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Adaptive Control Strategy for Treatment of Hepatitis C Infection Sahar Zeinali and Mohammad Shahrokhi* Chemical and Petroleum Engineering Department, Sharif University of Technology, Tehran 11155-9465, Iran
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S Supporting Information *
ABSTRACT: In this work, an efficient treatment strategy for hepatitis C disease using interferon (IFN) has been proposed on the basis of the back-stepping control technique. The basic model of the hepatitis C virus (HCV) has been considered for controller design. To tackle the problem of model parameter variations, the adaptive version of the back-stepping method has been utilized. For applying the proposed treatment, all states should be available while only the viral load is measured. To solve this problem, a nonlinear Luenberger-like observer has been designed to estimate the unmeasured states. In the proposed treatment, limitation of the drug efficacy has been taken into account. Asymptotical stability of the closed-loop HCV control in the presence of observer dynamics and efficacy limitation has been established by the Lyapunov stability theorem. The effectiveness of the proposed treatment has been illustrated by applying it to a HCV model. Simulation results indicate that applying the proposed scheme leads to the reduction of infected cells to the desired level. ments.26−30 Chakrabarty and Joshi26 presented a treatment strategy for combination therapy based on offline optimal control strategy. An objective function has been established in order to minimize the viral load and side effects of drugs. Also Chakrabarty and et al.28 proposed optimal treatments for monotherapy and combination therapy. Objective functions have been utilized to minimize viral load, infected cells, and drug side effects. The above works are based on the offline optimal control method and are not affected by viral load measurement. Furthermore, the controller has been designed on the basis of a nominal model, and variations of model parameters from one patient to another have not been taken into account. Therefore, adaptive nonlinear control techniques should be utilized in order to consider the model parameter uncertainties. However, implementation of most nonlinear control strategies requires system full state measurements that make it impractical in most applications. To overcome this problem, a nonlinear observer should be used to estimate the unknown states. Another constraint that should be considered is the input limitation. In the case of HCV treatment, the drug efficacy limitation should be taken into account. Recently, an adaptive nonlinear controller has been proposed for the treatment of hepatitis disease.31 In that work practical limitations of treatment implementation such as unavailability of states and bounds on the efficacy have not been considered. Also the proposed control strategy suffers
1. INTRODUCTION Hepatitis C is an infectious disease caused by the hepatitis C virus (HCV), and its transmission is through bloodstreams. Cirrhosis, liver transplantation, cancer, and death are the results of progression of this disease. Due to lack of a vaccine, antiviral therapy has been used to cure chronically infected patients. For several years, interferon α (IFN-α) has been used as an antiviral drug for HCV treatment.1 Combination therapy including pegylated-inteferon (PEG-IFN) and ribavirin (RBV) or using newly direct-acting antivirals (DAAS) are other ways of treatment.2,3 Mathematical modeling has been used as an effective way for getting a deep insight into biological systems. Many different mathematical models have been proposed for the dynamics of the HCV. Neumann et al.4 proposed the basic model for hepatitis C disease under treatment with IFN. They found that the main role of IFN is blocking the production of the virus from infected cells. Dahari et al.5 extended the basic model by considering the proliferation of uninfected and infected cells in the result of homeostatic mechanism. Several mathematical models have been proposed to describe the effect of combination therapy on the dynamics of HCV infection.6−8 These models suggest that ribavirin has a little role in the first phase of viral load reduction, but it enhances the second phase of viral load slope where the IFN efficacy is low. Also many studies have been carried out to develop new models by considering the role of the immune system in stimulating therapeutic cells that enhance the viral load reduction.9−12 Recently, new dynamical models have been proposed for hepatitis C under treatment with DAAS.13−17 Control theories have been extensively utilized in biological systems,18,19 for example for HIV20−25 and HCV treat© XXXX American Chemical Society
