Optimal control of anti-HBV treatment based on combination of Traditional Chinese Medicine and Western Medicine

Biomedical Signal Processing and Control 15 (2015) 41–48

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Optimal control of anti-HBV treatment based on combination of Traditional Chinese Medicine and Western Medicine Yongmei Su ∗ , Deshun Sun School of Mathematics and Physics, University of Science and Technology Beijing, Beijing 100083, PR China

a r t i c l e

i n f o

Article history: Received 13 May 2014 Received in revised form 5 September 2014 Accepted 22 September 2014 Keywords: Optimal control Traditional Chinese Medicine Combination treatment

a b s t r a c t In this paper, a Hepatitis B virus model with standard incidence rate and logistic proliferation of healthy and infected cells is presented. Based on this model, we study an optimal control problem about antiHBV infection combination therapy of Traditional Chinese Medicine and Western Medicine, the optimal strategies of taking medicine are given by simulation. Two optimal strategies with or without the impact to the infection rate by treatment are compared, simulation shows the impact to the reduction of infection may be omitted when mathematical model is used to study the anti-HBV therapy which is consistent with some references. What is more, optimal control strategy with other constant control strategies are also compared, and the simulation shows the optimal control strategy is better than constant control strategies. © 2014 Elsevier Ltd. All rights reserved.

1. Introduction Hepatitis B is one of the serious infectious diseases which threaten to global human health, and has become an important social and public health problems. Most patients with chronic hepatitis B virus (HBV) infection require long-term therapy [1,2]. The effective treatment of chronic HBV patients aims to prevent progression of chronic hepatitis B (CHB) to cirrhosis, hepatocellular carcinoma and eventually death. The most used Western Medicine (WM) to treat the HBV include interferon, nucleotide analogs (NA) and adenosine analogs (AA). The interferon, such as interferon alpha-2b, interferon alpha-2a, peginterferon alfa-2a is to kill the virus and activate the immune, the NA, for example, lamivudine, entecavir, telbivudine is to inhibit the activity of viral DNA polymerase and reverse transcriptase, and has the inhibited function to the virus DNA synthesis, when it comes to AA, such as adefovir dipivoxil and tenofovir disoproxil fumarate, its main function is to inhibit the replication of viral DNA [3]. Papers [4–7] showed that the combination therapy has a greater advantage over mono-therapy, both in terms of biochemical and virological response, paper [5] reported that combination therapy of lamivudine and adefovir can produce longer-lasting effects than mono-therapy in treating chronic hepatitis B virus infection patients.

∗ Corresponding author. Tel.: +86 13681592133. E-mail addresses: [email protected], [email protected] (Y. Su). http://dx.doi.org/10.1016/j.bspc.2014.09.007 1746-8094/© 2014 Elsevier Ltd. All rights reserved.

Now, Traditional Chinese Medicines (TCM) therapy is also used in treating HBV [8–10], the most commonly used herbal ingredients, Abrus cantoniensis, Ganoderma lucidum and Atractylodes macrocephala, are not specifically anti-viral agents. These are herbs that have been used as to improve the immune function and maintain normal physiological activities of the internal organs [9]. Paper [11] studied the HBV combination treatment by TCM and acyclovir, and a better treatment effect was obtained. Paper [12] analyzed 30 years’ data of hepatitis B liver cirrhosis by the combination therapy of TCM and WM, and the conclusion showed that the combination therapy is beneficial to reduce liver injury and improve liver function. The use of mathematical models to interpret experimental and clinical results has made a significant contribution to the fields of (anti-)human immunodeficiency virus (HIV), HBV and hepatitis C virus (HCV) infections [13–16], paper [17] discussed an adefovir anti-HBV infection therapy immune model with ALT

⎧ ˇxv ⎪ x˙ =  − dx − , ⎪ ⎪ x+y ⎪ ⎪ ⎪ ⎪ ˇxv pyz ⎪ ⎪ y˙ = − dy − , ⎪ ⎪ x+y x+y ⎨ v˙ = (1 − u)ky − εv,

⎪ ⎪ ⎪ z˙ = (g + k2 yz)(1 − z/zmax ) − ız, ⎪ ⎪ ⎪ ⎪ ⎪ w˙ = s + a1 x + b1 y + k1 yz − d1 w, ⎪ ⎪ ⎪ ⎩

x(0) > 0, y(0) ≥ 0, v(0) ≥ 0, z(0) ≥ 0, u(0) ≥ 0.

