Modeling and Control of Drug Delivery Systems

Modeling and Control of Drug Delivery Systems provides comprehensive coverage of various drug delivery and targeting systems and their state-of-the-art related works, ranging from theory to real-world deployment and future perspectives. Various drug delivery and targeting systems have been developed to minimize drug degradation and adverse effect and increase drug bioavailability. Site-specific drug delivery may be either an active and/or passive process. Improving delivery techniques that minimize toxicity and increase efficacy offer significant potential benefits to patients and open up new markets for pharmaceutical companies.

This book will attract many researchers working in DDS field as it provides an essential source of information for pharmaceutical scientists and pharmacologists working in academia as well as in the industry. In addition, it has useful information for pharmaceutical physicians and scientists in many disciplines involved in developing DDS, such as chemical engineering, biomedical engineering, protein engineering, gene therapy.

• Presents some of the latest innovations of approaches to DDS from dynamic controlled drug delivery, modeling, system analysis, optimization, control and monitoring
• Provides a unique, recent and comprehensive reference on DDS with the focus on cutting-edge technologies and the latest research trends in the area
• Covers the most recent works, in particular, the challenging areas related to modeling and control techniques applied to DDS

• Language: English
• Publisher: Academic Press
• Release date: Feb 6, 2021
• ISBN: 9780128211953

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Preface

Ahmad Taher Azar, Faculty of Computers and Artificial Intelligence, BenhaUniversity, Benha, Egypt, College of Computer & Information Sciences (CCIS), Prince Sultan University, Riyadh, Saudi Arabia

Drug delivery is the method or process of administering a pharmaceutical compound to achieve a therapeutic effect in humans or animals. Delivering drugs at controlled rate, slow delivery, and targeted delivery are other very attractive methods and have been pursued enthusiastically. Various drug delivery and targeting systems have been developed to minimize drug degradation and adverse effect, and to increase drug bioavailability. Site-specific drug delivery may be either an active and/or passive process. In the past few years, researchers have appreciated the potential benefits of nanotechnologies in providing vast improvements to drug delivery and targeting. Improving delivery techniques that minimize toxicity and increase efficacy offer great potential benefits to patients and also open up new markets for pharmaceutical companies. The book is intended to provide a comprehensive coverage of various drug delivery and targeting systems and their state-of-the-art related works, ranging from theory to the real-world deployment. Also, it discusses the future perspectives of DDS.

About the book

The new Elsevier book, Modeling and Control of Drug Delivery Systems, consists of 20 contributed chapters by subject experts who are specialized in the various topics addressed in this book. The special chapters have been brought out in this book after a rigorous review process in the broad areas of modeling, simulation, control, and drug delivery systems. Special importance was given to chapters offering practical solutions and novel methods for the recent research problems in drug delivery systems.

Objectives of the book

This book will attract many researchers working in the area of drug delivery Systems. This book presents some of the latest innovative of approaches to DDS from a point of view of dynamic modeling, system analysis, optimization, control, and monitoring, and so on. This book will be an important source of information for pharmaceutical scientists and pharmacologists working in the academia as well as in the industry. It has useful information for pharmaceutical physicians and scientists in many disciplines involved in developing DDS such as chemical engineering, biomedical engineering, protein engineering, gene therapy, and so on. This will be an important reference for executives in charge of research and development at several hundred companies that are developing drug delivery technologies. Thus the purpose of this book is to provide the community with a unique, recent, and comprehensive reference on DDS with the focus on cutting edge technologies and the recent research trends in the area.

Organization of the book

This well-structured book consists of 20 full chapters.

Book features

•The book chapters deal with the recent research problems in the areas of modeling, simulation, control, and drug delivery systems.

•The book chapters present various techniques of drug delivery systems such as Hepatitis C Virus Epidemic Control Using a Nonlinear Adaptive Strategy, Integral Sliding Mode Control of Immune Response for Kidney Transplantation, Smart Drug Delivery Systems, Polymeric Transdermal Drug Delivery Systems, Nanoparticle Drug Delivery, Targeted Drug Delivery, and so on.

•The book chapters contain a good literature survey with a long list of references.

•The book chapters are well-written with a good exposition of the research problem, methodology, block diagrams, and mathematical techniques.

•The book chapters are lucidly illustrated with numerical examples and simulations.

•The book chapters discuss details of different applications and future research areas.

Audience

•The book is primarily meant for researchers from academia and industry, who are working on modeling, simulation, control and drug delivery research areas—biomedical engineering, control engineering, computer engineering, computer science, nanotechnology, applied sciences, pharmacy, and medicine. The book can also be used at the graduate or advanced undergraduate level as a text book or major reference for courses such as Concepts of Drug Delivery, Advanced Drug Delivery systems, Controlled Drug delivery systems, and many others.

