Simulation Study on Closed Loop Control Algorithm of Type 1 Diabetes Mellitus Patients

Transcription

1 Simulation Study on Closed Loop Control Algorithm of Type 1 Diabetes Mellitus Patients Surekha Kamath, V. I. George and Sudha Vidyasagar 1 Department of Instrumentation and Control Engineering, MIT Manipal, 1 Department of Medicine, K.M.C Manipal, Manipal, India ABSTRACT It is challenging to maintain normoglycemic range of glucose concentration in type I diabetic patients. In this study H` control is applied for insulin delivery to prevent the hyperglycemic levels in a type I diabetic patient. From a control theory point of view, the blood glucose regulation problem is reformulated as a tracking one. A glucose tolerance curve (GTC) validated from several patients is used as reference model. Intra- and inter-patient variability poses a challenging task to control blood glucose concentration in diabetic patients. We develop a data based robust controller to control blood glucose concentration in type I diabetic (TIDM) patients in the presence of meal disturbances under patient model mismatch. Simulation studies are performed on the diabetic patient model under feedback control. It is seen that the proposed control strategy is able to control blood glucose concentration well within the acceptable limits and also compensate for slow parametric drifts. Keywords: Glucose insulin modeling, H control, Insulin dependent diabetic mellitus, Insulin modeling, Mathematical modeling, Robust control. 1. INTRODUCTION MANY control algorithms have been developed to deliver insulin in IDDM type I (Insulin dependent diabetic mellitus) [1,]. (Ollerton, 1989; Parker et al., 1999a) Although the control algorithms show good performanceit is very difficult to use a practical insulin pump because of their own complexity and the needs of computation devices. A robust controller is inherently simple algorithm and composed of more simple devices than the other control algorithms (MPC, EMPC, etc). A physiologic model of diabetes patients was developed with the mathematical analysis of insulinglucose interactions (Bolie, 1961; Ackerman et al., 1965). It has 19 states of differential equations composed of - 11 states for glucose dynamics, seven for insulin dynamics, and one for glucagons dynamics. This model is a nonlinear system. Although it is well established, practical patients have various uncertainties. Uncertainty causes differences between an actual patient and diabetic patient model. The parameter sensitivity analysis found the most sensitive metabolic parameters affecting glucose and insulin dynamics (Parker et al.,1998). Control and Complications Trial (DCCT) [3] showed that an intensive insulin therapy can reduce the incidence of long term illnesses. Therefore, an intensive therapy is encouraged for TIDM (type 1 diabetes mellitus) patients prescribed either by a continuousinfusion pump (CIP), or a multiple daily injection regimen (MDIR). On the other hand, it was also noticed in (DCCT, 199), that a possible side effect of an intensive therapy is the propensity to hypoglycemic scenarios in the patient. With this consideration, if the patient follows an intensive therapy, the prescribed insulin treatment must be carefully studied by the physician and should be constantly updated according to results achieved. This work deals with the design of controller for the delivery of insulin to the TIDM patients. In this contribution, the control problem is reformulated by considering the rate of the blood glucose level. That is, the control problem is reformulated as a tracking problem. The aim is to make blood glucose level of a TIDM patient track, under closed- loop configuration, the path of the glucose level for a healthy person. Glucose tolerance curves (GTC) were experimentally obtained from healthy subjects to validate a reference model. The controller is designed from H control theory. There are four major sites for insulin delivery subcutaneous, intramuscular, intravenous and intraperitoneal. While the subcutaneous site is the simplest and safest in the long term, the absorption of insulin from the subcutaneous tissue is delayed. The 30 IETE JOURNAL OF RESEARCH Vol 55 ISSUE 5 SEP-OCT 009