(1)

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Y. Su, D. Sun / Biomedical Signal Processing and Control 15 (2015) 41–48

where x, y and v represent the numbers of uninfected (susceptible) cells, infected cells and free viruses, respectively. z represents the number of CTL, w represents the levels of alanine aminotransferase (ALT). u represents the efficacy of treatment. Paper [18] pointed that the liver is an organ which can regenerate cells and any loss of infected hepatocytes would be compensated by the proliferation of hepatocytes, due to homeostatic mechanisms, and paper [18] gave the extended virus model:

⎧   dx x+y ⎪ = s + rx 1 − − dx − ˇxv, ⎪ ⎪ Xmax ⎪ dt ⎪ ⎪ ⎨   dy x+y = ˇxv + ry 1 − Xmax dt

⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ dv = ky − εv.

− ıy,

(2)

where x, y, v have the same meaning as those in model (1), the logistic function rx(1 − (x + y)/Xmax ) and ry(1 − (x + y)/Xmax ) represent the proliferation of healthy and infected cells respectively, where Xmax is the maximum hepatocyte count in the liver. On the other hand, optimal control theory has been applied extensively in case of virological models, especially in case of HIV models [19–26]. Joshi [25] built an optimal control model about HIV based on an ordinary differential equation. The optimal drug strategies are determined for various stages of treatment. Later, the HIV models become more and more complicated because more factors were considered, such as activated and resting CD4 + cells were contained [27]. Paper [28] presented a delay-differential equation model with optimal control that described the interactions between human immunodeficiency virus which is more closer to actual situation. Pachpute and Chakrabarty [29] considered an optimal therapy model for HCV with interferon and ribavirin.

⎧   dT T +I ⎪ = s + rT 1 − − dT − ˇTVI ⎪ ⎪ dt Tmax ⎪ ⎪ ⎪ ⎪   ⎪ ⎪ dI T +I ⎪ ⎨ = ˇTVI + rI 1 − − ıI Tmax

⎪ ⎪ dVI ⎪ ⎪ = (1 − )(1 − εp )pI − cVI ⎪ dt ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ dVNI dt

⎧   x+y ˇxv ⎪ ⎪ ˙ , =  − dx + rx 1 − − (1 − u4 ) x ⎪ Xmax x+y ⎪ ⎪ ⎪ ⎪   pyz ⎪ ⎪ ˇxv x+y ⎪ ⎪ − + ry 1 − , y˙ = (1 − u4 ) ⎪ ⎨ x+y Xmax x+y v˙ = (1 − u1 )(1 − u2 )ky − (1 + u3 )εv, ⎪ ⎪ ⎪   ⎪ ⎪ z ⎪ ⎪ − ız, z˙ = (1 + u3 )(g + k2 yz) 1 − ⎪ ⎪ zmax ⎪ ⎪ ⎪ ⎩

(4)

w˙ = s + a1 x + b1 y + k1 yz − d1 w.

dt

dt

proposed an optimal control of a delayed system, but the incidence rate was also bilinear. From what has been discussed above, in this paper, based on [17,29], we propose an anti-HBV treatment optimal model with two different WM (NA and AA) and TCM as follows:

(3)

= (1 − εp )pI − cVNI .

where εp is the efficacy of interferon and  is the efficacy of ribavirin. T, I, VI , VNI represent the uninfected hepatocytes, infected hepatocytes, infectious virion and non-infectious virion, respectively. An objective functional is formulated to minimize the viral load, as well as the drug side-effects and the optimal system is solved numerically to determine optimal efficacies of the drugs. Obviously, the above model used the bilinear incidence rate ˇTVI , but paper [30] has pointed that the standard incidence rate ˇxv/x + y would be more reasonable than bilinear incidence rate when used for dynamics model about hepatitis virus infection. Based on the standard incidence rate ˇxv/x + y, Paper [31] and Paper [32] both discussed a HBV infection model and gave the stable analysis, but Paper [32] considered the return of the infected cells to uninfected cells by loss of all covalently closed circular DNA, and also based on the standard incidence rate, paper [33] investigated a five dimension model with CTL cells response and the antibody response, and gave the existence condition of five steady states and full analysis of such steady states’ local stability. On the other hand, seldom papers have been concerned with the HBV optimal control model, Hattaf and Rachik [34] proposed a three dimension optimal controls model, and the optimal control represented the efficiency of drug therapy in inhibiting viral production and preventing new infections. Besides, Mouofo [35]

