Simulation Study on Type I Diabetic Patient

Transcription

1 Simulation Study on Type I Diabetic Patient Surekha Kamath, V. I. George and Sudha Vidyasagar 1 Department of Instrumentation and Control Engg., Manipal Institute of Technology, Manipal , 1 Department of Medicine, K.M.C Manipal, Manipal, India ABSTRACT Maintaining glucose concentration in normoglycemic range in Type I diabetic patients is challenging. In this study H` control is applied for insulin delivery to prevent 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 a reference model. Intra-and inter-patient variability poses a challenging task to control blood glucose concentration in diabetic patients. A data based robust controller is developed to control blood glucose concentration in type I diabetic patients in the presence of meal disturbances under patient-model mismatch. Simulation studies are performed on the diabetic patient model under feedback control which revealed 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). Although those control algorithms show good performance, they find it difficult to use a practical insulin pump because of their own complexity and need of computation devices. Robust controller is inherently simple algorithm composed of more simple devices than the other control algorithms (MPC, EMPC, etc). We developed a physiologic model of diabetes patients with mathematical analysis of insulin-glucose interactions. It has 19 states of differential equations, is composed of 11 states for glucose dynamics, seven for insulin dynamics and one for glucagon 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 the diabetic patient model. Through parameter sensitivity analysis, most sensitive metabolic parameters were found to affect glucose and insulin dynamics. Meanwhile, in a healthy patient, insulin released by the pancreas maintains the basal blood glucose concentration around euglycemic (normoglycemic) levels mg/ dl. Hence pancreas provides a basal rate of mu/ min and increases this amount during meal intakes (postprandial peak) to process the glucose absorbed from the gut. Consequently, in the absence of insulin, blood glucose level for a TIDM patient can decrease or increase above euglycemic levels (hypoglycemia and hyperglycemia, respectively) for long periods of time. In fact, the TIDM patient requires external insulin for survival. However, the diabetes control and complications trial (DCCT) [1] showed that an intensive insulin therapy can reduce the incidence of long-term illnesses. Therefore, an intensive therapy is encouraged for TIDM patients prescribed either by a continuous-infusion pump (CIP), or a multiple daily injection regimen (MDIR). On the other hand, it was also noticed in [1] that a possible side effect of an intensive therapy is the propensity to hypoglycemic scenarios in the patient. With this consideration, if an intensive therapy is followed by the patient, the prescribed insulin treatment must be carefully studied by the physician, and it should be constantly updated according with the results achieved. A wearable artificial pancreas in a feedback control system regulating insulin delivery, according to real time glycaemic changes, is not yet available because of many difficulties; part of them related to the development of efficient and reliable control algorithms []. It has been a challenge since 30 years to cope with the large variability of response from patient to patient and complexity of physiological regulation while avoiding hyperglycemia and hyperinsulinization. The present availability of short acting insulin and subcutaneous insulin pumps plus advanced studies in the field of subcutaneous glucose monitoring now open the possibility of a new generation 87

2 of algorithms for artificial wearable pancreas [3]. Simulation is almost a necessity for the state of the art in this field: it can reduce expensive experimental activities in the first phase of control systems development and can give comparison facilities. This work deals with the design of controller for delivery of insulin to type I diabetic 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 that the blood glucose level of a TIDM patient tracks, under closed-loop configuration, the path of the glucose level for a healthy person. Glucose tolerance curves (GTC s) were experimentally obtained from healthy subjects in order to validate a reference model. The controller is designed from H control theory. There are four major sites for insulin delivery [4]; 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 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 the other substances. Biocompatibility can be avoided by using tissue sampling techniques such as micro dialysis and reverse ionophoresis; expected problem under investigation are prolonged time for sample collection, quality of the 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. Ultimately, the objective is to develop a closed-loop glucose control system [1,5-8] consisting of three components: glucose sensor, control algorithm, and mechanical pump figure 1. In this system, glucose concentration is measured by the glucose sensor and based on measurement. The control algorithm computes the optimal insulin delivery rate. The mechanical pump then infuses the computed amount of insulin.. BLOOD GLUCOSE CONTROL PROBLEM Since the 80s, from the paper by [9,10], several models have been proposed to reproduce the glucose-insulin dynamics. The proposed models include: i) glucagon effect and threshold functions representing metabolic processes [11], ii) nonlinear terms for pharmacokinetic- pharmacodynamic effects [1] and iii) physiologic equations toward compartmental representations [13]. More recently, exercise effects have been included [7]. A physiological-based compartmental model has the advantage that the simulations can yield insight into the physiological parameters [14]. Such a model offers a powerful tool for generating predictions and clinical decision support in diabetes care [1,5]. Here, a physiological-based compartmental model is used to design and test the tracking control. The model includes equations for the main organs of the glucoregulatory system [1] and involves glucose uptake as well as the effects of glucose on hepatic glucose production, EGHGP, hepatic glucose uptake, EGHGU, the 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 ODE s, which comprises of 19 equations and has three dynamical subsystems: i) glucose, ii) insulin and iii) glucagon..1 Glucose: Insulin Modeling A variety of methods have been mentioned in literature to model the glucose- insulin dynamics. The most primitive one was a low order system reported by [9]. Some little more complex models which include glucagon effects and threshold functions that represent metabolic processes [4,15] had been reported. Sorensen 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 sub compartments Figure 1: A closed loop glucose control system. 88

