A fuzzy logic based closed-loop control system for ...

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On the other hand, those biomedical systems are often correlated with vagueness, ... Journal of Medical Engineering & Technology, Vol. 29, No. 2, March/April ...
Journal of Medical Engineering & Technology, Vol. 29, No. 2, March/April 2005, 64 – 69

A fuzzy logic based closed-loop control system for blood glucose level regulation in diabetics M. S. IBBINI*1 and M. A. MASADEH{2

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Department of Electrical Engineering, Jordan University of Science and Technology, PO Box 3030, 22110 Irbid, Jordan; e-mail: [email protected] In this study, a closed-loop system to control the plasma glucose level in patients with diabetes mellitus type I is proposed. This control scheme is based on fuzzy logic control theory to maintain a normoglycaemic average of 4.5 mmol l71 and the normal conditions for free plasma insulin concentration in severe initial state; in particular, when the diabetic patient is subjected to a glucose meal disturbance or fluctuations in the measured glucose level due to error in the measuring instrument. The proposed controller has demonstrated superiority over other conventional controlling therapies. While fuzzy logic controllers have shown promising results in many fields, a comparative study is presented with well-known conventional controllers such as Proportional-IntegralDerivative (PID) and continuous insulin infusion control strategies. The simulated results, for the proposed controller, are presented and discussed.

Introduction Diabetes is one of the most widely spread human diseases of our time, and diabetes mellitus type I represents approximately 10% of all Americans diagnosed with diabetes [1]. This disorder is characterized by insufficient secretion of insulin from the pancreas due to a malfunction in b-cells. Furthermore, it may result in an increase in the glucose concentration in plasma beyond the normoglycemic average of 4.5 mmol l71 or the loss of control of blood glucose levels. It is estimated that 16 million people in the United States and over 100 million people worldwide suffer from this chronic disease and its complications. It is reported that diabetes was the seventh largest cause of death in the United States, responsible for 140 cases of 1996 reported death certificates [2]. The effects of this disease on the everyday life of patients can be quite serious, ranging from the need to administer regular medication doses with injections, to being put at risk of heart attacks. In order to improve therapy and develop an appropriate medication for diabetes, the main characteristics of human glucose/insulin kinetics need to be studied on the basis of simulations of an appropriate mathematical model. In the last three decades, a practical model of the nonlinear dynamics of blood glucose and insulin interaction in diabetic individuals has been devel-

oped [3, 4]. The model consists of a set of nonlinear differential equations that need, in general, to be solved numerically. The conventional method of evaluating the model as a crisp model, however, does not prove to be reasonable since biological models are extremely subject to uncertainties. First, the parameters of the model are uncertain; they exhibit a large variability depending on the individual physique of the patient as well as on the extent and the duration of the disease. And second, the model input, consisting of the nutritional contents of the ingested sugared food, can usually only be quantified with a high degree of imprecision. A semi-closed loop control scheme, using optimal control techniques, for correcting an initial state of hyperglycaemia was proposed in [5]. Simulated results showed the effectiveness of the proposed control scheme to maintain not only the normoglycaemic average of plasma glucose level but also other normal conditions, such as free plasma insulin concentration. In general, biomedical systems are inherently complex and nonlinear, and may have variable timing, with possible time delays. Consequently, conventional control techniques can prove insufficient for controlling such systems. On the other hand, those biomedical systems are often correlated with vagueness, imprecision, and uncertainty in model parameters [6].

*Corresponding author. Email: [email protected] 1 Current address: Dean of Al Huson University College, Balqa Applied University, Jordan. 2 Current address: Electrical Engineering Department, College of Engineering Technology-Balqa, Applied University, Jordan Journal of Medical Engineering & Technology ISSN 0309-1902 print/ISSN 1464-522X online # 2005 Taylor & Francis Group Ltd http://www.tandf.co.uk/journals DOI: 10.1080/03091900410001709088

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Blood glucose regulation in diabetics

A promising approach to alleviate those problems is the use of fuzzy logic controllers (FLC) to take into account the uncertainties in the human glucose/insulin kinetics. In detail, uncertain model parameters and model inputs are represented by fuzzy numbers with their shape derived from experimental data or expert knowledge. The block diagram for the proposed fuzzy control system is shown in figure 1. In the next section, a physiological model of glucose/ insulin interaction for type I diabetic patients with exogenous insulin and glucose infusion is presented. Implementation of the fuzzy logic control scheme will then be outlined. Numerical simulations with practical parameters values are presented and analysed. In a first set of simulations, severe initial state of hyperglycaemia is assumed and the proposed fuzzy logic logic scheme is used to correct the case. In a second set of simulations, a sudden glucose meal disturbance is assumed and the fuzzy logic controller (FLC) is used to bring the plasma glucose level to normal conditions. In order to simulate real situations, system parameters are assumed uncertain with variations up to 70% of their adopted nominal values. Finally, the model output is assumed to be corrupted with white noise to simulate the influence of imprecise glucose measurement. It should be mentioned that those measurements can be corrupted due to equipment, adopted methods and patient conditions.

