Control of the Blood Glucose Level in Diabetic Patient Using Predictive Controller and Delay Differential Equation

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1 Journal of Informatics and Computer Engineering (JICE) Vol. 2(4), Jul. 216, pp Control of the Blood Glucose Level in Diabetic Patient Using Predictive Controller and Delay Differential Equation Mojgan Esna Ashari Department of Electrical Eng. Azad University of Khomeini share Isfahan, Iran Maryam Zekri Department of Electrical and Computer Eng. Isfahan University of Technology Isfahan, Iran Masood Askari IUT Branch, Culture and Research Academic center for education Isfahan, Iran Abstract: Diabetes is known as the sixth leading cause of death in the world leading to kidney and cardiovascular diseases as well. As a result, controlling the disease is of particular importance. To control blood glucose level in diabetes mellitus type I, insulin must be injected into the body. This injection must be strictly controlled any excessive increase in insulin injection can cause human death. Various controllers have been introduced to control diabetes type I with their own disadvantages. Diabetic patient model used in previous methods is not in compliance with physiological data, uncertainties are not considered in the model, or the controller has a very limited range of stability. In this paper, a Nonlinear Model Predictive Controller is introduced which not only resolves previous problems like lack of consideration of model uncertainty, sharp drop in blood glucose, and low stability, but also specifies insulin injection dosage based on blood glucose level momentarily so that. Keywords: Diabetes, Insulin- Glucose Regulation System Delays, Blood Glucose Level, Nonlinear Model Predictive Controller. I. INTRODUCTION Diagnosis and treatment of diabetes has been a controversial issue in recent years because based on World Health Organization report in 211 about 346 million people worldwide are suffering from the disease, and the number is expected to reach more than 552 million people till 23. As a result, reducing diabetic disorders will increase patients' quality of life and decline health care costs [1]. Diabetes is a metabolic disorder in which blood glucose level measures more than normal range of 8 to 14 milligrams per deciliter. In diabetic patients, β cells producing insulin in the body are destroyed and the body cannot control blood glucose levels by itself [2]. So blood sugar needs to be adjusted using insulin injections. In general, control strategies on blood sugar regulation fall into three categories: open loop, semi-closed loop, and closed loop control among which closed loop control is used in this paper.the closed loop control method acts like an artificial pancreas: blood glucose concentration is continuously measured by sensors and after comparing with the desired amount, the required amount of insulin is determined using an appropriate control algorithm, followed by a subcutaneous injection through insulin infusion pump. The overall schema of closed loop glucose control system is shown in Fig. 1. In recent decades, various control methods including Norm [4,5], adaptive control [6], proportionalderivative controller [7], proportional-integral-derivative (PID) controller [8], fuzzy controller [9,1], etc., have been proposed to regulate blood glucose levels in type-i diabetic patients. Fig. 1. Closed loop glucose control system [3] These methods differ in terms of employing control strategies, use of constraints, mathematical models and ease of implementation each with its own advantages and disadvantages. In this paper, the proposed control system is capable to predict blood glucose levels on a real-time basis and inject insulin so that blood glucose level always lies within normal range, showing suitable performance against meal disturbances and uncertainties in the model. After introduction, the second section introduces mathematical model of glucose-insulin system. Nonlinear model predictive control (NMPC) is presented and designed in section three, followed by simulation results in section four. Conclusion is finally given in section five. II. MATHEMATICAL MODEL OF GLUCOSE-INSULIN SYSTEM So far various mathematical models have been presented to model glucose-insulin system. The first simple models were developed by Chorbajian [11] and Ackerman [12]. In these 144 Article History: JICE DOI: / , Received Date: 11 Mar. 216, Accepted Date: 28 Jun. 216, Available Online: 3 Jul. 216