Received: Revised: Accepted: Published: A
June 2, 2019 July 9, 2019 July 24, 2019 July 24, 2019 DOI: 10.1021/acs.iecr.9b02988 Ind. Eng. Chem. Res. XXXX, XXX, XXX−XXX
Article
Industrial & Engineering Chemistry Research
production of virus by the factor (1 − u) where u is the efficacy of IFN which is the control input in the treatment strategy. To reduce the number of model parameters, the following dimensionless state variables, parameters, and time are introduced:
from controller singularity problem. In the present study these problems have been solved. In this work, the basic model of Neumann,4 which is in the nonstrict feedback form has been considered as the dynamic model of the disease. To take into account variations of model parameters from one patient to another, the model with parametric uncertainties has been utilized. Also the drug efficacy limitation has been considered in the proposed strategy. To make the treatment realistic, it is assumed that only viral load is measured. It should be noted that for the system in the nonstrict feedback form with parametric uncertainty in the presence of input saturation, no asymptotical observer-based controller has been proposed in the literature. Proposing such a controller is one of the main contributions of the present work. Asymptotic stability of the closed-loop disease control has been established using the Lyapunov stability theorem. The effectiveness of proposed strategy has been demonstrated by applying it to a HCV infection model via simulation study. In what follows novelties and approaches for handling the existing constraints have been summarized: • An asymptotically stable strategy for hepatitis C based on a nonstrict feedback model with parametric uncertainty has been proposed. • The adaptive version of the back-stepping technique has been utilized to take into account the disease model parameter uncertainties. • A nonlinear adaptive Luenberger-like observer has been designed to estimate unmeasured states of the HCV dynamics. In the design of the adaptive observer the Lipschitz condition has been relaxed. • To make the treatment realistic, drug efficacy limitation has been taken into account. • Asymptotic stability of the closed-loop HCV control has been established using the Lyapunov stability theorem. • Simulation results indicate that under the proposed treatment strategy, the infected cell concentration reaches the desired value and the cure condition is achieved. The paper is organized as follows. In section 2, the mathematical model of HCV is presented. The control objective, treatment strategy and closed-loop stability of HCV control, are proposed in section 3. Simulation results are given in section 4. Finally, in section 5 the conclusion is drawn.
β d d δ T , I ̅ = I , V̅ = V , ρ1 = , s s d d pβ s c ρ2 = 3 , ρ3 = , τ = d × t d d
T̅ =
Using the above variables, eq 1 can be transformed into the following form: l o dT ̅ o o = 1 − T̅ − TV ̅ ̅ o o o dτ o o o o o o dI ̅ m = TV ̅ ̅ − ρ1 I ̅ o o dτ o o o o o o dV ̅ o o o o dτ = (1 − u)ρ2 I ̅ − ρ3 V̅ n
(3)
l 0 υ≤0 o o o o o υ 0 < υ < um u = sat(υ) = m o o o o o n um υ ≥ um
(4)
Due to efficacy limitation, u has the following restriction:
where υ is the required efficacy dictated by the control treatment. Remark 1. In the present work, the models proposed by Neumann4 and Dahari5 (will be presented later) have been used to describe the dynamics of hepatitis C disease for Interferon therapy. Both of these models have been validated by using clinical data of patients under Interferon therapy. To check the accuracy of Dahari5 model, Snoeck7 has compared the predicted values obtained from the model with the clinical data and showed that it has a good prediction capability. Also many references26−30 have been proposed treatment strategies for hepatitis C infection based on the Neumann4 or Dahari5 model as a suitable representation of disease dynamics. Remark 2. It has been shown in Appendix A that the relative order of the HCV system is one and by transforming the system to the normal form, the same model is obtained. By setting the output to zero, it can be easily shown that the zero dynamics are stable. Also in Appendix A, it has been proved that the Neumann4 and Dahari5 models are observable.
