International Simposium on Robotics and Automation 24 August 25-27, 24 Self-tuning Adjustment Algorithm for Type I Diabetic Patients Based on Multi-Doses Regime. D.U. Campos-Delgado Facultad de Ciencias UASLP ducd@fc.uaslp.mx R. Femat Dept. Matemáticas Aplicadas y Sistemas Computacionales IPICYT rfemat@ipicyt.edu.mx M. Hernández Ordoñez Facultad de Ingeniería UASLP A. Gordillo-Moscoso Facultad de Medicina UASLP Abstract A self-tuning algorithm is presented for on-line insulin dosage adjustment in Type I diabetic patients. The algorithm suggested does not need information of the insulin-glucose dynamics (model free). Three doses are programmed daily, where a combination of two types of insulin: rapid (Lispro) and slow (NPH) is injected into the patient through a subcutaneous route. The doses adaptation is performed by reducing the error in the blood glucose level from euglycemics. In this way, a total of six doses are tuned per day: three rapid and two slow, where there is large penalty to avoid hypoglycemic scenarios. Closedloop simulation results are illustrated using a detailed nonlinear model of the subcutaneous insulin-glucose dynamics in a Type I diabetic patient with meal intake. 1. INTRODUCTION The diabetes mellitus is a disease that has largely exceeded its growth expectative, and its impact in the global health care problem has raised the interest of the scientific community for providing control algorithms that could be implemented for real-time patient treatment [1], [4], [13]. In a Type I diabetes mellitus (TIDM) patient, the pancreas is not able of producing insulin at all. This problem can produce short and long term illnesses (diabetes coma, nephropathy, retinopathy, and other tissue damage) due to the variations in the blood glucose level (BGL). Therefore, the BGL has to be monitored externally to maintain it regulated by applying insulin infusions in a regular scheme. Meanwhile, in a healthy patient, the insulin released by the pancreas maintains the basal blood glucose concentration around euglycemic (normal) levels 7 12 mg/dl. Hence the pancreas provides a basal rate of 22 mu/dl [17], and it increases this amount during meal intake in order to process the glucose absorbed from the gut. Consequently in the absence of insulin, the blood glucose level for a TIDM patient can decrease or increase above euglycemic levels (hypoglycemia and hyperglycemia respectively) for long periods of time. As a first step in the treatment of this illness, it is necessary to understand the insulin-glucose dynamics in diabetic patients. For this reason, several research efforts have focused on the mathematical modelling of these interactions [14], [17], [2]. These models can also be used as educational simulators for demonstration and self-learning [9]. There are two overall approaches for glucose control, and they depend on the location of the insulin infusions: (a) subcutaneous [1] and (b) intravenous [13]. For the intravenous approach, a continuous pump is used to deliver a variable insulin infusion rate to the patient, according with a control algorithm that processes the glucose measurements. Several control methodologies have been suggested: H robust control [8], [12], [15], model predictive control [1], [11], and optimal control [7]. However, due to the size of mechanical pumps, this approach is now limited to patients under a hospital treatment. On the other hand, the subcutaneous approach relies on several therapeutic regimes based on combinations of different types of insulin: rapid (Lispro), regular, slow (NPH) or ultra-lente [3], [5]; delivered to the patient through a subcutaneous route on multiple daily dosing regimes. The doses are programmed according with the information gathered by implanted glucose sensors (MiniMed R), picks of blood glucose concentration (Accu- Chek R) or non-invasive blood glucometers (GlucoWatch R) [18], and physician advice. Thus, algorithms for the optimal time and amount of insulin have been suggested in [6], [16]. Furthermore, in [19] a strategy based on neural predictive controllers was adopted. One attractive insulin combination is rapid (Lispro) and slow (NPH) types, where the NPHtype provides the basal insulin and the transient effects after each meal (post-prandial peaks) are regulated by the Lispro. The subcutaneous approach is indeed a challenging control problem since the insulin absorption has to be considered, and consequently a time-lag is present in the plasma insulin concentration. Nevertheless, this is the most common therapeutic regime for Type I diabetic patients in a chronical stage. In the overall picture, the blood glucose regulation in TIDM patients presents a challenging interdisciplinary problem that can be approached from different points of view: systems, control and medicine. However, in order to provide significant advances in the treatment of this disease, the knowledge from these areas must be merged together with a common and clear objective: effective patient treatment. The paper is organized as follows. The problem statement is detailed in