2 intramuscular site is usually preferred for people affected by brittle diabetes who have a subcutaneous barrier to insulin absorption but may result in muscle fibrosis and disconnection of cannula. The issues being addressed are invasiveness and error due to other substances. Biocompatibility can be avoided by using tissue sampling techniques such as micro dialysis and reverse ionophoresis; expected problems under investigation include prolonged time for sample collection, quality of sample collected and skin infection. Another way of dealing with the issue of biocompatibility is the development of noninvasive sensors. These are based on the principles of near infrared spectroscopy. The closed loop device consists of three components: 1) glucose sensor, ) control algorithm, and 3) mechanical pump [Figure 1]. In this system, the glucose concentration is measured by the glucose sensor and based on the measurement, control algorithm. It computes the optimal insulin delivery rate. The mechanical pump then infuses the computed amount of insulin.. BLOOD GLUCOSE CONTROL PROBLEM Several models have been proposed to reproduce the glucose-insulin dynamics,since the 1980s paper paper by Bergman et al. [4]. The proposed models include: (i) glucagon effects and threshold functions representing metabolic processes (Lehman and Deutsch) [5] (ii) nonlinear terms for pharmacokineticpharmacodynamic effects (Nalecz M et al.) [6] and (iii) physiologic equations toward compartmental representations (Parker RS. []) A physiology-based compartmental model has the advantage that the simulations can yield insight into the physiological parameters (Campos-Delgado DU et al.) [7] A physiology-based compartmental model is used to design and test the tracking control. The model includes equations for the main organs of the glucoregulatory system and involves glucose uptake as well as effects of glucose on the hepatic glucose production, EGHGP, hepatic glucose uptake, EGHGU, effects of insulin on peripheral glucose uptake, EIPGU and finally glucagon effect on hepatic glucose production, EGHGP. The glucose insulin system is governed by nonlinear ODEs, which includes 19 equations and has three dynamical subsystems: i) glucose, ii) insulin and iii) glucagon. Mathematical modeling of physical systems is not an easy job because of the components involved in the system. Physical behavior may also change due to transitions between two metabolic states. Physiological control of a nondiabetic human being is taken care of by the pancreas that functions normally. For a diabetic patient, glucose is controlled through insulin delivery which involves feedback methods..1 Glucose: Insulin Modeling Several methods in literature model the glucose-insulin dynamics. The most primitive one was reported by Bergman et al. [4]. It is a low order system. A few slightly more complex models had been reported, which include glucagon effects and threshold functions that represent metabolic processes Lehman and Deutsch [5], Sorensen [1]. Sorensen [1] departed from experimental results to formulate and validate metabolic processes of the model on the whole organ and tissue level. It was concluded that the model is nonlinear. Each compartment is divided into two subcompartments where mass balances were derived. Sorensen [1] departs from experimental evidence to formulate and validate metabolic processes of the compartmental model on the whole organ and tissue level. In this sense, the glucose- insulin model is nonlinear and has the following subsystems: Glucose, insulin and glucagon. This nonlinear model was used for control purposes by Pacini and Cobelli [8]. This process is based on the compartmental technique.. Patient Model Uncertainty The diabetic model used for patient simulations in this work is taken from Parker et al. (1999). This Figure 1: Closed loop glucose control system. IETE JOURNAL OF RESEARCH Vol 55 ISSUE 5 SEP-OCT

3 pharmacokinetic-pharmacodynamic compartmental model of the human glucose-insulin system was initially developed by Guyton et al. (1978) and Sorensen [1] and then modified by Parker et al. (1999) to include meal and exercise disturbances. This model has 19 state equations and 47 physiological parameters. Utilizing compartmental modeling techniques, the diabetic patient model is represented schematically in Figure 1. In this model the human body is divided into six compartments (brain, heart/lungs, gut, liver, kidney, and periphery). Individual compartment models are obtained by performing mass balance around tissues important to glucose or insulin metabolism. Subcompartments (namely, capillary and tissue), such as those in the brain and periphery, were included in case of significant transport resistance (e.g., time delay). The periphery represents the combined effects of muscle and adipose tissue while