2. HCV MODEL The mathematical model given by Neumann et al.4 has been considered to develop the treatment strategy l dT o o = s − dT − βTV o o o dt o o o o o o dI o = βTV − δI m o o dt o o o o o dV o o o = (1 − u)pI − cV o o n dt
(2)
3. CONTROL OBJECTIVE AND TREATMENT STRATEGY The objective of treatment is reducing the number of infected cells below a specified level called the cure boundary. The cure boundary was based on the assumption that virion production should cease when all infected cells are cleared, i.e., when there is less than one infected cell in the total plasma and extracellular fluid volume of ∼13.5 × 103 mL.7 In practice, if after 48 weeks no virion is detected in the blood sample of the patient, the treatment is stopped. During the next 24 weeks, the patient blood samples are tested and, if no virus is detected, it is assumed that the cure boundary has been reached. The clinical data indicate that all of the patients who experienced the above conditions have reached the sustained viroligic response (SVR). The purpose of this study is proposing a treatment strategy that reduces the infected cells to the cure boundary level under
(1)
where T, I, and V represent healthy target cells, infected cells, and viral load, respectively. Healthy hepatocytes are produced at constant rate s, die at rate d per healthy cell, and are infected at constant rate β. The produced infected hepatocytes are decreased by rate δ due to natural death. Viruses are produced by rate p per infected cell and are lost at a constant rate c per virus. Administration of IFN as the antiviral drug decreases B
DOI: 10.1021/acs.iecr.9b02988 Ind. Eng. Chem. Res. XXXX, XXX, XXX−XXX
Article
Industrial & Engineering Chemistry Research IFN therapy within 24−48 weeks. In order to achieve this goal, a nonlinear adaptive control strategy based on the back-stepping technique has been proposed. It should be noted that if the infected cell concentration approaches zero, the viral load also becomes zero. To implement the proposed strategy, all states should be available while only the viral load is measured. To tackle this problem, a Luenberger-like observer has been designed. In what follows, the observer design will be explained. Equation 3 can be rewritten into the following form: l dx1 o o o = x 2 + x 2(x3 − 1) − ρ1x1 o o o dτ o o o o o o dx 2 m = x3 + 1 − x 2 − x3(x 2 + 1) o o dτ o o o o o o dx 3 o o o o dτ = (1 − u)ρ2 x1 − ρ3 x3 n
V0̇ ≤ −e TQe + 2x 2,max 2 x32 + 2x 2̂ 2x3̂ 2 + ρ2 2 (1 + um)2 e32 + ρ1̃ 2 x1̂ 2 + (1 + um)2 ρ2̃ 2 x1̂ 2 + ρ3̃ 2 x3̂ 2 ÉÑ ÄÅ ÑÑ ÅÅ ÑÑ ÅÅ ρ − 1 0 k1 ÑÑ ÅÅ 1 2 ÑÑ ÅÅ ÑÑ ÅÅ ÑÑ ÅÅ k 1 ÑÑ Å 2 Q = ÅÅÅ 0 ÑÑ ÑÑ ÅÅ 2 2 ÑÑ ÅÅ ÑÑ ÅÅ Ñ ÅÅ k1 k 1 2 ÅÅ k 3 + ρ3 − ÑÑÑÑ ÅÅ 2 2 2 ÑÑÖ ÅÇ
where
In what follows, a nonlinear controller design based on the backstepping technique will be proposed. The following variables are introduced:
(5)
where x1 = I, x2 = T, and x3 = V̅ . In order to estimate the unknown states, the Luenberger-like observer in the following form has been used: l dx1̂ o o o ̂ o o dτ = x 2̂ + x 2̂ (x3̂ − 1) − ρ1 x1̂ + k1(x3 − x3̂ ) o o o o o o dx 2̂ o m o dτ = x3̂ + 1 − x 2̂ − x3̂ (x 2̂ + 1) + k 2(x3 − x3̂ ) o o o o o o o dx3̂ o o o o dτ = (1 − u)ρ2̂ x1̂ − ρ3̂ x3̂ + k 3(x3 − x3̂ ) n