3 where mass balances were derived. Sorensen 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 glucoseinsulin model is nonlinear and has following subsystems: glucose, insulin and glucagon. This nonlinear model was used for control purposes by [16]. This process is based on the compartmental technique. One of the features of this technique is that the model design is based on an understanding of the physiology; also, these models offer a powerful tool for generating predictions and clinical decision support in diabetes care [5,6]. Hence this model was chosen by us to simulate the glucose insulin dynamics and test the different controllers to validate the model and also compare the performance of the controllers.. Patient Model Uncertainty In this work the diabetic model used for patient simulations is taken from Parker et al. This pharmacokinetic-pharmacodynamic compartmental model of the human glucose-insulin system was initially developed by Guyton et al and Sorensen, and then modified by Parker et al, 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. This model divides the human body 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 where significant transport resistance (e.g., time delay) exists. 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 which is regulated by the manipulated variable, insulin infusion rate [17]. 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 sub model is described in Lehmann and Deutsch. Due to the inevitable patient-model mismatch there are some uncertainties; these uncertainties between the actual patient and the 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 the metabolic parameters of the liver and the periphery. In the patient model, glucose metabolism is mathematically described by threshold functions with the following structure: { } e = A B C xi + D tanh[ ( )] (1) The subscript is the state vector element involved in the metabolic effect, and subscript e denotes specific effects within the model: The effect of glucose on hepatic glucose production (EGHGP), the effect of glucose on hepatic glucose uptake (EGHGU), and the effect of insulin on peripheral glucose uptake (EIPGU). the control study. Nominal values of the above eight parameters (three sets of D and E, F LC and F PC ) are listed in table VALIDATION OF REFERENCE MODEL As blood glucose level is used like a reference, the transfer function P ref is validated from the glucose tolerance curves (GTCs) of six healthy subjects. The GTCs were obtained in the classical method, at t 5 0 the blood glucose level is measured and the subjects drank, a solution of 75 g of dextrose dissolved in 300ml of water (glucose load) and later, blood samples for blood glucose concentration determinations were obtained using a Dt 5 30 min sample interval, at least during 150 min. The Table shows the specifications of six healthy subjects. 3.1 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 Figures -8. Table 1: Values of uncertain parameters which determine patient dynamics Parameter Patient-1 Patient- Patient-3 Er EIGU Dr ECHGU Er EIPGU Dr EIPGU Er EGHGP Dr EGHGP F LC F PC Table : Specifications of healthy subjects Age Weight

4 Figure : Glucose tolerance curve for subject 1. Figure 3: Glucose tolerance curve for subject. Incremental blood glucose (mg/dl) The transfer function for reference model is validated from the mean of six healthy subjects data as the GTC shows that the BG response to a meal in a healthy subject behaves like a second-order system. The P ref has the following representation [18] P r ef Time(min) Figure 4: Glucose tolerance curve for subject 3. = s Kω n + ξωns + ωn K = 3900, j 5 0.7; v n Therefore the impulse response of P ref resembles the curve shown in Figure RESULTS AND DISCUSSIONs 4.1 H Glucose Control in Type I Diabetes Mellitus The order of the physiological model for TIDM (type I diabetes mellitus) patient is 19 th, hence the generalized plant G(s) is of the (19 1 m) order (where m is the sum () of the order of weighted transfer functions).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. Figure 10 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 the blood glucose control is given by [18] G s ( ) = Figure 5: Glucose tolerance curve for subject 4. W ppm Wm 0 WpP d1 0 0 W u d Wm Pm Wn P d 3 (3) 90