stomach and intestine effects are lumped into the gut compartment. This model was constructed to represent a sedentary 70-kg male diabetic patient. Controlled output for this system is the arterial glucose concentration regulated by the manipulated variable and insulin infusion rate. A disturbance variable, glucose uptake from the gut compartment is added to the model to simulate the diabetic patient ingesting a meal. The mathematical representation of the meal submodel is described in Lehmann and Deutsch [9]. There are some uncertainties due to the inevitable patient-model mismatch. These uncertainties between the actual and normal patient model could be translated to variations in the model parameters which represent glucose or insulin metabolism. Glucose and insulin dynamics were found to be most sensitive to variations in metabolic parameters of the liver and the periphery. In the patient model, glucose metabolism is mathematically described by threshold functions with the following structure: { ( ) } Γe = EΓ AΓ B e e Γ C e Γ x e i + DΓe tan h (1) The subscript is the state vector element involved in the metabolic effect and the subscript e denotes specific effects within the model: Effects of glucose on hepatic glucose production (EGHGP), glucose on hepatic glucose uptake (EGHGU) and insulin on peripheral glucose uptake (EIPGU). Inter- or intrapatient uncertainty were classified physiologically as either a receptor (D t ) or a post receptor (E t ) defect; these two parameters were estimated to fit the actual patient data. Differences in insulin clearance (metabolism) between patients also exist and were modeled as deviations in the fraction of clearance (i.e. insulin utilized) by a given compartment, such as the fraction of hepatic clearance (F LC ) or fraction of peripheral insulin clearance (F PC ). This uncertainty formulation essentially focuses on the liver (variability in five parameters) and peripheral (muscle/fat) tissues (variability in three parameters) as they are considered to be more relevant to the control study..3 Validation of Reference Model Blood glucose level is used like a reference; the transfer function P ref is validated from the glucose tolerance curves (GTC s) of six healthy subjects. First, the GTCs are obtained by the classical method at t equal to zero, blood glucose level is measured and the subjects then drink a solution of 75 g of dextrose dissolved in 300 ml of water (glucose load). Later, blood samples for blood glucose concentration determinations are obtained using a Dt = 30 min sample interval, at least during 150 min. The Table 1 shows the specifications of six healthy subjects..4 Glucose Curve Fitting Approach for Healthy Subjects Sample interval: Dt 5 30 min, length min. The overall response of all the six subjects is as shown in Figure. The GTCs (Glucose Tolerance Curve) permit to see that the BG response to a meal in a healthy subject behaves like a second-order system. Therefore the transfer function for reference model is validated from mean of six healthy subjects data. The P ref has the following representation Ruiz-Velaquez et al. [10] P ref = Kωn s + ξω s + ω n n K , j5 0.7, v n ; therefore the impulse response of P ref resembles the curve shown in Figure RESULTS AND DISCUSSION 3.1 H` Glucose Control in Type 1 Diabetes Mellitus The order of the physiological model for the 19 th TIDM patient generalized plant G(s) is of the (19 1 m) order (where m is the sum of the order of weighted transfer functions). Therefore, the controller synthesis is structured by reduced-order controller methodology. Here, the order reduction is implemented on the plant to obtain a reduced order model. () 3 IETE JOURNAL OF RESEARCH Vol 55 ISSUE 5 SEP-OCT 009

4 Figure 4 shows the standard feedback configuration for TIDM patient with weights. Weights involved in the block diagram correspond to the input weight, performance weight, meal disturbance weight and weight due to noise disturbance. The reduced-order plant is used to synthesize the controller. The generalized plant for blood glucose control is given by [11] WPPm Wm 0 WpP d1 G( s) = 0 0 Wu d W P W P d m m n 3 Table 1: Specifications of healthy subjects Age Weight (3) Where the (i) P is transfer function of reduced model for TIDM patient, P m is the transfer function of the meal model. (ii) W p corresponds to performance weight and is a first order transfer function and was chosen such that the frequency content of P m was captured (output disturbance attenuation). Moreover, W p was selected