by control v j (t) , j = 1, 2, . . . , . Obviously,

τ z (τ, ξ ) = −

e−A(τ −θ) Bv (θ, ξ ) dθ =

0

=−

m j=1

τ ξj

e−A(τ −θ) Bv j (θ ) dθ = −

m

ξ j z j (τ ) ,

(40)

j=1

0

∂ v (τ, ξ ) = v j (τ ) , j = 1, . . . , m, ∀ξ ∈ R, ∂ξ j ∂ ∂ ∂ v (τ, ξ ) = v (τ, ξ ) , . . . , v (τ, ξ ) ∂ξ ∂ξ1 ∂ξm

= v 1 (τ ) , . . . , v m (τ ) , ∀ξ ∈ R, v (τ, 0) = 0 ⇒ z (t, 0) = 0.

(41)

(42) (43)

Consider the differentiable operator y (t1 , ξ ) :=C z (t1 , ξ ) , ξ ∈ Rm , y (t1 , 0) = C z (t1 , 0) = 0, where z (t, ξ ) is a solution of nonlinear system ). (37 Using the theorem on implicit functions, we prove that K t1 , x 1 = Range (y (t1 , ξ )) contains an open neighborhood of the origin. ∂ z (τ, ξ ) , where z (τ, ξ ) is a solution of nonlinear system Denote w (τ ) := ∂ξ ξ =0

∂ z (τ, ξ ) = −g (z (τ, ξ ) , v (τ, ξ )) , z (0, ξ ) = 0. Obviously, z (τ, 0) is a (38), i.e., ∂τ ∂ solution of the system ∂τ z (τ, 0) = −g (z (τ, 0) , 0) , z (0, 0) = 0, so from the theorem of the uniqueness and g (0, 0) = 0 it follows, that z (τ, 0) = 0. Hence,

w˙ (τ )

∂ ∂ ∂ ∂ ∂ z (τ, ξ ) z (τ, ξ ) = = =− (g (z (τ, ξ ) , v (τ, ξ ))) ∂τ ∂ξ ∂ξ ∂τ ∂ξ ξ =0 ξ =0 ξ =0 ∂ ∂ = − gz (z (τ, ξ ) , v (τ, ξ )) z (τ, ξ ) − gv (z (τ, ξ ) , v (τ, ξ )) v (τ, ξ ) ∂ξ ∂ξ ξ =0

123

ξ =0


337

∂ ∂ z (τ, ξ ) v (τ, ξ ) = − gv (0, 0) (z (τ, 0) , 0) ∂ξ ∂ξ ξ =0 ξ =0 ∂ ∂ z (τ, ξ ) v (τ, ξ ) = −gz (0, 0) − gv (0, 0) ∂ξ ∂ξ ξ =0 ξ =0 ∂ ∂ ∂ z (τ, ξ ) v (τ, ξ ) v (τ, ξ ) = −A −B = −Aw (τ ) − B . ∂ξ ∂ξ ∂ξ ξ =0 ξ =0 ξ =0

−gz

Recall that A = gz (z (τ, 0) , v (τ, 0)) = gz (0, 0) = f x x 1 , 0 , B = gv (z (τ, 0) , v (τ, 0)) = gv (0, 0) = f u (0, 0) . From (40)–(42) it follows that w (τ ) = w 1 (τ ) , . . . , w m (τ ) is defined by

w˙ (τ ) : = −Aw (τ ) − B v 1 (τ ) , . . . , v m (τ ) , w˙ j (τ ) : = −Aw j (τ ) − Bv j (τ ) , j = 1, . . . , m. The controls v j (t) ,

j = 1, 2, . . . , m are chosen such that the vectors Cw 1 (t1 ) , Cw 2 (t1 ) , . . . , Cw m (t1 )

are linear independent [see (39) and two rows before]. Hence,

∂ 1 m rank Cw (t1 ) = rank Cw (t1 ) , . . . , Cw (t1 ) = rank C z (t1 , ξ ) = m, ∂ξ ξ =0 and z (t1 , 0) = 0. By the theorem on implicit functions, K t1 , x 1 = Range (y (t1 , ξ )) = Range (C z (t1 , ξ )) contains an open neighborhood of the origin. It proves that nonlinear system (34) is locally relatively y ∗ -controllable on [0, t1 ] . Theorem 5.3 Nonlinear system (30)–(31) is locally relatively y ∗ -controllable . Proof For given nonlinear system (30)–(31), we have ∂ f1 f 1 (0, 0, 0, 0) = 0; (x1 , x2 , u 1 , u 2 ) ∂ x1 ∂ K β1 (x1 + K ) ln = − γ1 (x1 + K ) x2 − γ3 (x1 + K ) u 1 ∂ x1 x1 + K K ∂ f1 = β1 ln − 1 − γ3 u 1 − γ 1 x 2 , (0, 0, 0, 0) = −β1 , K + x1 ∂ x1