International Simposium on Robotics and Automation 24 August 25-27, 24 Section 2. Section 3 defines the self-tuning dosage control structure, and Section 4 presents briefly the mathematical modelling of the insulin-glucose dynamics. Finally, Section 5 introduces the closed-loop simulation results and Section 6 presents concluding remarks and future work. 2. PROBLEM STATEMENT AND CONTROL METHODOLOGY According to Mexican customs, three major meals are taken per day: breakfast (desayuno) (7:-1: hrs), lunch (comida) (13:-15: hrs) and dinner (cena) (2:-22: hrs); where the comida meal is the major one of the day. Roughly, there is a time interval of 6 hrs. among each meal of the day. The approach presented in the paper relies in a three daily injections using rapid (Lispro) and slow (NPH) types of insulin. These doses are programmed 1 15 minutes before taking a meal. In Figure 1, the plasma insulin concentration for both types of insulin are shown for a subcutaneous injection of 1 U. The characteristics of these two types of insulin are depicted in Table I [5]. Due to the delayed action of the NPH insulin, the doses for lunch-time is omitted, and only rapid insulin is injected. Concentration ( µ U/ ml ) 9 8 7 6 5 4 3 2 1 TABLE I CHARACTERISTICS AND TYPES OF INSULIN Type Onset (hour) Peak (hour) Duration (hour) Lispro.1.25.25.5 3 4 NPH 2 4 4 1 1 18 Lispro NPH 2 4 6 8 1 12 14 16 time (hours) Fig. 1. Time evolution of plasma insulin concentration after a subcutaneous injection (1 U) of Lispro and NPH insulin Thus, the control objective is stated as: regulate the BGL around a normal level, defined as NGL 7 12 mg/dl (1) using three daily doses of a preparation of Lispro and NPH insulin. Furthermore, it is desired to reach this objective minimizing the amount of insulin by the patient. In this control scheme, several glucose measurements are available daily which could be derived from blood samples, in-vivo sensors or non-invasive means [18]. The control problem posed is very demanding since the doses given by a physician can vary abruptly from patient to patient. Moreover, the insulin-glucose dynamics for a TIDM patient are highly non-linear and can be modified by different parameters like diet, exercise, etc. [14], [17], [2]. According with the glucose measurements, the systemic blood glucose deviation from the NGL can be measured as: J 1 T T φ(t)dt 1 N N T s φ(kt s ) (2) k1 where N is the number of measurements during the measured interval T, T s represents the sampling interval in the glucose measurements, 8< and φ(t) (pointwise deviation from NGL) is defined as :G(t) 12 mg/dl G(t) > 12 mg/dl φ(t) P {G(t) 7} mg/dl G(t) < 7 mg/dl 7 G(t) 12 mg/dl (3) In definition (3), P 1 represents a constant that imposes a larger penalty in reaching hypoglycemic scenarios. Note that φ( ) is positive in the case of an hyperglycemic (above NGL) condition, it is negative in the case of an hypoglycemic (below NGL) one, and it should be close to zero ideally. Thus, (2) measures the area outside the NGL during the glucose evolution in a given time interval. However, due to the multi-doses control regime and the absorption process of the subcutaneous insulin infusions, the BGL cannot be completely regulated into the interval [7, 12] mg/dl, and there will be time instants where the BGL lies outside the desired interval. As a result, the doses should increased if the index (2) is positive (hyperglycemia), and reduced if the index is negative (hypoglycemia). A total of six insulin dosages are tuned: 1) Ir: b breakfast dose of rapid (Lispro) insulin. 2) Ir: l lunch dose of rapid insulin. 3) Ir d : dinner dose of rapid insulin. 4) Inph b : breakfast dose of slow (NPH) insulin. 5) Inph d : dinner dose of slow (NPH) insulin. by using the information of the performance index in (2), and the characteristic peak and duration time of each insulin type. 3. SELF-TUNING INSULIN ALGORITHM The self-tuning algorithm for each dose is computed by an evaluation of the performance index (2) based on the following criteria: The breakfast dose of slow insulin Inph b is evaluated by measuring (2) during the day until the dinner dosage (dinner), and the dinner dose Inph d is evaluated during the night and until breakfast. Thus, if the doses are properly chosen, they should regulate the BGL in an euglycemic level (J ).