taking into consideration the characteristic frequency of P ref (for tracking). (iii) Weight W m represents the effect of meal model, and permits to induce the maximum carbohydrate content into the meal. (iv) The sensor noise effects are weighted by W n which emulates any possible error generated by the glucose and W u stands for the weight for the control input. Figure 5 shows the simulation of the BG response of a TIDM patient under a meal, at t is equal to zero and 370 minutes. The controller used for this simulation is the resulted by H` approach. The meal contains 100 g of glucose. The maximum difference between the BG reference and the glucose level of the diabetic patient is 5.6 mg/dl. Figure 3: Output of reference model. Figure : Responses of six subjects. Figure 4: Standard feedback configuration with weights. Figure 5: BG response of type 1 diabetes mellitus patient under feedback control. IETE JOURNAL OF RESEARCH Vol 55 ISSUE 5 SEP-OCT

5 Figure 6: Block diagram for BG control including parametric uncertainties. Here, as it was mentioned above, the insulin delivered to patient i(t) (mu/min) is given by I(t) 5 u(t) 1 (4) where u(t) is the insulin portion calculated by the controller. 3. Robust Control Design in Type 1 Diabetes Mellitus The block diagram [Figure 6] contains the modified block feedback scheme when parameter variations are included into the model; which are incorporated in the form of weighted transfer functions. The generalized plant, G(s), has the following representation Ruiz,Velaquez et al. [10] z1 d1 z d z3 = G( s) d3 = z4 d4 y u WmGmWp 0 WpP GmWP WpP d W u d Wi d3 WmWim d4 WmGm Wn P Gm P u Where, weighted transfer functions are as described in the above equation of generalized plant for TIDM patient. W i and W im represent the weighting functions corresponding to multiplicative uncertainty for the input weight and for the input disturbance (meal). The order of both these weights is second order. Blood glucose response including the above mentioned uncertainties is as shown in the Figure 7. (5) Figure 7: BG response by adding parametric uncertainties. Simulation represents the worst case on the parametric variations. Since minimum BG level is higher than 70 mg/ dl, hypoglycaemia effects cannot be presented in the closed-loop. 4. CONCLUSION This paper discusses diabetes management as one of the challenging control problems in human regulatory systems. Treatment of the disease via a robust feedback control design has been considered and two types of algorithms applied for feedback control. A design stabilizes the blood glucose concentration of a diabetic patient at the desired level in the presence of external disturbances such as food intake and model parametric uncertainties which affect high accuracy and robustness of the entire system. A high order nonlinear model represents the dynamics of glucose-insulin in TIDM patients. The proposed linear controller showed acceptable performance when interconnected to the model. It was also shown that a reduced order controller can be designed when parametric variations are considered on the nonlinear model. The robust high accuracy performance of the controller is checked and confirmed by computer simulations. 5. Nomenclature G e Threshold function. AΓ, B C e Γ, e Γ Constants whose values are given by threshold e functions. DΓ e Receptor defect. EΓ e Post receptor defect. x i Component of the state vector. EIPGU Effect of insulin on peripheral glucose uptake. EGHGU Effect of glucose on hepatic glucose uptake. EGHGP Effect of glucose on hepatic glucose production. 34 IETE JOURNAL OF RESEARCH Vol 55 ISSUE 5 SEP-OCT 009

6 FHIC FPIC I P T G L C P P p P ref W p W u W n W m G m y u REFERENCES Fraction of hepatic insulin clearance. Fraction of peripheral insulin clearance. Insulin value of peripheral with respect to tissue space. Glucose value of liver with respect to capillary space. Nominal plant. Perturbed plant. Transfer function of the reference model. Performance weight. Weight for the control input. Weight for the sensor noise effects. Weight for the meal model. Meal model. Vector of measurements available to the controller. Vector of control signals. 