z 3 = x3̂ − α2 − h
(15)
(16)
(17)
Consider the following Lyapunov function: V1 = V0 +
1 2 z1 2
(18)
Taking the time derivative of V1 and using (12) and (17) yield V1̇ ≤ −e TQe + 2x 2,max 2 x32 + 2x 2̂ 2x3̂ 2 + ρ2 2 (1 + um)2 e32 (7)
+ ρ1̃ 2 x1̂ 2 + (1 + um)2 ρ2̃ 2 x1̂ 2 + ρ3̃ 2 x3̂ 2 + z1(z 2 + α1 + x 2̂ (x3̂ − 1) − ρ1̂ x1̂ + k1e3)
(19)
Adding and subtracting the term z1ρ̃1x̂1 in the right-hand side of (19) yields
(8)
V1̇ ≤ −e TQe + 2x 2,max 2 x32 + 2x 2̂ 2x3̂ 2 + ρ2 2 (1 + um)2 e32
(9)
+ ρ1̃ 2 x1̂ 2 + (1 + um)2 ρ2̃ 2 x1̂ 2 + ρ3̃ 2 x3̂ 2 + z1(z 2 + α1 + x 2̂ (x3̂ − 1) − ρ1̂ x1̂ + k1e3 + ρ1̃ x1̂ ) − z1ρ1̃ x1̂
(10)
(20)
Using Young’s inequality, we obtained the following inequalities:
Also the drug efficacy is bounded as given below 0 ≤ u ≤ um
(14)
z1̇ = z 2 + α1 + x 2̂ (x3̂ − 1) − ρ1̂ x1̂ + k1e3
The number of healthy cells is limited, which means it has an upper bound expressed by x2,max and therefore we have5 x 2 ≤ x 2,max
z 2 = x 2̂ − α1
Since the desired infected cell concentration, xr, is constant, the last term of the above equation becomes zero. Substituting (14) into (16) yields
The time derivative of the above Lyapunov function is given by V0̇ = e1e1̇ + e 2e 2̇ + e3e3̇
(13)
z1̇ = x 2̂ + x 2̂ (x3̂ − 1) − ρ1̂ x1̂ + k1e3 − xṙ (6)
Consider the following Lyapunov function: 1 1 1 V0 = e12 + e 2 2 + e32 2 2 2
z1 = x1̂ − xr
where αi’s are called virtual controls and h is the output of a firstorder filter that will be determined later. It should be noted that the reference signal xr denotes the cure condition. Step 1. The time derivative of z1 is given by
where x̂i and ρ̂i are the estimates of xi and ρi, respectively. ki’s are the observer gains that should satisfy a condition that will be given later. Let ei = xi − x̂i for i = 1, 2, 3 be the state estimation errors. The error dynamics is given by l de1 o o o = x 2x3 − x 2̂ x3̂ − ρ1x1 + ρ1̂ x1̂ − k1e3 o o o dτ o o o o o o de 2 m = −e 2 − x 2x3 + x 2̂ x3̂ − k 2e3 o o dτ o o o o o de 3 o o o = (1 − u)(ρ2 x1 − ρ2̂ x1̂ ) − ρ3 x3 + ρ3̂ x3̂ − k 3e3 o o n dτ
(12)
(11)
z1x 2̂ (x3̂ − 1) ≤ z12 +
where um is the maximum drug efficacy. As shown in Appendix B, the three terms on the right-hand side of (9) satisfy three inequalities given by (B10). Using these inequalities in (9) results in
−z1ρ1̂ x1̂ ≤ z12 + C
1 2 x 2̂ (x3̂ − 1)2 4
1 2 2 ρ ̂ x1̂ 4 1
(21)
(22) DOI: 10.1021/acs.iecr.9b02988 Ind. Eng. Chem. Res. XXXX, XXX, XXX−XXX
Article
Industrial & Engineering Chemistry Research 1 2 e3 4
(23)
1 2 z1 + ρ1̃ 2 x1̂ 2 4
(24)
z1k1e3 ≤ z12k12 + z1ρ1̃ x1̂ ≤
The time derivative of V3 is expressed as V3̇ ≤ −e TQe +
+ (1 + um)2 ρ2̃ 2 x1̂ 2 + ρ3̃ 2 x3̂ 2 − c1z12 − c 2z 2 2 + z 2z 3 1 + H2 − z1ρ1̃ x1̂ + z 3(H3 − ρ2̂ x1̂ u − h)̇ + ρ1̃ ρ∼1̇ γ1 1 1 + ρ2̃ ρ∼2̇ + ρ3̃ ρ∼3̇ γ2 γ3 (33)
Using inequalities (21)−(24) in (20), we have V1̇ ≤ −e TQe +
1 2 e3 + 2x 2,max 2 x32 + 2x 2̂ 2x3̂ 2 + ρ2 2 4