Acknowledgments

•As the editors, we hope that the chapters in this well-structured book will stimulate further research in modeling, simulation, control and drug delivery systems, and utilize them in real-world applications.

We hope sincerely that this book, covering so many different topics, will be very useful for all readers.

We thank all the reviewers for their diligence in reviewing the chapters. Special thanks to Elsevier, especially the book Editorial team.

Chapter 1: Hepatitis C Virus Epidemic Control Using a Nonlinear Adaptive Strategy

Javad K. Mehra,b; Samaneh Tangestanizadehc; Mojtaba Sharifia,b; Ramin Vatankhahc; Mohammad Eghtesadc    a Department of Electrical and Computer Engineering, University of Alberta, Edmonton, AB, Canada

b Department of Medicine, University of Alberta, Edmonton, AB, Canada

c Department of Mechanical Engineering, Shiraz University, Shiraz, Iran

Abstract

Hepatitis C is a viral infection that appears as a result of the hepatitis C virus (HCV), and it has been recognized as the main reason for liver diseases. HCV incidence is growing as an important issue in the epidemiology of infectious diseases. In the present study, a mathematical model is employed for simulating the dynamics of HCV outbreak in a population. The total population is divided into five compartments, including unaware and aware susceptible, acutely and chronically infected, and treated classes. Then, a Lyapunov-based nonlinear adaptive method is proposed for the first time to control the HCV epidemic considering modeling uncertainties. A positive definite Lyapunov candidate function is suggested, and adaptation and control laws are attained based on that. The main goal of the proposed control strategy is to decrease the population of unaware susceptible and chronically infected compartments by pursuing appropriate treatment scenarios. As a consequence of this decrease in the mentioned compartments, the population of aware susceptible individuals increases and the population of acutely infected and treated humans decreases. The Lyapunov stability theorem and Barbalat's lemma are employed in order to prove the tracking convergence to desired population reduction scenarios. Based on the acquired numerical results, the proposed nonlinear adaptive controller can achieve the above-mentioned objective by adjusting the inputs (rates of informing the susceptible people and treatment of chronically infected ones) and estimating uncertain parameter values based on the designed control and adaptation laws, respectively. Moreover, the proposed strategy is designed to be robust in the presence of different levels of parametric uncertainties.

Keywords

Nonlinear adaptive control; Hepatitis C epidemic; Infectious disease dynamics; Uncertainty and stability; Lyapunov stability theorem; Robust performance

1: Introduction

The hepatitis C virus (HCV) is a blood-borne virus identified as the main cause of liver diseases [1–3]. Globally, about 3% of the world population (170 million) are dealing with HCV and 71 million people have chronic hepatitis C infection [1, 4–6]. Several studies showed that the chronic stage of HCV will develop cirrhosis and liver cancer in the case of no treatment and approximately 339,000 people die every year due to these diseases [1, 7]. Despite previously mentioned statistics which makes HCV infection one of the important health threats, this disease received little attention especially in the regions with a higher rate of infectiousness [4].

Although fatigue and jaundice were mentioned as symptoms of the HCV, this disease often has no considerable symptom, even in the advanced stages. This is the reason that the HCV outbreak is called the silent epidemic [4, 8]. Several different ways were reported for HCV prevalence, which includes sharing injection equipment, unsafe sexual contacts, inadequately sterilization of syringes and needles especially for health-care personnel, and transfusion of polluted blood [1, 9]. Even though these are the main causes of the HCV epidemic, but some other reasons may also be critical in some societies based on special conditions. For instance in the developed countries, since there is precise control on the blood transfusion, the importance of injecting drug use in transmission of the disease has increased compared to the transfusion of polluted blood and its products [2, 9].

Natural cure at the chronic stage of HCV is not common, but it can happen for about 10%–15% of patients that the RNA of HCV is indistinguishable in their serum [5, 6]. For the rest of the patients (80%–85%) that the HCV could not be healed by their immune system response, some drug therapy regimes should be employed. Hepatitis C drugs have recently had some developments. Available safe, highly effective, and endurable combinations of oral antivirals that act directly have currently developed for this disease [4, 10]. Although vaccination is the most vital way of controlling different viral diseases, but unfortunately there is no vaccine for the HCV yet [5]. Therefore, preventing this disease has an important role in stopping the extension of its epidemic.