5 Figure 6: Glucose tolerance curve for subject 5. Figure 7: Blood glucose curve for subject 6. Figure 8: Overall response of all six subjects. Figure 9: Output of reference model. carbohydrate content into the meal. iv) The sensor noise effects are weighted by, W n i.e it emulates possible error generated by the glucose and W u stands for the weight for the control input. Figure 10: Standard feedback configuration with weights. 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 the performance weight, is a first order transfer function and was chosen such that the frequency content of was captured (output disturbance attenuation). Moreover W p was selected taking into consideration the characteristic frequency of P ref (for tracking). iii) The weight W m represents the effect of meal model, and permits to induce the maximum Figure 11 shows the simulation of the BG response of a TIDM patient under a meal, at t 5 0 min and at t min. The controller used for this simulation is the resulted by H approach [19]. 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. 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. 91

6 4. Robust Control Design in Type I Diabetes Mellitus The block diagram in Figure 1 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 [18] W m z 1 d1 GmW p 0 WpP GmWp WpP d1 z d W u d z3 = G ( s) d3 = Wi d3 z4 d4 WmWim d4 y u Wm Gm Wn P Gm P u Where weighted transfer functions are the same 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 above mentioned uncertainties is as shown in Figure 13. The simulation represents the worst case on the parametric variations. Since minimum BG level is higher than 70 mg/dl, hypoglycemia effects cannot be presented in the closed-loop. (5) Blood Glucose (mg/ dl) TIDM REFERENCE Time (min) Figure 11: BG response for type I diabetes mellitus patient under feedback control. Figure 1: Block diagram for BG control including parametric uncertainties. 5. CONCLUSION From the above discussion we can conclude that the order of H controller matches that of the diabetic patient model. Uncertainty within the reduced model, derived from physiological conditions within the nonlinear representation, is characterized in a control-relevant manner using relative uncertainty. Both nominal and robust controller performance satisfies the design criteria. Further, we can conclude that blood glucose response to a meal in a healthy subject behaves like a second order system having impulse as input and incremental glucose level as the output. NOMENCLATURE P ref Reference value K Gain j Damping factor v n Natural frequency W p Performance weight Figure 13: BG response by adding parametric uncertainties. W u Input weight W m Effects of meal model P Transfer function of diabetic patient model P m Transfer function of meal model 9

7 REFERENCES 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, Vol. 39, pp , B R Nucci G, and G Cobelli, The subcutaneous route to insulin-dependent diabetes therapy, IEEE Eng Med Biol, Vol. 0. pp , E R Carson, and T Deutsch, A spectrum of approaches for controlling diabetes, IEEE Control Syst Mag, Vol. 1. pp. 5-31, E D Lehmann, and T Deutsch, A physiological model of glucose-insulin interaction in type I diabetes mellitus, J Biomedical Engineering, Vol. 14, pp F C Erzen, G Birol, and A Cinar, Simulation studies on the dynamics of diabetes mellitus, Proc. IEEE Int. Symp. Bio-Informatics and Biomedical Eng, pp. 3-5, 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., pp , H E Lebovitz, Therapy for Diabetes Diabetes Mellitus, and Related Disorders, 3 rd edn, American Diabetes Association Alexandria, virgenia R Bergman, L Phillips, and C Cabal, Physiologic evaluation of factors controlling glucose tolerance in man, J Clinic Investigation, Vol. 68, pp , S Skogestad, and I Postlethwaite. Multivarible Feedback Control, Wiley, New York: E D Lehmann, and T Deutsch, Compartmental models for glycaemic prediction and decision-support in clinical diabetes care: Promise and reality, Computer Methods and programs in Biomedicine, Vol. 56, , M Nalecz, J Wojeieki, and I Zawicki, Control in artificial pancreas, IFAC control Aspects biomed. Engg. Pp R S Parker, F J Doyle 3rd, and N A Peppas, Robust H glucose control in diabetes using a physiological model, AICHE J, Vol. 46(1), pp , D U Campos-Delgado, R Femat, and E Ruiz-velaquez, Knowledgebased 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 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., MassachusettsInst. Tecnol. (MIT). Cambridge, G Pacini, and C Cobeli, Estimation of Beta-cell secretion and insulin hepatic extraction by the minimal modeling technique, Computational Methods Programs in Biomedicine. Vol. 3, pp. 41-8, P J Lenart, and R S Parker, Modeling exercise effects in type I diabetic patients, Proc. 15 th Triennial World Congress IFAC, Barcelona, Spain, 1-6 Jul 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 June, Milan, Italy, 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, Vol. 34, 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 _54_09; Copyright 009 by the IETE 93