(1 + um)2 e32 + 2ρ1̃ 2 x1̂ 2 + (1 + um)2 ρ2̃ 2 x1̂ 2 + ρ3̃ 2 x3̂ 2
i9 y 1 1 + z1jjj z1 + z 2 + α1zzz + x 2̂ 2(x3̂ − 1)2 + ρ1̂ 2 x1̂ 2 4 k4 { 4 + z12k12 − z1ρ1̃ x1̂
By adding and subtracting term z3ρ̃2x̂1 and z3ρ̃3x̂3 in the righthand side of (33), we have
(25)
V3̇ ≤ −e TQe +
The following virtual control is proposed: (26)
+ H2 − z1ρ1̃ x1̂ + z 3ρ2̃ x1̂ − z 3ρ3̃ x3̂ 1 + z 3(H3 − ρ2̂ x1̂ u − h ̇ − ρ2̃ x1̂ + ρ3̃ x3̂ ) + ρ1̃ ρ∼1̇ γ1 1 ∼̇ 1 ∼̇ + ρ2̃ ρ2 + ρ3̃ ρ3 γ2 γ3 (34)
Using (26) in (25) yields 1 2 e3 + ρ2 2 (1 + um)2 e32 + 2ρ1̃ 2 x1̂ 2 4
+ (1 + um)2 ρ2̃ 2 x1̂ 2 + ρ3̃ 2 x3̂ 2 − c1z12 + z1z 2 + H1 − z1ρ1̃ x1̂
Applying Young’s inequality, the following inequalities are obtained:
(27) 1
1
where H1 = 2x 2,max 2x32 + 2x 2̂ 2x3̂ 2 + 4 x 2̂ 2(x3̂ − 1)2 + 4 ρ1̂ 2 x1̂ 2 2
−z 3ρ2̃ x1̂ ≤
2
+ z1 k1 and c1 is a positive design parameter. Step 2. Consider the following Lyapunov function: V2 = V1 +
z 3ρ3̃ x3̂ ≤
1 2 z2 2
+ H2 − z1ρ1̃ x1̂ 1 4
2
( ) ∂α1 ∂x1̂
2
(x 2̂ x3̂ − ρ1̂ x1̂ + k1e3) and c2 a positive design parameter. Step 3. The time derivative of z3 is given by
(30)
The above equation can be rewritten in the following form:
where H3 = ρ2̂ x1̂ − ρ3̂ x3̂ + k 3e3 −
(31) ∂α2 ̇ x̂ ∂x1̂ 1
−
1 2 1 2 1 2 1 2 z3 + ρ̃ + ρ̃ + ρ̃ 2 2γ1 1 2γ2 2 2γ3 3
ρ1̂ ̇ = γ1( −z1x1̂ − 4x1̂ 2ρ1̂ )
(38)
ρ2̂ ̇ = γ2(z 3x1̂ − 2(1 + (1 + um)2 )x1̂ 2ρ2̂ )
(39)
ρ3̂ ̇ = γ3( −z 3x3̂ − 4x3̂ 2ρ3̂ )
(40)
Note that by setting the last three terms of derivative of Lyapunov function (37) to zero, the first parts of (38)−(40) are obtained. The second parts in (38)−(40) are considered to cancel terms 2ρ̃12x̂12, (1 + (1 + um)2)ρ̃22x̂12, and 2ρ̃32x̂32 in (37). By the way, adding these terms to adaptive laws (38)−(40) makes the controller more robust because they play the σ modification role in the update laws.
∂α2 ̇ x̂ . ∂x 2̂ 2
Consider the following Lyapunov function: V3 = V2 +
(37)
The following adaptive laws are proposed:
∂α2 ̇ ∂α2 ̇ z 3̇ = (1 − u)ρ2̂ x1̂ − ρ3̂ x3̂ + k 3e3 − x1̂ − x 2̂ − h ̇ ∂x1̂ ∂x 2̂
z 3̇ = H3 − ρ2̂ x1̂ u − h ̇
1 2 e3 + ρ2 2 (1 + um)2 e32 + 2ρ1̃ 2 x1̂ 2 2
1 i y + H2 + z 3jjjz 2 + z 3 + H3 − ρ2̂ x1̂ u − hzzż 2 k { 1 1 + ρ1̃ ( −γ1z1x1̂ − ρ1̂ ̇ ) + ρ2̃ (γ2z 3x1̂ − ρ2̂ ̇ ) γ1 γ2 1 + ρ3̃ ( −γ3z 3x3̂ − ρ3̂ ̇ ) γ3
(29) 1
(36)
+ (1 + (1 + um)2 )ρ2̃ 2 x1̂ 2 + 2ρ3̃ 2 x3̂ 2 − c1z12 − c 2z 2 2
+ (1 + um)2 ρ2̃ 2 x1̂ 2 + ρ3̃ 2 x3̂ 2 − c1z12 − c 2z 2 2 + z 2z 3
1
(35)
1 2 z 3 + ρ3̃ 2 x3̂ 2 4
V3̇ ≤ −e TQe +
1 2 e3 + ρ2 2 (1 + um)2 e32 + 2ρ1̃ 2 x1̂ 2 2
where H2 = H1 + 4 h2 + 4 x3̂ 2(x 2̂ + 1)2 + z 2 2k 2 2 +
1 2 z 3 + ρ2̃ 2 x1̂ 2 4
Using (35) and (36) in (34) yields
(28)
In Appendix C it has been shown that the time derivative of the above Lyapunov function satisfies the following inequality: V2̇ ≤ −e TQe +
1 2 e3 + ρ2 2 (1 + um)2 e32 + 2ρ1̃ 2 x1̂ 2 2
+ (1 + um)2 ρ2̃ 2 x1̂ 2 + ρ3̃ 2 x3̂ 2 − c1z12 − c 2z 2 2 + z 2z 3
9 α1 = −c1z1 − z1 4
V1̇ ≤ −e TQe +
1 2 e3 + ρ2 2 (1 + um)2 e32 + 2ρ1̃ 2 x1̂ 2 2
(32) D
DOI: 10.1021/acs.iecr.9b02988 Ind. Eng. Chem. Res. XXXX, XXX, XXX−XXX
Article