I_ ¼ nI þ ð1=VI ÞIexo

ð3Þ

with initial conditions G(0) = Go, I(0) = Io, and X(0) = Xo. The sign (  ) denotes the derivative with respect to time t. In the above equations; G(t) represents the plasma glucose concentration, I(t) is the free plasma insulin concentration, and X(t) is proportional to insulin concentration in the remote compartment. p1, p2 and p3 represent the minimal model parameters describing the dynamics of plasma glucose and insulin interaction. V1 is the insulin

Mathematical model of diabetic patients

Figure 2.

Input membership function.

The minimal model of Bergman et al. [3, 4] was adopted by many authors [7, 8] for its minimum number of parameters, simplicity, and compatibility with known physiological facts. This model basically describes the nonlinear dynamics of glucose and insulin kinetics for patients with diabetes mellitus type I. The model consists of a singleglucose compartment in which patient insulin is assumed to act through a remote compartment to influence net glucose uptake, as represented in equation (1). On the other hand, inflow of glucose in the blood as a consequence of the ingested food Gext(t) is assumed and best presented by (2). Infused exogenous insulin Iext(t) is finally modelled by a nonlinear differential equation representing the insulin inflow into the blood, as shown in (3). ð1Þ G_ ¼ p1 G  X G þ p1 GB þ Gexo X_ ¼ p2 X þ p3 I  IB P3

Figure 3.

ð2Þ

Table 1.

Output membership function.

FLC rules for the proposed system. dG/dt

G(t)

Figure 1. Block diagram of the fuzzy control insulin delivery system.

PL PB PM PS NOM NS NM NB

Negative

Zero

Positive

PL PB PM Z Z NB NB NB

PL PB PM PS Z NS NM NB

PL PB PB PB Z Z NM NB

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M.S. Ibbini and M.A. Masedeh

distribution volume and n is the fractional disappearance rate of insulin. Gexo(t) and Iexo(t) are the rate of exogenous infusion of glucose and insulin, respectively. Gb and Ib denote the basal value of plasma glucose and free insulin level, respectively.

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Fuzzy logic controller (FLC) design A block diagram of the system with the proposed fuzzy logic controller (FLC) is illustrated in figure 1. The FLC is based on two input-linguistic variables, the plasma glucose concentration signal G(t) and its rate of change G_ (t). In addition, insulin infusion rate is considered as the outputlinguistic variable. The plasma glucose level and its rate of change input membership functions (MFs) are illustrated in figure 2. The output membership function is then shown in figure 3. The input and output membership functions are chosen to be triangular for design simplicity. The main objective of using a fuzzy controller is to maintain the normoglycaemic average of plasma glucose concentration and other model variables concentration (e.g. plasma insulin) within a certain desired interval in spite of the complex physiological model, sudden glucose meal-disturbances, or error in glucose measurements. A table of fuzzy IF-THEN rules that links the input and output MFs is built based on the desired plasma glucose dynamic behaviour. Those linguistic rules are given in table 1. Each rule output is demonstrated using MIN-MAX law and each crisp output is computed using CENTROID (COG) defuzzification method.

Figure 4.

Table 2.

Constant parameters for 70 kg body weight [7].

Parameter

Value

VI [L] n [min71] GB [mmol l71] IB [mU l71]

12 5/54 4.5 15

Table 3. Parameter 71

Control action surface.

p1 [min ] p2 [min71] p3 [min72 mU71 l ]

Model parameters. Values 0.028 0.025 0.000013

Figure 5. A severe initial state of G(0) = 15 mmol71 correction with insulin infusion program.

Blood glucose regulation in diabetics

Figure 4 shows the control action surface. It represents the input/output relation in three-dimensional configuration for the proposed fuzzy logic controller. Clearly, controller inputs change with the output in a piecewise linear fashion. The controller action demonstrates the superiority of using fuzzy logic control and compares favorably with the results discussed in [7]. Numerical simulation and discussion

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For simulation purposes, Bergman model parameters and constants are adopted from [8] and given in table 2. The same numerical values were used in the simulations of [5]

Figure 6. Glucose meal disturbance subjected after 8 hours given by Pexo(t) = ec xp(7k t).