where x, y, v, z, w have the same meaning as those in model (1), u1 and u2 stand for the influences of NA and AA respectively. (1 + u3 ) represents the improvement of the cellular immune function by the TCM. Noting that humoral immunity and cellular immunity both have their own unique role, and can cooperate with each other. When viruses enter the body, it induces the humoral immune firstly, because T cells cannot identify the invading virus antigen, only when the viruses invade host cell, the small molecule protein antigens from virus would appear on the cell surface, and only after the combination with cell surface receptor into complex, T cells can recognize antigens and T-cell immune can be activated. Since humoral immunity and cellular immunity have the indivisible relation, when the cellular immunity is improved, the humoral immunity function should be reflected in our model, so we use (1 + u3 ) to embody the influence without using a single variable to stand for the humoral immunity. Noting that, paper [36] pointed that nucleoside analogs may also interfere with de novo infection of hepatocytes by hindering the transformation of relaxed circular DNA into cccDNA. If this is the case, then treatment can reduce the rate of infection. And paper [36] used (1 − u4 ) to reflect the reduction of infection rate. So when we omit the reduction of infection rate by treatment, we choose u4 = 0. We will consider two cases: u4 = 0 and u4 = / 0 to test how the reduction of infection rate by treatment can infect the treatment effect. The organization of the paper is as follows. In Section 2, the control problem with u4 = 0 is formulated. The necessary conditions for an optimal control and the corresponding states are derived using Pontryagin’s Maximum Principle. In Section 3, the resulting optimality system about combination of Traditional Chinese Medicine and Western Medicine is numerically solved with u4 = 0. The corresponding analysis and simulation with the case u4 = / 0 are given in Sections 4 and 5. Section 6 compares the results with other constant design strategies. Section 7 draws the conclusion. 2. Optimal control problem for combination of Traditional Chinese and Western Medicines with u4 = 0 In this section, without considering the reduction of infection rate by treatment, that is u4 = 0, we present an optimal control problem motivated by biomedical considerations. The control goal is not only to formulate an objective functional which lowers the levels of HBV and the infected hepatocytes during and at the end of therapy, but also to minimize the therapeutic side-effects and the cost of drugs, so the objective function is defined as: J(u) =

1 1 (S22 y2 + S33 v2 ) + 2 2



tf

(Q22 y2

t0

+ Q33 v2 + R11 u21 + R22 u22 + R33 u23 )dt,

(5)

Y. Su, D. Sun / Biomedical Signal Processing and Control 15 (2015) 41–48

where t0 represents the beginning time of the treatment, tf represents the terminal time of the treatment. S22 , S33 , Q22 , Q33 , R11 , R22 and R33 represent the cost coefficients for the respective variables. R11 , R22 and R33 come from the cost of corresponding drugs and side-effect. For convenience, we define the state vector X = T T ( x y v z w ) , and the control vector U = ( u1 u2 u3 ) , the above Eq. (3) can be written as:

where

⎧ pf ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ pf ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ pf ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ pf

(6)

And the corresponding cost functional is as follows: 1 T 1 X (t)SX(t) + 2 2

J=



tf

[X T (t)QX(t) + U T (t)RU(t)]dt.

(7)

t0

⎪ ⎪ ⎪ pf ⎪ ⎪ ⎪ ⎪ ⎪ pf ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ pf ⎪ ⎪ ⎪ ⎪ ⎩

ˇxv

11

=

13

=−

21

=

22 =

X˙ = f (X(t), U(t), t), X(t0 ) = X0 .

