Linear Quadratic Control Problem in Biomedical Engineering

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1 European Symposium on Computer Arded Aided Process Engineering 15 L. Puigjaner and A. Espuña (Editors) 5 Elsevier Science B.V. All rights reserved. Linear Quadratic Control Problem in Biomedical Engineering Irma Y. Sánchez Chávez a, Rubén Morales-Menéndez b, Sergio O. Martínez Chapa c a Mechatronics and Automation Department, b Center for Industrial Automation and c Electrical Engineering Department, ITESM Campus Monterrey Eugenio Garza Sada 51 Sur, 64,849 Monterrey NL, México Abstract Optimal control allows the incorporation of functional constraints and requirements as a departure point for the design process. The glucose continuous control problem in a diabetic patient is addressed with the linear quadratic optimal control design principles. A glucose control system is proposed. The glucose-insulin dynamics is represented by the Ackerman's model. This model considers the glucose level as the single output, so a state space observer is used to estimate the blood insulin level. The cost function is defined in quadratic terms of the exceeding glucose level and the amount of supplied insulin. An optimal control law is generated under servocontrol and regulatory approaches. Robustness of the control law in each case is examined by a Monte Carlo simulation. Results demonstrate the suitability of the optimization and regulatory approaches in biomedical engineering problems. Keywords: glucose optimal control 1. Introduction A control system for continuous drug delivery can reach or keep a desired physiological condition in a sick person. Diabetes mellitus is a chronic disease that affects the ability of the body to produce insulin or to be sensitive to insulin. The insulin allows the absorption of glucose in the cells to be used as energy. There are two major types of diabetes mellitus: juvenile-onset diabetes (type I) and maturity-onset diabetes (type II). In type I diabetes the cells of the pancreas that produce insulin are destroyed, which suggests a treatment based on introduction of insulin (Seeley et al. 1995). Insulin infusion can be done subcutaneously and intravenously. The subcutaneous route is easier and safer to manage, which is an advantage for closed systems implementation since it is the route used in traditional open loop diabetes treatment. The intravenous route avoids time delays to reach blood stream and to produce body response, which is convenient for continuous closed loop performance. Both insulin delivery types have been considered in closed loop treatment systems (Bellazi et al. 1, Parker et al. 1). Lispro insulin may combine the advantages of both routes because of its fast absorption after subcutaneous injection. Author to whom correspondence should be addressed: isanchez@itesm.mx

2 An open loop treatment system may be seen as a partially closed loop system because medical prescription of insulin is based on home glucose monitoring among other information of the patient (Bellazi et al. 1). Decision support systems have been designed for diabetes management for this kind of therapy systems. PID type controllers have been used for blood glucose control. A PID based on a sliding scale approach, tested in patients in intensive care unit, is reported (Chee et al. 3). A PD controller has been derived with a pole assignment strategy and tested in patients (Bellazi et al. 1). Robustness of closed loop systems has been achieved by adaptive and predictive mechanisms to account for sparse glucose measurement (Woodruff et al. 1988) and time variations of the glucose-insulin process. The model predictive control algorithm implements a self-tuning controller that has been studied by simulation (Parker et al. 1999) and tested in vivo (Dudde and Vering 3). A quadratic performance criterion is usually considered in glucose advanced control algorithms in terms of glucose deviation and amount of exogenous insulin. In this work, a Linear Quadratic Control (LQC) problem is stated for designing an optimal controller in such terms. The formulation approach of this criterion may lead to tracking problems (Parker et al. 1). Servocontrol and regulatory approaches are discussed. The paper is organized as follows. Section presents the LQC problem and the Ackerman s glucose-insulin model. In section 3, the blood glucose level optimal controller is designed. Section 4 presents simulation results. Section 5 discusses the results. Finally, in section 6, the paper is concluded.. Fundamentals.1 Linear quadratic control problem The LQC problem consists of determining a control law u( to minimize the cost function given by equation (1): 1 J u) = e T T [ e ( Q( e( u ( R( u( ] 1 t f ) + + t T ( ( t f ) Se( t f dt (1) where S is a constant matrix; Q and R may vary with time; Q, R and S are symmetric matrices; S and Q are positive semidefinite, and R is positive definite. The control law u( is the input of the state-space model x ( t ) = A( t )x( t ) + B( t )u( t ), x ( ) = x defined in [t, t f ] where A( is the state transition matrix and B( is the input matrix. An optimal control law u*( is assumed to exist for this problem in [t, t f ]. An optimal trajectory x*( is associated with u*(. The control signal and the state vector are u( = u*( + ευ( and x( = x*(+ ε y( where ε is a small positive number and υ( is arbitrary. The optimal control law is obtained when ε =, which makes dj(ε)/dε =. The control law is specified as u(=-k c x(+k, Kc=R -1 B T P, K=-R -1 B T μ (Naidu 3, Vinter ) where P is a symmetric positive semidefinite matrix and μ is a column vector, both used to define a costate vector λ(=px(+μ.