1. J T Sorensen, A physiologic model of glucose metabolism in man and its use to design and access improved insulin therapies for diabetes, Ph.D. Thesis, dept chem. Eng., Massachusetts Inst. Tecnol. (MIT). Cambridge, R S Parker, F J Doyle 3 rd, and N A Peppas, Robust H` glucose control in diabetes using a physiological model, AICHE J 46(1), , DCCT- The Diabetes Control and Complications Trial Research Group. The effect of intensive treatment of diabetes on the development and progression of long-term complications in insulin-dependent diabetes mellitus. N Engl F Med 39:977-86, R Bergman, L Phillips, and C Cabal, Physiologic evaluation of factors controlling glucose tolerance in man, J Clin Investig 68, , E D Lehmann, and T Deutsch, Compartmental models for glycaemic prediction and decision-support in clinical diabetes care: Promise and reality. Comput Met Progr Biomed 56, , M Nalecz, J Wojeieki, and I Zawicki, Control in artificial pancreas, IFAC control Aspects biomed. Engg. pp , D U Campos-Delgado, R Femat, E Ruiz-Velaquez, et al, Knowledge-based controllers for blood glucose regulation in type I diabetic patients by subcutaneous route. In: Proceedings of the International Symposium on Intelligent control, Houston., 3-5 Oct G Pacini, and C Cobeli, Estimation of Beta-cell secretion and insulin hepatic extraction by the minimal modeling technique, Comput Met Progr Biomed 3, 41-8, E D Lehmann, and T Deutsch, A physiological model of glucose-insulin interaction in type I diabetes mellitus, J Biomed Engg 14,35-4, E Ruiz-Velaquez, R Femat, and D U Campos-Delgado, Blood glucose contreol for type I diabetes mellitus: A robust tracking H`, Control Engg Pract 1 pp , E Ruiz-Velaquez, R Femat, and D U Campos- Delgado, A robust approach to control blood glucose level: Diabetes Mellitus Type I, Proceedings of the fourth IFAC symposium on robust control design (IFAC- ROCOND)5-7 Jun, Milan, Italy, R Bellazi, G Nucci, and G Cobelli, The subcutaneous route to insulin-dependent diabetes therapy, IEEE Eng Med Biol, 0:54-64, B W Bode, Medical management of Type I diabetes, 4 th edn, American diabetes Association, Alexandria, Virginia, E R Carson, and T Deutsch, A spectrum of approaches for controlling diabetes, IEEE Control Syst Mag 1(6): 5-31, J C Doyle, K Glover, P P Khargonekar, and B A Fracis, State space solutions to standard H and H control problems, IEEE Trans. On Automatic Control 34, , F C Erzen, G Birol, and A Cinar, Simulation studies on the dynamics of diabetes mellitus, Proc IEEE Int Symp Bio-Informatics Biomed Eng 3-35, F P Kennedy, Recent developments in insulin delivery techniques: Current status and future potential, Drugs, Vol. 4, pp , M Halim, E R Carson, S Andreassen, O Hejelsen, and R Hovorka, The role of a diabetic advisory system (dias) in the management of insulin-dependent diabetes mellitus, Proc Int Conf IEEE Eng In Medicine and Biology Soc , H E Lebovitz, Therapy for diabetes mellitus and related disorders, 3 rd edn. American Diabetes Association Alexandria, Virginia, P J Lenart, and R S Parker, Modeling exercise effects in type I diabet ic patients, Proc. 15 th Triennial World Congress IFAC,Barcelona,Spain 1-6 July S Skogestad, and I Postlethwaite, Multivarible Feedback Control, Wiley, New York R S Parker, G L Bowlin, and G Wnek, Eds. Insulin Delivery Encyclopedia of Biomaterials and Biomedical Engineering. New York: Markel Dekker, pp , AUTHORS Surekha Kamath received her B.E. in Electrical and Electronics from Mysore University in 1993 and M. Tech in 003 from MIT Manipal. She is working as a lecturer in ICE Department of MIT Manipal since 004. Her research interests include biological control system, biological signal processing etc. She has registered for Ph.D. under MAHE in surekakamathk4@yahoo.com V. I. George received graduate degree in Electrical Engineering from university of Mysore in M. Tech. degree in Instrumentation and Control engineering from NIT Calicut in Received Ph.D. from Bharathidasan university, Tiruchirappall in 004. He is currently Prof. and Head, in the department of Instrumentation and Control engineering at MIT Manipal. His research interest are Instrumentation and control systems, Multivariable robust control, Optimization. vig_rect@yahoo.com Sudha Vidyasagar received her undergraduate training at Stanley Medical College at Chennai, completing it in Dec She further did a diploma course in pediatrics in the same college, then proceeded to do a degree in internal medicine from Kasturba Medical College, Manipal. She was awarded the best outgoing student medal in M.D. general medicine in Dec She has joined KMC Manipal in July 1986, and have been teaching in the academic institution since then. Presently she is working as a Professor in Medicine Department of KMC Manipal. svsagar33@yahoo.com DOI: / ; Paper No JR 75_08; Copyright 009 by the IETE IETE JOURNAL OF RESEARCH Vol 55 ISSUE 5 SEP-OCT