In the present study, a nonlinear adaptive method is developed for treatment and control of the HCV epidemic. For this purpose, the recently published nonlinear HCV epidemiological model in [4] is employed and different parametric uncertainties are taken into account, despite the previous optimal strategies [4]. The main goal of the proposed control scheme is the population decrease in the unaware susceptible and chronically infected compartments in the existence of parametric uncertainties. Accordingly, two control inputs (efforts to inform susceptible individuals and treatment rate) are employed to track descending desired populations of the previously mentioned compartments. The asymptotic stability and tracking convergence of the closed-loop system having uncertainties are proven using the Lyapunov stability theorem and Barbalat's lemma. Innovations of this research are as follows:

•For the first time, a nonlinear adaptive method is developed to control the HCV epidemic by defining a novel Lyapunov function candidate that provides the tracking convergence proof.

•Due to the lack of accurate information about HCV model parameters in each society, parametric uncertainty is taken into account in this research for the first time, and the defined control objectives are achieved in the presence of these inaccuracies.

•In all of the previous studies that have been conducted on the control of the HCV outbreak, populations of some undesired compartments at the end of the investigation period were considered as the criterion for designing control inputs [4, 11–14]. However, in this study for the first time, the populations of two unaware susceptible and chronically infected classes during the entire treatment period are considered as the criterion and control inputs are designed in a way to track desired values instead of focusing on their final populations at the end of process.

The rest of this chapter is organized as follows. In Section 2, related research work will be explained. Description of the dynamic model and the proposed control scheme will be presented in Sections 3 and 4, respectively. The simulation results will be depicted and discussed in Section 5, and the concluding remarks will be mentioned in Section 6.

2: Related Research Work

Previous related studies are presented in this section and are divided into three parts, including mathematical modeling, optimal control for HCV, and adaptive control strategies for different biological systems.

Several analytical analyses were conducted on the dynamic modeling of the HCV epidemic, which are described here. Martcheva and Castillo-Chavez [15] presented a simple mathematical model with three compartmental variables including susceptible, acutely infected, and chronically infected. They considered different epidemiological observations in the model. Yuan and Yang [8] added the exposed class to the previous model [15]. They considered that the susceptible individuals transmit to the exposed compartment in the case of having contact with the infected compartments. Zhang and Zhou [5] added a new term in the model of Yuan and Yang [8], which denotes the death rate due to the HCV. Hu and Sun [16] proposed another epidemiological model for the HCV with four classes in which the recovered compartment was taken into account for the first time. Naturally, the recovered people transmit to this class from the acutely infected and chronically infected compartments and become immune against this. Ainea et al. [17] extended the previous model [16] by adding the exposed class. Both these models [16, 17] considered the HCV disease-induced death rate for both acutely infected and chronically infected classes. Shen et al. [18] proposed a dynamical model with six classes including susceptible, exposed, acutely infected, chronically infected, treated, and recovered populations. They propounded treatment influence for the first time and classified treated people in a distinct class. Shi and Cui [19] improved the model in [18] and divided the treated class into two different classes by defining the treatment for chronic infection and aware reinfection.

Some researches have been conducted for optimal control of the HCV outbreak. Okosun [11] employed a SITV (susceptible, acutely infected, treated, and chronically infected) model for the HCV that was an extended form of the dynamics presented in [8]. This model [11] included the treatment compartment and considered movement for susceptible, treated, and acutely and chronically infected people among their compartments. Some time-dependent optimal control strategies are proposed, in order to control the HCV disease. A cost function is calculated for these strategies in order to evaluate the effectiveness of the control methods and select the most efficient one. Okosun and Makinde [12] employed a SEITV (susceptible, exposed, acutely infected, treated, and chronically infected) dynamical model for the HCV outbreak considering the screening rate and drug efficacy as control inputs for acutely and chronically infected populations and used the Pontryagin's principle to solve the optimal control problem. Another epidemiological model was investigated in [4] for the HCV outbreak in which the susceptible class was divided into aware and unaware classes. Moreover, they considered two control inputs including screening and treatment rates for the HCV epidemic model, which was determined by an optimal control law. In [4], the dynamics was formulated with the susceptibility reduction due to the publicity and the treatment process to identify the feasible effect of public concerns and treatment on the HCV. An optimal neuro-fuzzy strategy was also introduced in [13] in order to control the HCV epidemic. They [13] employed the mathematical model proposed in [12] as a deterministic model and utilized the genetic algorithm to obtain optimal control inputs.