43

2

(x + y)

−r

x+y Xmax



−1

−r

ˇv ˇxv x x − d, pf12 = − −r , 2 x+y Xmax Xmax (x + y)

ˇx , pf14 = pf15 = 0. x+y

ˇv ˇxv y pyz −r − + , 2 2 x+y Xmax (x + y) (x + y) pyz (x + y)

2

−r

ˇxv y pz − −r − 2 x+y Xmax (x + y)

x+y



− 1 , pf23 =

Xmax

ˇx py , pf24 = − , x+y x+y

25

= 0.pf31 = 0, pf32 = −k(u2 − 1)(1 − u1 ), pf33 = −(1 + u3 )ε, pf34 = pf35 = pf41 = 0,

42

= −(1 + u3 )k2 z

44

= −ı −

z

Z



− 1 , pf43 = 0,

(1 + u3 )g + (1 + u3 )k2 yz − (1 + u3 )k2 y Z

z Z



− 1 , pf45 = 0.

pf51 = a1 , pf52 = b1 + k1 z, pf53 = 0, pf54 = k1 y, pf55 = −d1 .

where the diagonal matrices

⎡

0

0

0

0

0

⎤

⎡

0

0

0

0

0

⎤

⎡

⎢ 0 S22 0 0 0 ⎥ ⎢ 0 Q22 0 0 0 ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ S = ⎢ 0 0 S33 0 0 ⎥ , Q = ⎢ 0 0 Q33 0 0 ⎥ ⎢ ⎥ ⎢ ⎥ ⎣0 0 ⎣0 0 0 0 0⎦ 0 0 0⎦ 0

⎡ R=

R11

⎢ ⎣ 0 0

0

0

0

0

0

R22

0

0

R33

0

0

0

0

0

40

0

0

⎥ ⎦ 50

X˙ = f (X(t), U(t), t)

⎧   x+y ˇxv ⎪ x˙ =  − dx + rx 1 − , − (1 − u4 ) ⎪ ⎪ Xmax x+y ⎪ ⎪ ⎪ ⎪   pyz ⎪ ˇxv x+y ⎪ ⎪ ⎪ ⎨ y˙ = (1 − u4 ) x + y + ry 1 − Xmax − x + y ,

⎡ 30

0

0

(12)

w(tf )

⎤



⎥ ⎥ ⎥ ⎥ =0. ⎥ ⎦ (13)

So we get the optimal control:

    ⎧ 3 R22 ky − 23 k2 y2 ⎪ ∗ ⎪ u1 = min max ,a ,b , ⎪ ⎪ R11 R22 − 23 k2 y2 ⎪ ⎪ ⎪     ⎨ max

3 (R11 R22 − 3 R22 ky)

,a

,b

,

R22 (R11 R22 − 23 k2 y2 ) ⎪ ⎪ ⎪     ⎪ ⎪ ⎪ 3 εv − 4 (g + k2 yz)(1 − (z/zmax )) ⎪ ⎩ u∗3 = min max ,a ,b . R33

(14)

(9)

⎪ v˙ = (1 − u1 )(1 − u2 )ky − (1 + u3 )εv, ⎪ ⎪ ⎪ ⎪ z ⎪ ⎪ z˙ = (1 + u3 )(g + k2 yz)(1 − ) − ız, ⎪ ⎪ zmax ⎪ ⎪ ⎩

X(t0 ) = X0 ,

0

0

z ∂H = R33 u3 − 3 εv + 4 (g + k2 yz) 1− zmax ∂u3

u∗2 = min

w˙ = s + a1 x + b1 y + k1 yz − d1 w.

20

0

⎤ ⎡ x(tf ) ⎤ y(tf ) ⎥ 0 0⎥⎢ ⎥ ⎥⎢ ⎥ ⎢ ⎥ ⎥ v 0 0⎥⎢ (t ) f ⎥, ⎢ ⎥⎢ ⎥ 0 0 ⎦ ⎣ z(tf ) ⎦ 0

(8)

which L(X(t), U(t), t) = (1/2)[XT (t)QX(t) + UT (t)RU(t)]. Using Pontryagin’s minimum principle, the necessary conditions is as follows:

=

0

⎡ ∂H = R11 u1 − 3 (1 − u2 )ky ⎢ ∂u1 ⎢ ∂H ⎢ ∂H =⎢ = R22 u2 − 3 (1 − u1 )ky ∂U ⎢ ∂u2 ⎣

H(X(t), U(t), (t), t) = L(X(t), U(t), t) + T · f (X(t), U(t), t)

0

⎢ 0 S22 0 ⎢ ⎢ (tf ) = SX(tf ) = ⎢ 0 0 S33 ⎢ ⎣0 0 0

⎤

Based on the dynamic constraint f(X(t), U(t), t) and the Lagrangian L(X(t), U(t), t), the Hamiltonian is as follows:

10

0

(10)

⎤

1 · pf11 + 2 · pf21 + 3 · pf31 + 4 · pf41 + 5 · pf51

⎢ Q22 · y + 1 · pf12 + 2 · pf22 + 3 · pf32 + 4 · pf42 + 5 · pf52 ⎥ ⎢ ⎥ ⎥, ⎥ ⎥ ⎣ 1 · pf14 + 2 · pf24 + 3 · pf34 + 4 · pf44 + 5 · pf54 ⎦

⎢ ∂H = − ⎢ Q33 · v + 1 · pf13 + 2 · pf23 + 3 · pf33 + 4 · pf43 + 5 · pf53 ˙ = − ∂X ⎢ 1 · pf15 + 2 · pf25 + 3 · pf35 + 4 · pf45 + 5 · pf55

(11)

which a and b stand for the minimum and maximum effect. The optimal control system thus, is a coupled forward state equation and a backward adjoint equation, along with the regular control. This problem, being nonlinear and coupled in nature, needs to be solved using concurrent and iterative numerical procedures. In this paper, the optimal control strategy is obtained by solving the state system (9) and adjoint system (11) and the transversality conditions (12) in theory based on minimum principle, and optimality system is simulated by using the Runge–Kutta fourth order scheme and the steepest gradient method [29]. 3. Simulation for combination of Traditional Chinese Medicine and Western Medicine with u4 = 0 In this part, we simulated the optimal procedure without considering the reduction of infection rate by WM treatment, which means u4 = 0. In Table 1 are some parameters used in the model.

44

Y. Su, D. Sun / Biomedical Signal Processing and Control 15 (2015) 41–48

Table 1 Some parameters used in the simulation. Parameters

Value

Reference

 ε d ı r g d1 s a1 b1 k k2 Zmax ˇ p k1 Xmax 

2500 cell ml−1 day−1 2.9 day−1 4.7 × 10−3 day−1 0.5 day−1 0.45 day−1 10 cell ml−1 day−1 0.1 day−1 1 × 10−1 U ml−1 day−1 3 × 10−7 U cell−1 day−1 3 × 10−7 U cell−1 day−1 7.4 virions cell−1 day−1 6 × 10−10 ml cell−1 day−1 0.3 × 105 cell ml−1 6 × 10−1 ml vir−1 day−1 1 × 105 ml cell−1 day−1 4 × 10−7 ml cell−1 day−1 1 × 107 cell ml−1 0.5

[29] [29] [29] [17] [29] [17] [17] [17] [17] [17] [29] [17] [17] [17] [17] [17] [29] Assumed Fig. 2. State dynamics for uninfected, infected cells, and free virus with u4 = 0.

The initial condition is taken to be (4.4 × 106 2.4 × 106 2.3 × 106 150 200)T . S22 = S33 = Q22 = Q33 = 1, R11 = R22 = R33 = 1000. We chose 50 days as the therapy time, that is the beginning time t0 = 0 and terminal time tf = 50. The efficacy u1 , u2 , u3 can theoretically lie between 0 and 1, which 0 corresponds to no effectiveness of the drug and 1 corresponds to full effectiveness of the drug. However, because the people cannot absorb the drug for all possible values of drug doses, the perfect efficacy is unlikely to be achieved totally, so we suppose the maximum therapy effect is 0.98, the simulation results are shown in Figs. 1–3. Fig. 1 gives the optimal control strategy, which shows if one wants to get an optimal treatment effect at the minimum expense within a supposed time, he needs only take the full drug dose at the former stage, and then reduce the drug dose gradually until about half the full drug dose for the rest time of treatment. As for the therapy effect, we can see from Fig. 2 that the optimal treatment results are in a rapid first phase decline in the viral load during the first three days, and then has a slowly decrease from 4th to 12th, followed by a rebound from 13th day to 15th day, and finally declines slowly for the rest of the treatment period. The infected hepatocyte concentration decreases throughout the treatment period, whereas, the uninfected hepatocyte concentration increases during the first 12 days of therapy and remains close to Xmax for the rest of the period. Besides, the simulation in Fig. 3

Fig. 3. State dynamics for CTL and ALT with u4 = 0.

shows that the numbers of CTL and ALT are large also when the number of free virus and infected cells is very big at the beginning stage, and they also decline with the reduction of virus till the normal level at the end of the treatment.