2 models, only two components, i.e. glucose and insulin, were considered with a linear structure. Later, Bergman et.al offered minimal models that used more physical facts [13]. In the following years other models were presented with better matching with physiological data. For example, negative feedback ODE model (Tolic), DDE model with a time delay (Engelberghs) [14], DDE model with intermittent delay (Sturis, Tolic), DDE model with two time delays and constant insulin reduction rate (Wang, Li ), and DDE model with one time delay and varying insulin degredation rate (Wang, Li) [15]. Some drawbacks related to each of these four models are as following: time delay between the moment blood glucose level increases and the moment insulin secretion happens and also time delay in hepatic glucose production were not considered in the models and insulin reduction rate was also considered as constant. In the fifth model introduced by Wang and Li in 29, however, all these problems were fixed -- glucose-insulin regulation was modeled using variations of insulin secretion and degradation rate of insulin obeys Michaelis Menten kinetics. To have a suitable control in nonlinear model predictive controller, system model and diabetic patient model should be able to simulate insulin-glucose regulation system and also diabetic patient model so that more adaptation on physiological data is achieved. In this paper, the easier model of Engelberghs is used as model predictive control system and the more complex model of Wang as patient model, assuming that the body is unable to produce insulin and that subcutaneous injection of insulin is the only way to control glucose. The equations of the model are as follows: dg = E dt g + f 5 I(t τ 2 ) f 2 (G) + f 3 (G). f 4 (I) (1) di dt = αf 1 (G) I t 1 dg = G dt in f 2 G(t) f 3 G(t) f 4 I(t) + f 5 I(t) (2) di dt = αi in + βf 1 G(t TT 1 ) d 1 I(t) d 2 + I(t) Equation (1) is used as model predictive control system [14] and (2) as patient model [15]. Eg represents the glucose quantity supplied by the external medium, through injection at a constant rate (this term corresponds to food intake). The reference values of the parameters are: E g = 18 mg min, t 1= 6 min, τ 2 = 5 min, TT 1 = 15 min, α >, β [,1] For type 1 diabetes, β = (no insulin is secreted from the pancreas). f 1 (G) = R m /(1 + exp((c 1 G/Vg)/a 1 )) f 2 (G) = V b /(1 exp( G/(C 2 Vg))) f 3 (G) = G/(C 3 Vg) (3) f 4 (G) = U O + (U m U O )/(1 + exp( βln (1/Vi + 1/ (Et i )))) f 5 (I) = R g /(1 + exp(α(i/vp C 5 )) f 1 (G) Glucose-dependent insulin secretion. f 2 (G) Insulin-independent glucose consumption by the brain and nerve cells. f 3 (G)f 4 (I) Glucose-dependent insulin consumption by muscle cells and fat. f 5 (I) Glucose production controlled by insulin concentration. Parameters related to functions f 2 to f 5 are mentioned in Table I. TABLE I: PARAMETERS OF THE FUNCTIONS IN (3) Parameters Units Values Vg 1 1 Ub mg.min-1 72 C2 mg C3 mg.min-1 1 Vp 1 3 Vi 1 11 ti Min 1 Rm mu.min-1 21 C1 mg a1 mg U mg.min-1 4 Um mg.min-1 94 β C4 mu Rg mg.min-1 18 a 1.mU-1.29 C5 mu E 1.min-1.2 Va 1 1 The uptake of food glucose, modeled by (4) [18], is denoted by G in. G in (t): t < 15(min) t t < 45(min).5 45 t 24(min) In this model, G(t) is the output and G in and I in (t) are system inputs. Insulin reduction in body differs from person to person. In this paper, insulin reduction for the controller system model is defined as 1 and for varying patient model t 1 with Michaelis Menten kinetics. Our aim is to determine the control function I in (t) in order to stabilize the patient's blood glucose levels in a normal range despite portion distortion and uncertainty of parameters. III. NONLINEAR MODEL PREDICTIVE CONTROL Predictive controllers were originally developed by engineers in 197s. This method was introduced by Clerke to control unstable non-minimum phase and slow systems [16]. NMPC is formulated to solve a finite horizon optimal control problem on a real-time basis according to system dynamics and constraints involving states and controls. Fig. 2 depicts the basic principle of model predictive control. Based on measurements obtained at time t, the controller forecasts the future [16]. (4) 145

3 Dynamic manner of the system over a prediction horizon Tp determines the input such that a predetermined performance objective function is optimized. If there were no model-plant discrepancy, and if the optimization problem were solvable for infinite horizons, then the input function at time t = to the system is applicable for all times t. The resulting manipulated input function will be implemented only until the next measurement gets available. min J (x(t ), u(t); T c, T p ) (8) J(x(t), u; T c, T p ) = i+t p i F x(τ), u(τ) dτ The function F, hereinafter called cost function, specifies the favorable control performance that can arise. The standard quadratic form is the simplest and most often used one: F(x, u) = (x x d ) T Q(x x d ) + (u u d ) T R(u u d ) (1) Where u d and x d are the desired input and output respectively. Q and R are symmetric, definite positive and weighting matrices. T p is the horizon of the predicted output and T c is the control horizon. Equation (1 ) gives the error between desired output and model-predicted output. -- to obtain the output values in the next time interval, the optimal input values in time period T c are used, then the value of the control variable is set constant in the last calculated value [16]. The block diagram of nonlinear model predictive control is given in Fig. 3, composing of a system model and plant. (9) Fig. 2. Principle of model predictive control. The time gap between recalculations can vary; however, it is supposed to be fixed -- the measurement will occur every δ sampling time-units. Using the new measurement at time t + δ, the entire process-prediction and optimization will be repeated to find a new input function with the control and prediction horizons moving forward. As shown in Fig. 2, the input u is denoted as an arbitrary function of time. The calculation of the applied input based on the predicted system behavior allows the inclusion of constraints on states and inputs as well as the optimization of a given cost function. The stabilization problem for a class of systems introduced by the following nonlinear set of differential (5): X(T) = F X(T), U(T), X() = X (5) Is subject to input and state constraints of the form [16]: u(t) U, t (6) x(t) X, t where x(t) and u(t) represent the states and inputs vector respectively. U and X constraints are given in (7) where u min, u max and x min, x max are constant vectors [16]: u {u R m u min u u max } (7) x {x R n x min x x max } To distinguish between the real system and the system model used to predict the future within the controller, the internal variables in the controller are denoted by a bar (e.g. x, u ). The finite horizon optimal control problem commonly described above is mathematically formulated as following [16]: Fig. 3. Nonlinear model predictive control block diagram. IV. DESIGNING NONLINEAR PREDICTIVE CONTROLLER FOR INSULIN INJECTION SYSTEM To design predictive controller for the model, an objective function needs to be designed. Real-time optimization of objective function will lead to design of a control signal which will track the suitable reference path by predicting system behavior. According to studies, the optimal amount of blood glucose is 11 milligrams per deciliter; nevertheless, the 8-to-14 interval is also known as green or healthy zone [17]. Therefore, the objective function is defined as follows: N J = { G (t + j) G S (t + j) 2 Q 1 (j) N + G (t + j) G up (t + j) 2 Q 2 (j) N + G (t + j) G down (t + j) 2 Q 3 (j) M + W(Z 1 )u(t + j 1) 2 R(j)} (11) 146