As described, all of previous studies on the control of HCV epidemic were conducted on the optimal strategies. On the other hand, some other research works were performed on the adaptive control of different diseases as presented here. Moradi et al. [20] suggested a Lyapunov-based adaptive method to control three different hypothetical models of the cancer chemotherapy inside the human body and compared results among these models. In the next step of this research [21], a composite adaptive strategy has been developed for online identification of cancer parameters during the chemotherapy process. Boiroux et al. [22] employed a model predictive controller for the type 1 diabetes model and used an adaptive controller to balance the blood glucose. They determined the model parameters based on the clinical information of past patients. Aghajanzadeh et al. [23] suggested an adaptive control strategy for hepatitis B virus infection inside the human body by antiviral drugs. They considered model parameters uncertainties on model parameters and employed adaptive controller to control the dynamic despite uncertainties of the system. Sharifi and Moradi [24] designed a robust scheme with adaptive gains to control the influenza epidemic, considering its dynamic model's uncertainties. Padmanabhan et al. [25] proposed an optimal adaptive method to control the sedative drug in anesthesia administration. They employed an integral reinforcement learning method in order to overcome the uncertainty of parameter values.

3: Dynamic Model of Hepatitis C Virus Epidemic

Mathematical modeling is a useful way of analyzing the epidemiology of a disease. These models have two important capabilities: (1) finding out mechanistic understanding of the disease and (2) exploring potential outcomes of the epidemic under different conditions [26]. For assessment of the proposed method for the HCV prevalence control in a population, a nonlinear compartmental model is used with five different classes including unaware susceptible (Su), aware susceptible (Sa), acutely infected (I), chronically infected (C), and the treated (T) humans [4]. The susceptible compartment is divided into two classes, including aware and unaware people. Note that aware people have information about the HCV transmission ways and preventing methods despite the unaware population. Since there is no available vaccine for the HCV, informing people about preventing methods is a very important way to reduce the risk of infection for susceptible people [1]. Therefore, the unaware susceptible individuals (Su) will be infected in contact with the infected population (I, C, and T) with a higher rate in comparison with the aware susceptible individuals (Sa) [4]. Thus, the transmission rate for unaware susceptible humans (Su) should be considered larger than this rate for aware susceptible humans (Sa) in the dynamic model [4]. The nonlinear mathematical model of HCV epidemic is as follows:

   (1)

. u1 and u2 are control inputs and defined respectively as the effort rate to inform unaware susceptible individuals and the treatment rate for chronically infected class. N denotes the total population and will be calculated as

   (2)

The population of unaware susceptible (Su) increases with the rate of b, respectively. Infectiousness rate for acutely infected people is higher than chronically infected individuals, and the treated people have the lowest rate; thus, it is assumed that K1 > K2 [4, 5]. The total population (N) decreases with two different rates μ and θ, where μ denotes the rate of natural death that decreases populations of all compartments. However, θ is the rate of HCV-induced death and decreases the population of the chronically infected compartment (C).

During the acute stage (I), the HCV could have different behaviors for each patient based on his/her immune system response. For 15%–25% of cases in this stage, the RNA of HCV becomes indistinguishable in their blood serum and the ALT level returns to the normal range. This observation is defined by the term (1 − q)γI in the proposed HCV dynamics [4, 6]. Approximately, the immune system in 75%–85% of the patients could not remove the hepatitis C virus in the acute stage and their disease becomes advanced to the chronic stage. Note that if the HCV RNA remains in the patient's blood for at least 6 months after the onset of acute infection, the chronic level of the disease will appear which is defined by the term qγI in Eq. (1) [5, 6]. Finally, the defeat in the treatment process is defined by the term p. The treated population decreases by the rate of ξT and joins the chronic class by the rate of pξT in the case of treatment failure and the rest of this population (1 − p)ξT will join the aware susceptible class if the treatment is successful. The schematic diagram of the proposed nonlinear dynamics of the HCV epidemic is depicted in Fig. 1 and descriptions of the parameters are presented in Table 1 [4].

Fig. 1 Schematic diagram of population transmission among different classes of HCV epidemic.

Table 1

4: Nonlinear Adaptive Controller Formulation for Epidemiology of HCV

In this section, a new nonlinear adaptive controller is formulated for the uncertain hepatitis C virus epidemic. The main purpose of the control method is to minimize the populations of unaware susceptible (Su) and chronically infected (C) classes. Two control inputs u1(t) and u2(t) are considered in order to reach this objective. u1(t) denotes the effort rate to inform the susceptible individuals from the HCV by media publicity, educational campaigns, public service advertising, and so on, and u2(t) is employed to reflect the rate of treatment on chronically infected individuals [4].

Using the above-mentioned control inputs, the populations of unaware susceptible (Su) and chronically infected (C) classes will decrease by tracking some desired values. Moreover, due to the decrease of the mentioned components, the number of aware susceptible (Sa) and treated (T) individuals will increase and decrease, respectively. The Lyapunov theorem is employed to prove the stability of the closed-loop system. In addition, some adaptation laws are defined in order to update the estimated parameters of the system to guarantee the stability and robustness of the system against the uncertainties of the dynamic model. A conceptual diagram of the proposed nonlinear feedback controller with the adaptive scheme is illustrated in Fig. 2.