International Simposium on Robotics and Automation 24 August 25-27, 24 Blood Deviation Ji time t Adjustment Algorithm i Ir I i nph Subcutaneous Injection Diabetic Patient Measurement G Fig. 2. Block Diagram of the Self-tuning Scheme. The rapid insulin doses (Ir, b Ir, l Ir d ) are evaluated during the posterior time following the infusion, until the next dosage is prescribed. If at the time of the infusion, the BGL is above 3 mg/dl (hyperglycemic scenario), then to the next dosage an increase of 3 % of the previous dose is added. If at the time of the infusion, the BGL is below 5 mg/dl (hypoglycemic scenario), then to the next dosage a reduction of 3 % of the previous dose is subtracted. The last two criteria are included in order to avoid extreme situations of hyperglycemia and hypoglycemia. If the breakfast, lunch and dinner times are denoted by t b, t l and t d, then five error measures are used for tuning, assuming for simplicity of the notation of continuous variables: td Jnph b 1 φ(t)dt (4) t d t b t b Jnph d 1 tb φ(t)dt (5) t b t d t d Jr b 1 tl φ(t)dt (6) t l t b t b Jr l 1 td φ(t)dt (7) t d t l t l Jr d Jnph d (8) These integral can then approximated by summations of sampled values. The tuning rules are given by (9) and (1), where the index k is referred to the actual evaluation and k 1 to the past one, (α, γ) are positive constants related to the correction steps for each type of dose, and (β, δ) are also positive constants that involve a momentum correction that speeds up the convergence to the optimal values. The glucose measurement G i G(t i ) i b, l, d represents the glucose value at the time of the insulin infusion. The four parameters (α, β, δ, γ) could also be time varying, where they are adjusted according with the past history of the performance index. However, this strategy was not pursued in this work, since the simulation showed good performance in the scenarios tested (Section 6). The feedback block diagram of the self-tuning control algorithm is presented in Figure 2. It is important to point out that the tuning rules in (9) and (1) can also be visualized as a PID discrete structure with respect to the error, plus an expert rule to avoid hyperglycemic and hypoglycemic scenarios. 4. DIABETIC PATIENT MODELLING In this section, the mathematical modelling of a Type I diabetic patient is described. For simplicity, this model is presented in three parts: (1) - Compartmental Model: describes the interactions between the different compartments in the body that influence the insulin-glucose dynamics; (2) Input via Gastric Emptying: presents the modelling of the glucose liberation in the gut following the ingestion of a meal; (3) Subcutaneous Injection: describes the interactions in the insulin absorption due to subcutaneous injections, and different types of insulin (see Figure 3). The models presented in this section were initially developed by Sorensen [17], Berger and Rodbard [2], and Lehmann and Deutsch [9]. A. - Compartmental Model The insulin-glucose model for a Type I diabetic patient used in this work has a physiological structure based on a compartmental technique [14], [17]. This model departs from experimental evidence to formulate and validate metabolic processes of the compartmental model on the whole organ and tissue level including counter-regulatory effects. Thus, the insulin-glucose model is governed by 19 nonlinear ordinary differential equations, and is divided into three subsystems: 1), 2), and