3 . Ackerman glucose-insulin model Research on glucose control in diabetic patients depends on the development of accurate models. Models based on mass balances between different organs or compartments have been combined with models for gastric absorption of food and subcutaneous absorption of insulin to form a high order non-linear model. However, a parsimony principle leads to the management of simpler models in order to obtain a closed solution to the optimal control problem (Parker et al. 1999). The Ackerman's model is widely known because of its simplicity, since it considers one compartment that represents the global glucose-insulin dynamics in the human body (Ackerman et al. 1965). This model is based on the glucose tolerance test where the basal level is disturbed by the intake of glucose. The Ackerman model can be used as a single input single output (SISO) system for the discussion of control issues, which can be solved before using broader models. The non-linear interaction between glucose and insulin is described by G ( t ) f ( G,H ) + p( t ) and H ( t ) f (G,H ) + u( t ) with G(t=)=G, H(t=)=H, = 1 = p(t=)= and u(t=)=, where G( is the glucose level, H( is the hormone level, p( is the external glucose supply rate and u( is the insulin infusion rate at time t. Considering the deviation variables g(=g(-g and h(=h(-h, and applying a standard linearization procedure, the linear representation of model is g (t ) = m1g( t ) mh( t ) + p(t ), and h ( t ) = m4g( t ) m3h( t ) + u( t ). Parameters m 1, m, m 3, and m 4 have been obtained from experimental data (Yipintsol et al. 1973). In the case of a type I diabetic patient, m 4 =. The term p ( t ) is not considered. The final state-space model is given by x ( t ) = Ax( t ) + Bu( t ), x()=, y(=cx(+du( where x(=[x 1 ( x (] T =[g( h(] T, A=[-m 1 -m ; m 3 ], B=[ 1] T, C=[1 ] and D=. 3. Optimal controller Under the servocontrol approach, let x d be the desired glucose change in a diabetic person (x d = G d G ). Deviation of blood glucose level from its desired value (x 1 x d = G G d ) and insulin infusion must be minimized. The objective function is: [( x x ) + ρu ] = J( u ) 1 d dt () where ρ is a positive weighting factor. By comparing equations (1) and (), it can be ; Q = ;, R = ρ, t and t =. identified that S = [ ], [ ] The optimal control law is u(=-k c x(+k, where K c =[K 1 K ] and K=m x d /(ρξ). The following Ackerman model parameters in min -1 are used: m 1 =.9, m =.31 and m 3 =.415 (Yipintsol et al. 1973). With ρ =1 min and x d =- mg/dl (according to the simulation problem presented in section 4), the parameters of the control law are ξ=-9.811x 1-4 min -, K 1 =-.991 min -1 and K =.183 min -1. For the regulatory approach, a normal glucose level is referred as the initial steady state. Any deviation from this value is a disturbance. Therefore, x 1 is the deviation of glucose level from the desired value. From the performance criteria represented by equation (3), the control law is u(=-k c x( with K c =[ ]. = f

4 [ x + ρu ] = J ( u ) 1 dt (3) The control law in both approaches is a function of the two states of the system, glucose (x 1 () and insulin (x () concentrations. The glucose concentration measurement is supposed to be available while the insulin concentration needs to be estimated. The design of the control law and the design of the state observer are independent. The state observer is designed according to classical control theory. The Ackerman's model is an observable system. The observer model is given by x~ ( t ) = Ax ~ ( t ) + Bu( t ) + K [ y() t Cx ~ e () t ] where y () t Cx ~ () t is the observation error. The gain vector K e =[ ] T produces a faster observer response with respect to the closed loop behaviour. 4. Simulation The simulation problem consists of a situation of hyperglycaemia in a type I diabetic patient with an initial glucose level of 3 mg/dl. The closed loop system should reach a desired steady state level of 1 mg/dl. The sensor and actuator are assumed to be ideal systems and the open loop dynamics is supposed to be dominated by the model of the patient or physiological process to be controlled. Under a servocontrol approach, the initial deviation value is zero assuming an initial stable state in hyperglycaemic condition (3 mg/dl). The glucose change to be achieved is - mg/dl. Simulation results are shown in figure 1. The transitory elapses 4 hours (left-top graph); a steady state error of.3 mg/dl is detected (right-top graph); estimation of insulin concentration is reliable (left-bottom graph), and the cost function grows indefinitely (right-bottom graph). For the regulatory approach, the reference steady state is the normal condition of glucose level at 1 mg/dl, so the initial deviation of mg/dl is considered a disturbance to the closed loop system. No offset error is obtained, and the cost function converges. 5. Discussion The stated simulation problem has been presented in the literature (Kikuchi et al. 1978). Glucose levels at 1 and hours with the servocontrol approach are very similar to reported values. Glucose levels of and mg/dl are obtained while previous works report 183 and 119 mg/dl at these times respectively. For the regulatory approach, glucose concentrations of and mg/dl are reached at 6 and 1 min. These results differ more from the reported results, however the performance is more satisfactory. The situation simulated for performance comparison between the servocontrol and regulatory control approaches consists of an initial hyperglycaemic condition, instead of a meal disturbance managed in recent works (Parker et al., Kienitz and Yoneyama 1993) as enough proof for the adequacy of the regulatory formulation.