Fig. 2 Conceptual diagram of the nonlinear adaptive method developed to control the HCV epidemic in the existence of uncertainties on the model parameters.

4.1: Nonlinear Adaptive Control Laws

Control inputs (u1(t), u2(t)) could be calculated using dynamics of the unaware susceptible and chronically infected compartments from Eq. (1) as

   (3)

   (4)

Property

The right-hand sides of Eqs. (3), (4) can be linearly parameterized in terms of their available parameters. ϕ1 and ϕ.

Now, the right sides of the above equations can be rewritten as

   (5)

   (6)

where Y1 and Y2 are the regressor matrices, contain known functions of HCV epidemic variables. θ1 and θ2 are the parameter vectors, which contain unknown parameters of the dynamic (Eqs. 7, 8). Accordingly, these matrices and vectors are defined as

   (7)

   (8)

This regressor presentation is used to summarize the equations and define the adaptation and control laws. In order to design nonlinear control laws, two new variables ϕ1 and ϕ2 are defined as follows:

   (9)

   (10)

where λ1 and λ2 are the controller gains and considered to be positive and constant. Now, the nonlinear adaptive control laws are defined as

   (11)

   (12)

are the vectors of estimated parameters.

In the following section, taking advantage of the Lyapunov stability theorem, it will be proven that the control laws (11), (12) together with some adaptation laws guarantee the tracking convergence, stability, and robustness for the treatment of HCV outbreak.

4.2: Stability Proof and Adaptation Laws

The closed-loop dynamics of the system is achieved first by substituting the control laws (11), (12) into the dynamics of the HCV epidemic (1):

   (13)

   (14)

(for i .

The adaptation laws are designed to update parameters’ estimation to keep the system's robustness against uncertainties, as

   (15)

   (16)

where Γ1 and Γ2 are the adaptation gain matrices and considered to be positive definite.

Now, employing the Lyapunov stability theorem [27] and based on the previously derived closed-loop dynamics (13), (14) and the designed adaptation laws (15) and (16), the tracking convergence, stability and robustness for the aware susceptible and chronically infected classes will be proven. With this aim, a positive definite Lyapunov candidate function is selected as

   (17)

The Lyapunov function's time derivative is determined:

   (18)

because θ is zero). By substituting the adaptation laws (15) and (16) into Eq. (18), the time derivative of V is simplified to

   (19)

As mentioned in the previous descriptions, λ1 and λ2 are considered to be positive; thus, the Lyapunov function's time derivative is negative semidefinite. Thus, based on Barbalat's lemma (described in Appendix) and the Lyapunov stability theorem [as t →∞) in the presence of uncertainties. Thus, the numbers of unaware susceptible (Su) and chronically infected (Cand C → Cd).

5: Results and Discussion

For the effectiveness evaluation of the proposed method, some simulations are presented in this section. Note that computer simulations have proven to be useful for evaluating the spreading behavior of infectious diseases [28]. In the present study, the simulation process is performed in the Simulink-Matlab environment. The parameters’ values of the HCV epidemic model (1) are listed in Table 2.

Table 2

) and the treatment of chronically infected people (Cd):

   (20)

   (21)

where a1 and aare the initial and final (steady-state) populations of the unaware susceptible class, respectively. C0 and Cf are the initial and final (steady-state) populations of the chronically infected compartment, respectively.

The presented reduction and treatment scenarios (20), (21) are employed in these simulations as the desired decreasing behavior of the HCV epidemic control. However, other continuously decreasing fashion can be used as desired scenarios without loss of generality. The values of parameters in the desired HCV population reduction scenarios (20), (21) are listed in Table 3. These scenarios for unaware susceptible and chronically infected compartments are shown in Fig. 3.

Table 3

Fig. 3 Desired scenarios for the reduction of unaware susceptible and chronically infected compartments in the HCV epidemic.

In the absence of control inputs, the HCV infection will extend in the society based on Eq. (1). Accordingly, the treated population will decrease and reach zero exponentially due to the lack of treatment process. In that case (no control input), unaware and aware susceptible individuals will get the hepatitis C virus in contact with the infected people in I and C compartments and will join the acutely infected class (I). Since there is no treatment for acutely infected individuals (as seen in Eq. 1), the HCV disease will progress and reach the chronic stage. Thus, the population of the chronically infected compartment (C) will increase and the populations of all other compartments will decrease. Fig. 4 depicts the above-mentioned points about the HCV outbreak in the case of no control input.