4 In (11), G is anticipated blood sugar, G s is the optimal blood sugar level (11), and G up and G down are high (14) and low (8) blood sugar values respectively. Most important and above all, Q 1,2,3 values have to be regulated so that blood sugar levels do not lie in unhealthy conditions. Q 1 = 1 Q 2 = if G < G up. (12) 1 if G > G up Q 3 = 1 if G < G down if G > G down the body in standard conditions. The profile provided in this diagram is a standard one defined based on (4) is. In the absence of insulin injection, blood sugar is 142 milligrams per deciliter and insulin level is 18 units -- the initial system was deliberately put in unhealthy conditions. Results are presented in Fig.5, As seen, blood sugar level always remains in unhealthy situation. As depicted in Fig. 6, the amount of blood glucose level is immediately shifted from unauthorized to authorized level when controller is used G in 3 The R-value in (11), denoting objective function penalty for taking too much insulin, is taken as 1 in simulations. Prediction horizon is set as 45 minutes [17]. As for genetic algorithm parameters, maximum number of repetitions is 3, initial population size is 25, cross over/recombination value is 5%, and mutation rate is considered as 4%. In addition, the Rollet Wheel Selection is considered as the method for selecting members of recombination Time (min) Fig. 4. Glucose intake rate G in in (4). V. SIMULATIONS AND RESULTS The proposed simulation method was implemented in MATLAB software. Fig. 4, depicts amount of sugar entering (A) (B) 18 I 28 G Fig. 5. Open loop response system. The reta of (A) Imsulin (B) Glucose change in 24 hours. 147

5 6 G in 1.5 I in G I Fig. 6. Glucose intake rate G in, Insulin injection I in, Blood Glucose levels G, Blood Insulin levels I in 24 hours after controlling. 3 G in 1.5 I in G I Fig. 7. Intake. Glucose intake rate G in, Insulin injection I in, Blood Glucose levels G, Blood Insulin levels I with disturbances. Table II: Comparison the daily infused insulin under the optimal nmpc and that of the proposed in [18]. Controller Daily infused insulin (mu/kg) Fuzzy PD Fuzzy PI Genetics optimal Fuzzy PI 78.8 Genetics optimal Fuzzy PID 78.1 Genetics optimal NMPC Genetics optimal NMPC with disturbances 75.8 To evaluate system performance in the face of portion distortion, two cases involving hit distortion and rising glucose level in the body were studied. Fig. 7, shows that the controller does not allow any above-the-limit deviation for the system. However, in this case, 1.5% excessive insulin was injected into the patient to compensate for the distortion incurred to him/her. Based on Table 2, our proposed controller shows better performance in the regulation of blood glucose levels and also reduction of daily insulin dosage compared with other controllers [18]. VI. CONCLUSIONS In this paper, stabilizing blood glucose levels for type I diabetic patients was investigated using non-linear predictive controller over Wang's extended model. Based on obtained results, the designed controller enjoys a relatively good stability and can regulate blood glucose levels on a real-time basis. As a consequence, the risk of hypoglycemia and hyperglycemia do not threaten the patient. Moreover, the designed controller showed acceptable performance in dissipating portion distortion and model uncertainty as well, properly adjusting the patient's blood glucose levels with minimum insulin injection in a way quite favorable in medical science. 148

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