5 mg/dl Real glucose level Desired glucose level Insulin supply mg/dl Real glucose level Desired glucose level x 16 Observation error, mg/dl Cost function, J Figure 1. Servocontrol approach results A linearized and low order glucose-insulin model may cause uncertainty as non-linear high order models do. Although a more complete model may be suitable to manage characterized uncertainty (Parker et al. ), all effects may not be represented, which justifies a random variation of model parameters to analyze controller robustness. Monte Carlo simulation proofs the robustness of the control system. To illustrate the variability, figure presents box and whisker plots for different percentages of variation. These graphs were computed by 3 independent runs. Beyond 6% parameter variation, the servocontrol system shows unacceptable performance. The regulatory design performs with no significant difference with more than 5% variation. 6. Conclusions Blood glucose control in a diabetic patient is an example of a biomedical problem where optimal control theory (LQC) can be applied. The servocontrol system gives results similar to those reported for the same simulation problem. A small offset is observed in the final steady state in this work. The regulatory control approach is more appropriate for this biomedical problem because no specification of a desired change is required; instead, any blood glucose deviation from its normal value is managed as a disturbance to be solved to recover the normal state. The regulatory system is more robust to variations in plant dynamics since the offset does not appear. Given the natural time variation of physiological processes, predictive or adaptive control laws can have superior performance (Parker et al. 1999, Morales-Menéndez et al. 4). Closed loop diabetes treatments tend to be bloodless, painless and more precise than conventional treatments consisting of insulin injections with certain frequency and doses. Implementation of proposed closed loop treatments requires microsystems with reliable continuous sensors and actuators and embedded control algorithms.

6 Cost function Cost function Offset % Ackerman s model variation Offset % Ackerman s model variation Figure. Robustness tests. Left-graphs for servocontrol, right-graphs for regulatory approach References Ackerman, E., Gatewood, L., Rosevear, J. & Molnar G. 1965, Model Studies of Blood-Glucose Regulation, Bull. Mathem. Biophys., vol. 7(suppl). Bellazi, R., Nucci, G. & Cobelli, C. 1, The Subcutaneous Route to Insulin-Dependent Diabetes Therapy, IEEE Eng. in Medicine and Biology, vol., no. 1, pp Chee, F., Fernando, T. & Van Heeden, P. 3, Expert PID Control System for Blood Glucose Control in Critically Ill Patients, IEEE Trans. Info. Tech. in Biomed., vol. 7, no. 4, pp Dudde, R. & Vering, T. 3, Advanced Insulin Infusion using a Control Loop (ADICOL) Concept and Realization of a Control-Loop Application for the Automated Delivery of Insulin, 4 th International IEEE EMBS Special Topic Conf. on Info. Tech. App. in Biomed., pp Kienitz, K. & Yoneyama, T. 1993, A Robust Controller for Insulin Pumps Based on H Theory, IEEE Trans. on Biomedical Eng., vol. 4, no. 11, pp Kikuchi, M., Machiyama, N., Kabei, N., Yamada, A. & Sakurai, Y. 1978, Homeostat to Control Blood Glucose Level, Int. Symp. Med. Inf. Syst., pp Morales-Menéndez, R., Mutch, J., de Freitas, N., Poole, D. & Guedea-Elizalde, F. 4, Dynamic Modelling and Control of Industrial Processes with Particle Filtering Algorithms, Barbosa-Póvoa and Matos H editors ESCAPE-14, Lisbon Portugal, pp Naidu, D. 3, Optimal Control Systems, CRC Press, Boca Raton, Fl. Parker, R., Doyle III, F. & Peppas, N. 1999, Model-based Algorithm for Blood Glucose Control in Type I Diabetic Patients, IEEE Transactions on Biomedical Eng., vol. 46, no., pp Parker, R., Doyle III, F. & Peppas, N. 1, The Intravenous Route to Blood Glucose Control, IEEE Eng. in Medicine and Biology, pp Parker, R., Doyle III, F., Ward, J. & Peppas, N., Robust H Glucose Control in Diabetes Using a Physiological Model, AIChE Journal, vol. 46, no. 1, pp Seeley, R., Stephens, T. & Tate, P. 1995, Anatomy & Physiology, St. Louis. Vinter, R., Optimal control, Birkhöuser, Boston, Ma. Woodruff, E., Gulaya, S. & Northrop, R.1988, The Closed-Loop Regulation of Blood Glucose in Diabetes, Proc. of the 14 th Annual Northeast Bioengineering Conference, pp Yipintsol, T., Gatewood, L., Ackerman, E., Spivak, P., Molnar, G., Rosevear, J. & Service, F. 1973, Mathematical Analysis of Blood Glucose and Plasma Insulin Responses to Insulin Infusion in Healthy and Diabetic Subjects, Comput. Biol. Med., vol 3, pp Acknowledgements The authors thank Dr. Graciano Dieck Assad and Prof. Óscar Miranda Domínguez for their valuable comments and the Consejo de Ciencia y Tecnología del Estado de Nuevo León for financial support.