4. Optimal control problem for combination of Traditional / 0 Chinese and Western Medicines with u4 = In this section, considering that treatment can reduce the rate of infection, that is u4 = / 0, we present another optimal control problem, by the same means as in Section 2, the objective functional for combination therapy is defined as,

J(u) =

Fig. 1. Optimal control with u4 = 0.

1 1 (S22 y2 + S33 v2 ) + 2 2



tf

(Q22 y2

t0

+ Q33 v2 + R11 u21 + R22 u22 + R33 u23 + R44 u24 ) dt

(15)

Y. Su, D. Sun / Biomedical Signal Processing and Control 15 (2015) 41–48

Fig. 4. Optimal control with u4 = / 0.

where S22 , S33 , Q22 , Q33 , R11 , R22 , R33 and R44 represent the cost coefficients for the respective variables, the diagonal matrices are

⎡

0

0

0

0

0

⎤

⎡

0

0

0

0

0

⎤

⎢ 0 S22 0 0 0 ⎥ ⎢ 0 Q22 0 0 0 ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ S = ⎢ 0 0 S33 0 0 ⎥ , Q = ⎢ 0 0 Q33 0 0 ⎥ ⎢ ⎥ ⎢ ⎥ ⎣0 0 ⎣0 0 0 0 0⎦ 0 0 0⎦ ⎡

0

0

R11

⎢ ⎢ 0 R=⎢ ⎢ 0 ⎣

0

0

0

0

0

0

R22

0

0

0

R33

0

0

0

⎤

0

0

0

0

0

⎥ ⎥ ⎥. ⎥ 0 ⎦ R44

The responding vector former is J=

1 T 1 X (t)SX(t) + 2 2



tf

[X T (t)QX(t) + U T (t)RU(t)]dt

Fig. 5. State dynamics for uninfected, infected cells, and free virus with u4 = / 0.

(16)

t0

Then we get Hamiltonian function as follows: H(X(t), U(t), (t), t) = L(X(t), U(t), t) + T · f (X(t), U(t), t)

(17)

Using Pontryagin’s minimum principle, the necessary conditions (in terms of the Hamiltonian) for U∗ to be an optimal control is stated as follows:

⎧ 0 ˙ X = f (X(t), U(t), t) 1 ⎪ ⎪ ⎪ ⎪ 0 ⎪ X(t0 ) = X0 , 2 ⎪ ⎪ ⎪ ⎪ ⎨ ∂H 30

˙ = −

,

∂X ⎪ ⎪ ⎪ 0 (t ) = SX(t ), ⎪ 4 f f ⎪ ⎪ ⎪ ⎪ ⎪ ∂ H ⎩ 50 = 0. ∂U

(18)

Fig. 6. State dynamics for CTL and ALT with u4 = / 0.

45

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Y. Su, D. Sun / Biomedical Signal Processing and Control 15 (2015) 41–48

Fig. 7. State dynamics of uninfected, infected cells, free virus, CTL and ALT with strategy B.

And we get the optimal control strategy as follows:

   R ky − 2 k2 y2   ⎧ 3 22 3 ⎪ u∗1 = min max , 0 , 0.98 , ⎪ 2 2 2 R11 R22 − 3 k y ⎪ ⎪ ⎪ ⎪     ⎪  ⎪ 3 (R11 R22 − 3 R22 ky) ⎪ ⎨ u∗2 = min max R22 (R11 R22 − 2 k2 y2 ) , 0 , 0.98 , 3     ⎪ 3 εv − 4 (g + k2 yz)(1 − (z/zmax )) ⎪ ∗ , 0 , 0.98 , u3 = min max ⎪ ⎪ R33 ⎪ ⎪ ⎪     ⎪ ⎪ ⎩ u∗ = min max (2 − 1 )ˇxv , 0 , 0.98 . 4

(19)

R44 (x + y)

we use in Section 3, R44 = 1000, and simulation results are shown in Figs. 4–6. From the simulations in Fig. 4, we find that the WM treatment can reduce the infection rate by higher efficiency at the earlier stage of treatment which is consistent with the maximum treatment effect by WM. On the other hand, we could not find the obvious differences between Figs. 2 and 5, Figs. 3 and 6, by calculating the terminal values which are as follows: (xtf = 9.9 × 106 , ytf = 0.036, vtf = 0.00749, ztf = 33.549, wtf = 32.279), u4 = 0,