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where optimal techniques were suggested to control the glucose level. GB denotes the basal blood glucose concentration for normal subjects and is adopted for the diabetic model. The numerical values of p1, p2, p3 are those estimated by Bergman et al. [4] and used by many researchers, as in [5, 7, 8]. Those adopted numerical values are given in table 3. In a first series of simulations, the FLC is applied to the nonlinear model (equations (1) – (3)) and a severe initial state of 15 mmol l71 is corrected. For comparison purposes, a classical PID controller with tuned proportional, integral, and derivative gains (KP = 90, KI = 5, KD = 3) is also simulated. Moreover, a classical open-loop therapy technique is assumed and numerically simulated [7, 9]. The technique assumes a constant continuous insulin infusion rate of u = nVIIB = 1000 mU h71 as is often practiced in medical situations. Figure 5 illustrates the case of a severe initial state of hyperglycaemia with a faster response of the FLC with respect to the other two methods. The proposed controller shows faster responses to reach the basal values of plasma glucose and insulin concentrations. In a second set of simulations, a glucose meal disturbance is assumed after 8 hours of normal conditions. To depict a practical case, uncertainty of system parameters is also assume in the numerical simulations. Figure 6 illustrates the glucose meal disturbance, which is assumed to be exponential, occurring at time t = 8 hours after a long period of normal conditions [8]. Figure 7 illustrates the diabetic patient model variable responses when subjected to the glucose meal disturbance shown in figure 6, using basal values of plasma glucose and

Figure 7. Model variables responses when subjected to glucose meal disturbance.

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M.S. Ibbini and M.A. Masedeh

Figure 8.

Model variables responses when subjected to decrease in p1 up to 70% of basal value.

Figure 9. Model variables responses when subjected to error in measuring plasma glucose level. insulin concentration as initial conditions. The results clearly show that, of the three different techniques considered in the simulations, FLC seems to be the most effective short-term control.

On the other hand, a decrease in one system parameter p1 to 70% of its normal value is also considered. All three different control policies are then simulated. The obtained results are shown in figure 8. A similar variation of 50% in

Blood glucose regulation in diabetics

the value of p1 was also assumed and simulated in [9]. Figure 8 clearly demonstrates the superiority of the proposed FLC over other techniques. In fact, those results are expected, as the FLC does not require a mathematical model as opposed to all other assumed techniques. In a last set of simulations, a white-disturbance is assumed, as proposed in [9], to simulate errors in measuring plasma glucose level due to equipment, methods, or to the patient conditions. Figure 9 demonstrates that the FLC output remains closer to the normoglycaemic average with a faster response than other conventional controllers. For the proposed FLC, the results show that the measured plasma glucose and insulin levels reach the basal values one hour before PID controller does.

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Conclusions In this study, a fuzzy logic controller has been proposed to maintain the normoglycaemic average for diabetic patient of type I. This study demonstrates the stability of the system, specifically when it is subjected to a glucose meal disturbance, uncertainty in model parameters, and fluctuations in measured glucose level. Computer simulations of different conventional controllers are demonstrated using PID and constant insulin infusion that models the openloop method controllers. The results of the proposed controller are used to evaluate the effectiveness of this technique to improve the performance of the controlled system, and ensure its superiority over other conventional

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schemes in controlling hyperglycaemia. The control technique studied here is very convenient for real time implementation due to its design simplicity and ease of tuning. References [1] Endocrine Web INC. Diabetes Center: www.endocrineweb.com, Copyright # 1998. [2] McShane, M.J., Russell, R.J., Pishko, M.V., and Cote, G.L., 2000, Glucose monitoring using implanted fluorescent microspheres. IEEE Engineering in Medicine and Biology, November/December, 36 – 45. [3] Bergman, R.N., Ider, Y.Z., Bowden, C.R., and Cobelli, C., 1979, Quantitative estimation of insulin sensitivity. American Journal of Physiology, 236, 667 – 677. [4] Bergman, R.N., Phillips, L.S., and Cobelli, C., 1981, Physiologic evaluation of factors controlling glucose tolerance in man: measurement of insulin sensitivity and b-cell glucose sensitivity from the response to intravenous glucose. Journal of Clinical Investigation, 68, 1456 – 1467. [5] Ibbini, M.S., Masadeh, M.A., and Bani Amer, M.M., 2004, A semiclosed-loop optimal control system for blood glucose level in diabetics. Accepted for publication in this journal. [6] Ying, H., 2000, Fuzzy Control and Modeling: Analytical Foundations and Applications. (New York: Wiley–IEEE Press). [7] Furler, S.M., Kraegen, E.W., Smallwood, R.H., and Chisholm, D. J., 1985, Blood glucose control by intermittent loop closure in the basal mode: computer simulation studies with a diabetic model. Diabetes Care, 8, 553 – 561. [8] Fisher, M.E., 1991, A semiclosed-loop algorithm for the control of blood glucose levels in diabetics. IEEE Transactions in Biomedical Engineering. 38(1), 57 – 61. [9] Ollerton, R.L., 1989, Application of optimal control theory to diabetes mellitus. International Journal of Control, 50, 2503 – 2523.