Fig. 4 Populations of (A) unaware and aware susceptible, and (B) acutely infected, chronically infected and treated classes in the absence of control inputs.

However, applying the proposed strategy based on the designed nonlinear control laws (11), (12) with the obtained adaptation laws (15), (16), the population changes in different compartments in the presence of 20% parametric uncertainty are depicted in Fig. 5.

Fig. 5 Populations of (A) unaware and aware susceptible, and (B) acutely infected, chronically infected and treated classes in the presence of control inputs ( u 1 and u 2 ) based on the proposed laws ( 11), ( 12).

As seen, due to the employment of the first control input (u1), the population of unaware susceptible compartment (Su) reduces and they join the aware susceptible class (Sa). Since the rate of infection for the aware susceptible people is less than that of the unaware ones, and due to the effect of control input u1, the extension of the HCV infection decreases compared with the no-control-input case (shown in Fig. 4). Moreover, using the treatment as the second control input (u2), the population of the chronically infected compartment (C) decreases (Fig. 5) based on the described scenarios (Cd and C → Cd). Fig. 6 depicts the desired and real populations of unaware susceptible and chronically infected classes, which imply the appropriate convergence performance using the nonlinear controller. The corresponding tracking errors are presented in Fig. 7.

Fig. 6 Convergence of unaware susceptible and chronically infected populations ( S u and C and C d ).

Fig. 7 Tracking errors between the desired and real values of unaware susceptible and acutely infected compartments.

As described, two control inputs are adjusted according to the proposed nonlinear adaptive strategy in order to prevent the HCV outbreak. The first control input u1(t) denotes the effort rate to inform the susceptible individuals from the HCV and the second one u2(t) is the treatment rate for chronically infected individuals. These control inputs are considered to be normalized in Eq. (1) to be in the range of [0, 1]. The obtained values for these inputs using the proposed control strategy are shown in Fig. 8, which satisfy the physiological constraint (u) based on the designed adaptation laws (15), (16) in the presence of 20% uncertainty.

Fig. 8 Control inputs ( u 1 and u 2 ) during the treatment period of HCV epidemic.

Fig. 9 Estimation of parameters in (A) θ 1 and (B) θ 2 during the treatment period based on the adaptation laws ( 15) and ( 16), respectively.

5.1: System Response to Different Uncertainty Levels

In this section, the effects of different uncertainty levels are investigated for the HCV epidemic dynamics. For this purpose, 50%, 70%, and 90% uncertainties are considered on the initial guess of parameters in θ1 and θ2 (defined in Eqs. 7, 8). Performance of the adaptation laws (15), (16) on the tuning of estimated model parameters is investigated in Fig. 10. As discussed and proven in Section 4, these adaptation laws guarantee that the estimation errors of the HCV dynamic parameters remain bounded against different uncertainty levels.

Fig. 10 Adaptation of (A) θ 1 (1) and (B) θ 2 (1) using Eqs. ( 15) and ( 16), respectively, for different uncertainty levels.

).

Fig. 11 ) for different parametric uncertainty levels.

As observed in and C → Cd) in the existence of different levels of uncertainty.

6: Conclusion

In the present study, a new nonlinear adaptive strategy was developed to control the hepatitis C virus epidemic based on a mathematical model having uncertainties. For the first time, an adaptive feedback controller was employed to decrease the populations of unaware susceptible and chronically infected compartments based on the desired scenarios. Two control inputs were employed for this goal. The first one is the effort rate to inform the susceptible individuals from the HCV and the second one is the rate of treatment for chronically infected people. The Lyapunov stability theorem and Barbalat's lemma were used to prove the tracking convergence to desired treatment scenarios. The proposed control laws and adaptation laws provided the stability of the closed-loop HCV epidemic system in the presence of parametric uncertainties. Results of numerical simulations showed that by adjusting the control inputs and the estimated parameters based on this strategy, the number of the unaware susceptible and chronically infected individuals are decreased. As a result, the population of the aware susceptible was increased and the population of the acutely infected and treated classes reached out to zero at the end of the process. Moreover, the obtained results implied that the tracking convergence is achieved for a wide range of uncertainties. Designing optimal trajectories and employing unstructured uncertainties can be considered as the next steps of this research in the future.

Appendix: Barbalat's Lemma

The Lyapunov function V) in Eq. ()) in Eq. (19) is negative semidefinite. Thus, based on the Lyapunov stability theorem [27], V remain bounded.