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∂ f1 (x1 , x2 , u 1 , u 2 ) ∂ x2 ∂ K β1 (x1 + K ) ln − γ1 (x1 + K ) x2 − γ3 (x1 + K ) u 1 = ∂ x2 x1 + K ∂ f1 = −γ1 (K + x1 ) , (0, 0, 0, 0) = −γ1 K ; ∂ x2 ∂ f1 (x1 , x2 , u 1 , u 2 ) ∂u 1 ∂ f1 = −γ3 (x1 + K ) , (0, 0, 0, 0) = −γ3 K ; ∂u γ4 x22 ∂ f2 ∂ (x1 + K ) x2 − γ2 (x1 + K ) x2 − μx2 + u2 (x1 , x2 , u 1 , u 2 ) = ∂ x1 ∂ x1 η1 + (x1 + s) η2 + x 2 γ4 x22 ∂ ∂ ∂ (x1 + K ) x2 − u2 = (γ2 (x1 + K ) x2 + μx2 ) + ∂ x1 η1 + (x1 + K ) ∂ x1 ∂ x 1 η2 + x 2 1 ∂ f2 − γ2 x 2 , = η1 (0, 0, 0, 0) = 0; 2 ∂ x1 (η1 + (x1 + K )) γ4 x22 ∂ f2 ∂ (x1 + K ) x2 − γ2 (x1 + K ) x2 − μx2 + u2 (x1 , x2 , u 1 , u 2 ) = ∂ x2 ∂ x2 η1 + (x1 + K ) η2 + x 2 γ4 x22 x1 + K ∂ ∂ − γ2 (x1 + K ) − μ x2 + u2 = ∂ x2 η1 + (x1 + K ) ∂ x 2 η2 + x 2 x1 + K (2η2 + x2 ) x2 = − γ2 (x1 + K ) − μ + γ4 u 2 , η1 + (x1 + K ) (η2 + x2 )2 ∂ f2 K − γ2 K − μ; (0, 0, 0, 0) = ∂ x2 η1 + K γ4 x22 ∂ f2 ∂ f2 , (x1 , x2 , u 1 , u 2 ) = (0, 0, 0, 0) = 0; ∂u 2 η2 + x2 ∂u Therefore, the corresponding linearized system is: x˙1 = −β1 x1 − γ1 K x2 − γ3 K u 1 , K + γ2 K − μ x 2 . x˙2 = η1 + K

(44) (45)

One can write system (44)–(45) in form (38), where −γ1 K , K K +η1 − γ2 K − μ −γ3 K , 0 B:= , C:= (1, 0) . 0, 0

A:=

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−β1 , 0,

(46) (47)


339

Kβ1 γ3 , 0 , C B = (K γ3 , 0) , C AB = (Kβ1 γ3 , 0) , {C B, C AB} = 0, 0 {(K γ3 , 0) , (Kβ1 γ3 , 0)} = K γ3 {−1, 0, β1 , 0 } . Hence, rank {C B, C AB} = rankC = 1 if and only if γ3 K = 0. The numeric values of the parameters γ3 and K show that the last inequality holds true. The proof is finished by using Theorem 5.2. Here, AB =

6 Perspectives Imatinib therapy is currently considered a “gold standard” therapeutic protocol for CML. Many biomedical researchers have empirically improved this treatment protocol. In order to accelerate these improvements, there is an urgent need for a systematic comparative study that would assist the planning and management of more advanced protocols. In the current study, we provide a mathematical platform for the study of CML therapy with imatinib of improved protocols including IFN-α therapy. Theorem 5.3 asserts that a successful treatment exists for a sufficiently small number of CML cells and for any number of immune cells (CTLs) with no limit to the value of control function. We plan to impose constraints of the control function in our upcoming research, in order to be more consistent with the clinical reality, in which only less than forty percentages of patients completely recover from CML [37]. According to Theorem 5.3, the number of leukemic cells does not reach zero, but tends toward the acceptable level. Thus, if the number of leukemic cells x1 (t) reaches the acceptable level, the treatment can be stopped for a certain period of the time (to be determined individually by the clinician). If a concentration of cancer cells becomes greater than a acceptable level (several orders of magnitude), then a patient should repeat combine treatment by imatinib and IFN-α.

7 Conclusions Mathematical modeling of chronic myeloid leukemia (CML) has been the subject of intense interest in recent years. Usually, the current clinical practice is the treatment of patients with a single drug (usually imatinib). In the case of drug resistance, the combination strategy of several drugs is adopted [38]. The present research focused on the modeling of combined imatinib and IFN-α therapy. We established conditions of local and global stability and conditions of unconstrained local controllability allowing us to release the combined drug CML treatment. We demonstrated that the elimination of cancer cells over a given time period is attainable. However, the treatment dose may be extremely high for advanced disease, with clinically inadmissible side effects. Thus, the maximum tolerable dose should be administered. This can be accounted for in the current model by obtaining the controllability problem for the system (30)– (31) with the control constraints. In future study, we plan: to investigate the relative controllability problem of the system (30)–(31) with control constraints and to obtain a constructive solution of the controllability problem for the system (30)–(31) with

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control constraints. This would allow us to establish a treatment regimen suitable for any initial disease conditions. Acknowledgments The authors would like to thank the editors and the referees for their valuable comments and suggestions which improved the original submission of this paper.

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