Industrial & Engineering Chemistry Research Substituting (38)−(40) into (37) and using 1 1 ρĩ ρî = ρĩ (ρi − ρĩ ) ≤ − 2 ρĩ 2 + 2 ρi 2 , the time derivative of V3 satisfies the following inequality: 1 2 e3 − c1z12 − c 2z 2 2 + H2′ 2 1 i y + z 3jjjz 2 + z 3 + H3 − ρ2̂ x1̂ u − hzzż 2 k {
Vṙ ≤ −e TQ′e − c1z12 − c 2z 2 2 − c3z 32
Since the HCV model is observable, ki’s can be selected such that matrix Q′ becomes positive definite, indicating that the time derivative of Vr becomes negative semidefinite. Therefore, from inequality (48), it is concluded that all closed-loop variables are bounded, i.e., e(t), zi, ρ̂i, and r ∈ L∞. Since the disease dynamic is bounded input bounded state and the drug efficacy is bounded, xi’s are also bounded. From the boundedness of e(t) and zi, it can be inferred that x̂i, αi ∈ L∞. Also with (15) and (44), it is concluded that h and the control signal υ are also bounded and therefore the boundedness of all variables in the closed loop has been established. However, integrating (48) indicates that e(t), zi ∈ L2. Also from (7) and (16) it is inferred that ė, ż1 ∈ L∞. Therefore, with Barbalat’s lemma, it can be concluded that limt→∞e(t), z1 → 0. From asymptotic stabilities of the observer and controller, it is concluded that x̂1 → x1 and x̂1 → xr and consequently x1 → xr, indicating that the control objective has been achieved. In what follows, some guidelines for choosing the design parameters are provided. Select ki’s such that the matrix Q′ becomes positive. Choose some positive values for ci’s and γi’s. It should be noted that higher values of these design parameters lead to higher drug efficacy, which results in shorter time for reaching the cure boundary. However, higher efficacy means higher drug dosage, which increases the drug side effects and therefore a trade-off should be made in choosing these design parameters.
V3̇ ≤ −e TQe +
where H′2 = H2 + ρ22(1 um)2)ρ22x̂12 + 2ρ32x̂32.
+
um)2e32
+
2ρ12x̂12
(41)
+ (1 + (1 +
In order to take into account limitation of the drug efficacy, the following filter has been considered: τ h ̇ = −(ρ2̂ x1̂ )2 h − ρ2̂ x1̂ (u − v) + (ρ2̂ x1̂ )2 + τ 1 ij y jjz 2 + z 3 + H3 + c3z 3 + r 2z 3zzz 2 k {
(42)
In the above equation, τ has been added to the denominator to avoid singularity, c3 is a positive design parameter, and r is updated according to the following adaptive law: r ̇ = γ4r( −sign(r − 1)H2′′ + z 32)
(43)
where H2″ = H2 + ((1 + + 2x̂1 + (1 + (1 + um) )x̂12 + 2 2x̂3 )W and W is the upper bound of parameters norm, i.e., ρ12 + ρ22 + ρ32 ≤ W. From (43), it is concluded that r(t) ≥ 1. This can be proved as follows. If r(t) < 1, it is inferred from (43) that ṙ(t) > 0, indicating r(t) ≥ r(0) = 1, which contradicts with r(t) < 1. Therefore, r(t) ≥ 1 for t = [0,∞).32,33 The following singularity free controller is proposed: um)2e32
υ=
2
2
4. SIMULATION RESULTS In this section the effectiveness of the proposed treatment is discussed. The Neumann4 model parameters used by Dahari et al.5 in their simulation study have been utilized for simulation purposes. Model parameters and initial values of states are given in Table 1.