5. Simulation for western medicine combined treatment based on combination of Traditional Chinese Medicine and / 0 Western Medicine with u4 =

(xtf = 9.9 × 106 , ytf = 0.014, vtf = 0.00746, ztf = 29.992,

In this part, we will simulate the optimal procedure when u4 = / 0, all the parameters and initial condition are the same as

/ 0 can make the terminal value of infected We can see that u4 = cells and virus smaller than u4 = 0, and the terminal value of CTL is

wtf = 32.246), u4 = / 0.

Fig. 8. State dynamics of uninfected, infected cells, free virus, CTL and ALT with strategy C.

Y. Su, D. Sun / Biomedical Signal Processing and Control 15 (2015) 41–48 Table 2 Three strategies and corresponding number of objective function. Parameters

The objective function

Strategy A2 Strategy B Strategy C

1.0102 × 1012 2.5166 × 1014 2.4904 × 1014

47

optimization of our models can suggest improved therapies for treating HBV infection, we believe that the analysis and simulation presented in this paper, can play a role in developing an improved HBV treatment regimen. Acknowledgments

smaller than u4 = 0, but the difference is not large, on the other hand, the terminal value of uninfected cells and ALT are almost same. So when the therapy model of virus is set, the u4 = 0 which reflects the reduction of infection can be omitted. The result is consistent with the conclusion [17,37] when the main function of the medicine is to block viral production.

We would like to thank the anonymous referees which have improved the quality of our study. We also would like to thank the support of the Fundamental Research Funds for the Central Universities (No. 06108040), the National Natural Science Funds of China (No. 61074192) and the National Natural Science Funds of China, under grant (11101028).

6. The comparation with other design strategies

References

We have got the optimal therapy strategy A1 (u1 = u1 (t), u2 = u2 (t), u3 = u3 (t), u4 = 0) and A2 (u1 = u1 (t), u2 = u2 (t), u3 = u3 (t), u4 = u4 (t)) which is shown in Figs. 1 and 4, in the following part, we compare the optimal control effects of strategy of A2 with constant control strategy which was used in paper [17,37], the similar result can also be obtained with A1. We choose the constant control strategy is as follows:

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B(u1 = 0.95, u2 = 0.95, u3 = 0.95, u4 = 0.95), C(u1 = 0.45, u2 = 0.45, u3 = 0.45, u4 = 0.45) which correspond to the almost maximum therapy efficient and almost minimum therapy efficient of strategy A2 , we choose the other parameters are the same as those we use in strategy A2 to compare their difference by the simulation and the value of objective functions, the simulation are shown in Figs. 7 and 8, the value of objective functions are given in Table 2. From the simulation, we can see the hepatitis B could also be cured by strategy B just as strategy A2 , but the objective functions value of strategy B is much larger than that of strategy A2 with nearly 250 times. As for the strategy C, it could not cure the hepatitis B, and the objective functions value is also much larger than that of strategy A2 with nearly 250 times. So we can say that the strategy A2 is the optimal control in this system. 7. Conclusion In this paper, we established an optimal control model for HBV based on the combination therapy of TCM and WM. Two optimal control strategies A1 and A2 were given, one case neglected the impact of the infection rate by treatment, the other one concerned with the reduction of the infection rate by treatment. The simulation showed that the impact to the treatment effect by the reduction of the infection rate is hardly obvious, so when the therapy model of virus was set, the u4 which reflects the reduction of infection in the model can be omitted. On the other hand, by the simulation, we also find that in order to reach the optimal control effect which minimizes the viral load and the drug side-effects at a given stage, suppose the effectiveness of the drug is positive related to its dosage, the patient may only take the full dosage at the beginning stage of treatment, they can reduce the dosage at about 18th gradually until half dosage at about 30th day and keep the half dosage to the end of treatment. We also compare our optimal therapy strategy A2 and other constant control strategies by the simulation and the value of objective functions, the comparation results shows that the optimal control strategy A2 is better than constant control strategies in this system. Though the dynamics of infection are more complex than portrayed by the simple mathematical models used here, numerical

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