International Simposium on Robotics and Automation 24 August 25-27, 24 I i r(k) I i r(k 1) + αj i r(k) + β [ J i r(k) J i r(k 1) ] + Γ I i r(k 1) i b, l, d (9) Inph(k) i Inph(k i 1) + γjnph(k) i + δ [ Jnph(k) i Jnph(k i 1) ] + Γ Ir(k i 1) i b, d (1) 1 3 G i > 3 mg/dl Γ 5 G i 3 mg/dl G i < 5 mg/dl 1 3 Periphery Measurement TABLE II PARAMETERS IN ABSORPTION PROCESS FOR DIFFERENT TYPES OF INSULIN Meal Carbohydrates Intake Gastric Absorption Dynamics Lispro Regular NPH Lente Ultra Lente s 1.8 2. 2. 2.4 2.5 a(h/u).5.5.18.15 b(h) 1.3 1.7 4.9 6.2 13. Infusion Fig. 3. 3) Glucagon. Subcutaneous Absorption Dynamics Block Diagram of - Simulation Model. Glucagon Effects The first two subsystems were modelled for the brain, arterial system (heart/lungs), liver, gut, kidney, and periphery compartments, see Figure 3. The glucagon was modelled as a single blood pool compartment. The system output is the peripheral interstitial glucose, that permits to obtain accurate glucose levels. The system has two inputs: Subcutaneous insulin infusion, input via gastric emptying by a meal. The definition for the dynamic equations of each compartment model are detailed in [15]. B. Input via Gastric Emptying The amount of glucose in the gut following the ingestion of a meal containing Ch milimoles of glucose equivalent carbohydrate is modelled as a first order differential equation [9]. In this model, the rate of gastric emptying due to a meal is a function of the amount of carbohydrates intake Ch. This function can have two shapes: 1) Ch < 12 mmol of carbohydrate, then the rate of gastric emptying presents a triangular function, with equally raising and decreasing rates, and a peak value of 12 mmol/h. 2) Ch 12 mmol of carbohydrate, then a trapezoidal shape is observed, also with equally raising and decreasing rates, but it saturates to 12 mmol/h during some specific interval. Finally, the glucose input for a meal intake is given by a proportion of the glucose in the gut. C. Subcutaneous Injection Assume that a insulin dose of D units is injected subcutaneously at time t insulin. Hence the velocity of absorption can be described by da dt s(t t insulin ) s (ad + b) s D (t t insulin ) [(t t insulin ) s + (ad + b) s ] 2 k ea (11) where k e represents the plasma elimination constant, and the parameters a, b, s are defined in terms of the different types of insulin: Lispro, Regular, NPH, Lente or Ultralente (Table II) [2]. Finally, the plasma insulin concentration due to the subcutaneous injection is proportional to the absorbed insulin. It is assumed that the insulin effect of previous injections is additive [2], i.e. the insulin plasma concentration depends on the combined effect of the actual and previous dosages. This consideration is not significant for Lispro insulin since its duration is approximately from 3 to 4 hours, and the Lispro doses are programmed in periods of 6 hours during the day and 12 hours at night. However, it is important for NPH insulin since its duration is from 1 to 18 hours. 5. CLOSED-LOOP SIMULATION RESULTS The self-tuning control structure of Figure 2 was simulated using the non-linear model of the diabetic patient of Figure 3. The numerical simulation was implemented in MATLAB/Simulink c. A total of 12 (284 hrs.) were simulated with three meals per day: Breakfast: 8: hrs., Lunch: 14: hrs., Dinner: 2: hrs. Three infusions of insulin are programmed per day by a subcutaneous injection, where a combination of Lispro and NPH insulin is calculated according with the control algorithm.