jijz + 1 z + H + c z + r 2z zyz + ρ ̂ x ̂ h 3 3 3 3 3z j 2 2 1 2 (ρ2̂ x1̂ )2 + τ k { ρ2̂ x1̂
(44)
Table 1. Nominal Values of HCV Model Parameters and Initial States Used in the Simulation Study
Substituting (42) and (44) into (41) results in V3̇ ≤ −e TQ′e − c1z12 − c 2z 2 2 − c3z 32 − r 2z 32 + H2′ ÑÉÑ ÅÄÅ k1 ÑÑ ÅÅ ÑÑ ÅÅ ρ1 − 1 0 ÑÑ ÅÅ 2 ÑÑ ÅÅ ÑÑ ÅÅ ÑÑ ÅÅ k 1 2 ÑÑ Q′ = ÅÅÅ 0 ÑÑ ÅÅ 2 2 ÑÑ ÅÅ ÑÑ ÅÅ ÑÑ ÅÅ k1 k ÑÑ 2 ÅÅ k + ρ − 1 ÑÑ 3 ÅÅ 2 3 ÑÑÖ 2 ÅÇ
(45)
where
1 2 r 2γ4
value
parameter
value
state
value
s d β
2.6 × 104 2.6 × 10−3 2.25 × 10−7
δ p c
0.26 2.9 6
T(0) I(0) V(0)
4.4 × 106 1.4 × 106 106
Table 2. Nominal Values of Dimensionless HCV Model Parameters, Initial States, and Their Estimates
(46)
Taking time derivative of Vr and using (43) and (45) results in Vṙ ≤ −eT Q′e − c1z12 − c 2z 22 − c3z 32 + H2′ − H2′′r 2sign(r − 1)
parameter
The corresponding dimensionless parameters, initial states and their estimates are presented in Table 2. The controller design parameters are tabulated in Table 3. The maximum value of efficacy is set to 0.96. Figures 1 and 2 show the performance of the proposed treatment and its
In order to check boundedness of r(t), the following Lyapunov function is considered: Vr = V3 +
(48)
(47)
Since for finite time r(t) > 1 and H2′′ ≥ H2′ , it is inferred that H2′ − H′2′r2 sign(r − 1) ≤ 0 and therefore (47) can be written as E
parameter/state
value
parameter/state
value
ρ1 ρ2 ρ3 ρ̂1(0) ρ̂2(0) ρ̂3(0) h(0)
100 9.6524 × 105 2.3077 × 103 300 2.5 × 104 103 0
T̅ (0) I̅(0) V̅ (0) T̅̂ (0) ̂ I̅(0) V̅̂ (0) r(0)
0.44 0.14 86.5385 0.4 0.7 50 1
DOI: 10.1021/acs.iecr.9b02988 Ind. Eng. Chem. Res. XXXX, XXX, XXX−XXX
Article
Industrial & Engineering Chemistry Research Table 3. Controller and Observer Design Parameters parameter
value
parameter
value
c1 c2 c3 k1 k2 k3
1 2 × 104 104 300 20 5000
γ1 γ2 γ3 γ4 τ W
10 0.001 0.001 1 5 1014
corresponding drug efficacy. Also cells and virus concentrations and their estimates are shown in Figures 3−5.
Figure 3. Infected cell concentration and its estimate versus time.
Figure 1. Infected cell concentration and cure boundary versus time.
Figure 4. The healthy cell concentration and its estimate versus time.
Figure 2. Corresponding drug efficacy versus time.
As can be seen from Figure 1, under the proposed treatment strategy, the infected cell concentration has reached the desired level after approximately 80 days. From this figure it is inferred that the proposed strategy is effective and asymptotic tracking has been achieved. Figures 3−5 show that the estimated values of all states have converged to their actual values. 4.1. Robustness of the Proposed Treatment. In this section the robustness of the proposed treatment strategy against model uncertainties has been discussed. The uncertainties can be parametric or structural. To check the robustness of the proposed treatment, the model parameters have been changed to test the performance under parametric
Figure 5. Viral load and its estimate versus time.