Barbalat's lemma: If g exists and has a finite value, it is guaranteed that [27]

   (A.1)

In order to use this lemma for the controlled system of HCV outbreak, g(t) :

   (A.2)

By integrating both sides of Eq. (A.2), one can write:

   (A.3)

is negative, V (0) is larger than V (∞) and V (0) − V (∞) ≥ 0. Moreover, as mentioned previously, V in Eq. (A.3) exists and has a bounded value. Therefore, it is concluded using the Barbalat's lemma that

   (A.4)

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Chapter 2: Integral Sliding Mode Control of Immune Response for Kidney Transplantation

Pouria Faridia,c; Ramin Vatankhaha; Mojtaba Sharifib,c    a Department of Mechanical Engineering, Shiraz University, Shiraz, Iran

b Department of Electrical and Computer Engineering, University of Alberta, Edmonton, AB, Canada

c Department of Medicine and Dentistry, University of Alberta, Edmonton, AB, Canada

Abstract

This chapter presents a sliding mode control strategy for immune response of renal transplant recipients. When the renal transplantation is inevitable, it is necessary to suppress the immune system for suitable kidney performance after this operation. However, this suppression will allow latent viruses to become active. The most prevalent virus, in this case, is known as human cytomegalovirus (HCMV), which is considered as the main concern. In this study, a new controller is designed for an immunosuppressive drug to assure that kidney is working well, while viral load is observed to be not a serious threat. Defining a sliding surface for the sixth-order state-space model of this transplant, the serum creatinine of blood is controlled as a measure of the kidney performance.

Keywords

Human cytomegalovirus; Renal transplantation; Serum creatinine; Sliding mode control; Viral load

1: Introduction

Acute kidney injury (AKI) is a prevalent secondary disease for patients who are in the treatment process inside hospitals. Even those who passed the recovery period of AKI are still at the risk of other complications, such as chronic kidney diseases (CKD) [1]. According to the annual report of the United States Renal Data System in 2018, people with estimated Glomerular Filtration Rates (eGFR) less than 60 mL/min/1.73 m² are at the risk of kidney diseases. CKD has five stages characterized by eGFR magnitude: stage 1 with eGFR > 90 mL/min/1.73 m² is the least dangerous stage, while people in stage 5 with eGFR < 15 mL/min/1.73 m² are suffering from the end-stage renal disease (ESRD) that is mortal in the absence of dialysis or renal transplantation [2, 3]. eGFR that shows kidney performance is calculated by the CKD-EPI creatinine equation, based on the serum creatinine concentration in blood (a breakdown product resulting from the activity of muscles), race, sex and age [4].

In developing countries such as India and Pakistan, which together are home to one-sixth of the world's human population, the annual incidence of ESRD is estimated at around 100 per million people. This means 100,000 and 15,000 patients are at the risk of ESRD in 1-billion and 150-million populations of India and Pakistan, respectively. When dialysis equipment is not satisfactory, kidney transplantation is the best way of treatment [5]. In developed countries such as the United States, the number of people with kidney failure (in the ESRD stage) is continually growing and has the highest increase rate in the world. Research studies have shown that 75% of children in the United States with ESRD got renal transplants, which implies the significance of this treatment [2]. In renal transplantation, there is a possibility of kidney rejection in the absence of standard health care. Pharmacological immuno-suppression is currently the most effective way to reduce the chance of this rejection. However, using this therapy, patients’ bodies will be exposed to several viral loads and bacterial pathogens [6].

Human herpesvirus five or Human Cytomegalovirus (HCMV) is the most common and significant pathogen among renal transplant recipients. Primary HCMV infection is usually without any specific symptom, but the uncontrolled case will lead to a life-long infection in patients’ bodies. Moreover, HCMV disease can be caused by the reactivation of HCMV latent infectious viral load [7].

In recent years, a wide range of investigations and experiments have been conducted regarding the HCMV infection, its dynamics, diagnosis, risk factors and international guidelines on its management [8–12]. Emery et al. [8] opposed the popular belief that Cytomegalovirus (CMV) replicates slowly, which had been an accepted theory due to time-consuming in vitro experiments for showing up the cytopathic effects. Taking three different cases into account, they proved that CMV in vivo replication has tremendous dynamics (variation). Serological tests, standard tube cell culture technique, antigenemia assay, polymerase chain reaction (PCR), immunohistochemistry, nucleic acid sequence-based amplification (NASBA) and hybrid capture assay are recognized methods of CMV detection that each has some merits and demerits [9]. Bataille et al. [11] investigated modern immunosuppression and its risk factors on 300 people having the same trial therapy, and the case D +/R − (CMV-seropositive donor/CMV-seronegative recipient) had the highest risk factor. Also, based on their observations, it was mentioned that a patient with impaired early kidney function becomes a candidate at risk.