uncertainty, while the HCV model has been switched to evaluate the performance under structural uncertainty. F
DOI: 10.1021/acs.iecr.9b02988 Ind. Eng. Chem. Res. XXXX, XXX, XXX−XXX
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Industrial & Engineering Chemistry Research 4.1.1. Parametric Uncertainty. The parameters of the HCV model are patient dependent and can differ from one person to another. To check the performance of the proposed treatment, the controller with design parameters given in Table 3 is applied to another model with different parameters. According to the disease dynamic (3), a decrease of parameter ρ1 results in a decline of infected cell death rate and an enhancement of virus production. However, an increase of ρ2 and a decrease of ρ3 lead to a higher virus concentration. To consider the worst scenario, model parameters have been changed such that concentrations of virus and infected cells increased and the concentration of the healthy cells decreased. The applied parameter changes are as follows l ρ → 0.7 × ρ o 1 1 o o o o ρ2 → 1.3 × ρ2 m o o o o o ρ → 0.7 × ρ3 o n 3
Comparing the performances given in Figures 1 and 6 shows that the proposed treatment strategy is robust against model parameter variations. As can be seen, the performance of the treatment almost remains the same although the model parameters have changed 30%. 4.1.2. Structural Uncertainty. To check the performance of the proposed treatment under structural uncertainty, the controller with design parameters given in Table 3 is applied to the model of HCV proposed by Dahari et al.5 This model has been given by (50) l o ij dT ̅ T̅ + I ̅ yzz o o o = 1 + rT̅ jjj1 − zz − T̅ − TV ̅ ̅ o o j z o dτ Tmax ̅ o k { o o o o o o dI ̅ T̅ + I ̅ zyz ji m = TV zz − ρ1 I ̅ ̅ ̅ + rI ̅ jjjj1 − o o z o dτ Tmax ̅ o k { o o o o o o dV ̅ o o = (1 − u)ρ2 I ̅ − ρ3 V̅ o o o dτ n
(49)
Figures 6 and 7 show variations of the infected cell concentration and drug efficacy versus time, respectively.
(50)
In the above model there are two additional parameters namely, r and Tmax that have been set to 1.6154 × 103 and 1, respectively. Figures 8 and 9 show variations of the infected cell concentration and drug efficacy versus time, respectively.
Figure 6. The infected cell concentration and cure boundary versus time in the case of parametric uncertainties. Figure 8. Infected cell concentration and cure boundary versus time in the case of model mismatch.
Comparing the performances given in Figures 1 and 8 shows that switching from the Neumann4 model to (50) does not affect the proposed treatment performance. 4.2. Performance of the Proposed Treatment under Measurement Noise. In order to check the performance of the proposed treatment under measurement noise, a white noise with the magnitude of 10% of the output value has been added to the measurements. Figures 10 and 11 show variations of the infected cell concentration and the corresponding drug efficacy versus time. As can be inferred from Figures 10 and 11, the proposed control treatment can tolerate this level of noise and has an acceptable performance. Although the simulation study shows the effectiveness of the proposed treatment strategy, it is desirable to test it on real patients. So far all suggested treatment strategies have been evaluated via simulation study26−31 and to the best of authors’
Figure 7. Corresponding drug efficacy versus time in the case of parametric uncertainties.
G
DOI: 10.1021/acs.iecr.9b02988 Ind. Eng. Chem. Res. XXXX, XXX, XXX−XXX
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knowledge no real application has been reported. We hope that, in the future, there will be a chance to apply the proposed control treatments in the literature to real patients.
5. CONCLUSION In this paper, an observer-based adaptive back-stepping treatment strategy has been proposed for hepatitis C under IFN therapy. To simulate the disease behavior, the Neumann4 model has been used. To consider the model uncertainty, the adaptive version of the back-stepping technique has been utilized. In the proposed strategy the limitation of drug efficacy has been taken into account. The asymptotical closed-loop stability of the HCV control has been established using the Lyapunov stability theorem. Simulation results indicate that the infected cell concentration has reached the desired level approximately after 80 days. Also the viral load has approached almost zero concentration and the estimated states have converged to their real values. Simulation results also show that the control strategy is robust against model parameter and structural uncertainties.
Figure 9. Corresponding drug efficacy versus time in the case of model mismatch.
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FUTURE WORKS To make the treatment more realistic, a stochastic model of the disease should be used for designing a control scheme for hepatitis C. For this case a Kalman filter should be used to estimate the unmeasured states. Proposing such a treatment strategy is the subject of the future work.
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ASSOCIATED CONTENT
S Supporting Information *
The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.iecr.9b02988.
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Figure 10. Infected cell concentration and cure boundary versus time for noise corrupted measurements.
Appendix A, zero dynamics and observability analysis; Appendix B, inequalities used in observer design; Appendix C, second step of back-stepping design (PDF)
AUTHOR INFORMATION
Corresponding Author
*E-mail address:
[email protected]. ORCID
Mohammad Shahrokhi: 0000-0003-1283-1824 Notes
The authors declare no competing financial interest.
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REFERENCES
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