Mathematical modeling or analysis is a vital tool for investigating the dynamic behaviour of biological systems and making decisions about treatment methods and their duration. There has been considerable effort in employing various useful control strategies for a variety of diseases and illnesses. Some control strategies have been developed for the hepatitis B virus (HBV). For instance, Sheikhan et al. [13] compared three strategies (nonlinear feedback neural-type sigmoid, open-loop time-based fuzzy and closed-loop fuzzy) to achieve the optimum performance of the HBV control. Laarabi et al. [14] proved the existence of an optimal control method by combining two control laws to reduce the therapy cost and maximize the number of healthy cells in HBV infection. In another work [15], a fuzzy logic structure was used to solve an HBV optimal control problem. Two other control inputs were designed to have efficient drug therapy considering both hindering viral production and inhibiting new infection of HBV based on a robust adaptive Lyapunov-based control theory [16]. Hepatitis C virus (HCV) has also attracted the attention of researchers in recent years. Zhang et al. [17] developed an epidemiological model of HCV and used numerical simulations to study the influence of the model parameters based on available data obtained from China. In another study [18], the performance of an optimal controller was investigated based on an HCV model having acute-infected and chronic-infected individuals as its compartments. After that, Zhang et al. [19] utilized an optimal control measure to inhibit the prevalence of HCV while minimizing the cost and population of infected individuals. A novel optimal adaptive neuro-fuzzy controller was developed to decrease the number of HCV infected individuals using an additional genetic algorithm optimization [20]. Cancer tumor modeling and control have also been studied in this field of research. In 2001, a four-population model containing tumor and host cells, drug therapy and immune response was presented [21]. Accordingly, Babaei et al. [22] proposed a model reference adaptive control method to determine a personalized drug administration to treat cancer with parametric uncertainty. Moradi et al. [23] also developed an adaptive robust control method for three nonlinear mathematical cell-kill models of cancer in the presence of parameter uncertainties. They have extended the previous study [23] by enhancing their strategy to a modern composite adaptive control in which the model parameters were precisely identified online during the cancer chemotherapy [24]. Moreover, Khalili et al. [25] suggested an optimal open-loop control strategy for drug delivery in chemotherapy considering the human obesity effects.

Moradi and Sharifi [26] also proposed a nonlinear robust adaptive sliding mode control method to reduce the number of susceptible and infected humans to zero in an influenza outbreak regarding a five-state compartmental model of this disease. The employed model of influenza was developed by Arino et al. [27] with five state variables known as SEAIR (Susceptible-Exposed-Asymptomatic-Infectious-Recovered). That model was enhanced in Ref. [28] by defining the vaccination, social distancing and antiviral rates as three possible control inputs. In this modern era, HIV prevention and treatment is also of pivotal importance. Ngina et al. [29] presented an in vivo deterministic model of HIV and presented an optimal control scheme based on that. In a new study [30], a robust sliding mode controller was formulated to reduce the population of infected CD4+ T cells with antiviral therapy according to the acquired output information. In 2000, a differential SEIR model of malaria, containing both humans’ and mosquitos’ populations and their interaction was taken into account [31]. Another epidemiological model of malaria was formulated to consider personal protection, possible treatment and vaccination strategies in two latent periods [32]. After that, Rafikov et al. [33] employed an optimal control method for a mathematical malaria model by placing genetically modified mosquitos in the environment. Furthermore, a robust nonlinear controller with adaptive gains was proposed to inhibit the prevalence of malaria with seven variables for human and mesquite compartments [34].

To study the immune response of renal transplant recipients, there has been made considerable effort to predict, model and optimally control the HCMV infection in both primary and latent cases. Flechet et al. [35] analyzed AKI-predictor as an online machine learning-based prediction tool and compared it to physicians’ ability to predict AKI in clinical uses. Based on the obtained results, it was emphasized that AKI-predictor was beneficial in terms of successfully removing false HCMV positives (error in determining a patient at high risk) and reducing clinical costs. In addition, Parreco et al. [36] compared different machine learning algorithms for predicting AKI. Wodarz et al. [37] presented a model based on mice infected with murine cytomegalovirus (MCVM), which introduced important knowledge for explaining the growth of CMV specific CD8+ T cells in human by getting older. Kepler et al. [38] formulated a fifth-order state-space model for HCMV infection that was extended by Banks et al. [39] to six-order one, considering serum creatinine of the blood as a measure of the kidney performance. Their model contained antiviral and immunosuppressive drugs as two control inputs. Then, Kwon et al. [40] simplified the model in [39], considering immunosuppressive drug as the only control input, proved the local stability of the dynamics and determined both uninfected and infected steady-states. They utilized a model predictive control (MPC) method as an optimal strategy to achieve a balance between over-suppression and