International Simposium on Robotics and Automation 24 August 25-27, 24 Therefore, during the simulation period, a total of 36 doses are computed. A. Meals Description The meals carbohydrate intakes were calculated according with the following profile: male, 3 years old, 8 kg, 1.75 m, number of hours of sleep per day: 7, number of hours of very light activity: 4, number of hours of light activity: 9, number of hours of intense activity: 4, amount of calories per day: 3734 Cal/day. It is considered that 5 % of the calories are coming from carbohydrates, and take that 4 calories are equivalent to 1 gr. of carbohydrates (CH). Consequently, it is needed 466.7 gr. of carbohydrates per day. Assuming a distribution of this amount of carbohydrates in three meals: 35 % breakfast, 45 % lunch and 2 % dinner, results in the next meal distribution of carbohydrates: Breakfast: 163.34 gr. CH, Lunch: 21.1 gr. CH, and Dinner: 93.34 gr. CH. Therefore, the lunch is the heaviest meal of the day according to Mexican customs. During the simulation time (12 ), the amount of carbohydrate intake per meal was varied around the nominal values calculated previously ±1%, but looking to add up to 3734 Cal/day (see top plot in Figure 4). B. Simulations Scenarios First, in order to visualize the effect of lack of insulin infusions in a Type I diabetic patient, a simulation was carried out assuming no insulin dosage. The meal time and distribution described in the previous section were used. The simulation is presented in Figure 4. Thus, the BGL rapidly takes high values (hyperglycemic), and it even presents higher peaks during the absorption of each meal. Consequently, it is necessary to implement an external insulin dosage scheme to regulate the BGL in the patient. Two simulation scenarios were tested with the self-tuning adjustment: 1) Case 1: the patient starts with high doses for both types of insulin (I b r I l r I b r 6 U, and I b nph Id nph 16 U), as a result the BGL is low initially. 2) Case 2: consider an opposite scenario to Case 1, the patient starts with small doses for both types of insulin (I b r I l r I b r 2 U, and I b nph Id nph 8 U), producing initially high BGL. In Figures 5 and 6, the results for Case 1 are illustrated. Thus, a regular meal intake is simulated and due to the initial high dosage values, the BGL is below 7 mg/dl during long periods of time. However, the algorithms is able to decrease properly the insulin infusions (see Figure 6) in order to regulate the BGL into euglycemics almost constantly. Now, in Figure 7, the opposite scenario is devised. The doses are low and the BGL initially reaches a high value 22 mg/dl. In this case, the algorithm increases the insulin infusion to reach Rate of Absorption (mg/min) Plasma (mg/dl) 4 3 2 1 2 4 6 8 1 12 5 4 3 2 1 NGL 2 4 6 8 1 12 Fig. 4. Simulation for a 12-Days Period without Infusions: (Top) Meal intake, and (Bottom) Blood glucose level. Rate of Absorption (mg/min) Plasma (mg/dl) 4 3 2 1 2 4 6 8 1 12 15 1 5 NGL 2 4 6 8 1 12 Fig. 5. Simulation for Case 1: (Top) Meal intake, and (Bottom) Blood glucose level. a regulated BGL. It was observed that the tuning algorithm did not converge to the same insulin doses for Case 1 and 2, see Figures 6 and 7. This behavior has to be explored in future work to quantify its effect in the resulting therapeutic regime. 6. CONCLUSIONS In this paper a self-tuning algorithm is introduced for insulin dosage adjustment in TIDM patients. Two types of insulin: rapid and slow are considered into the formulation, which are applied in a daily multi-doses scheme. The doses are updated in order to regulate the BGL into euglycemics. The tuning algorithm presents a error correction where a momentum term
International Simposium on Robotics and Automation 24 August 25-27, 24 Doses (U) Plasma Concentration (mu/l) 15 1 5 2 4 6 8 1 12 8 6 4 2 2 4 6 8 1 12 NPH Lispro NPH+Lispro Fig. 6. Simulation for Case 1: (Top) infusions, and (Bottom) plasma concentration. Plasma (mg/dl) Doses (U) 2 15 1 5 2 4 6 8 1 12 15 1 5 NGL NPH Lispro 2 4 6 8 1 12 Fig. 7. Simulation Case 2: (Top) Blood glucose level, and (Bottom) infusions. is used to speed up the convergence. Moreover, the tuning does not need information of the insulin-glucose dynamics (model free), and it is simple to implement. Therefore, it has the potential of being implemented in a microcomputer for home treatment. Besides, it can be adapted to another combination of insulin, or another route of application to the patient, like inhaled insulin. Tthe approach and methodology introduced could also be visualized as a alternative for Type II diabetic patients, and it could be a valuable tool for educational purposes. REFERENCES [1] R. Belazzi, G. Nucci and G. Cobelli, The Subcutaneous Route to -Dependent Diabetes Therapy, IEEE Engineering in Medicine and Biology, Vol. 2, pp. 54-64, January/February 21. [2] M. Berger and D. Rodbard, Computer Simulation of Plasma and Dynamics after Subcutaneous Injection, Diabetes Care, Vol. 12, pp. 725-736, 1989. [3] D.U. Campos-Delgado, R. Femat, E. Ruiz-Velázquez and A. Gordillo- Moscoso, Knowledge-Based Controllers for Blood Regulation in Type I Diabetic Patients by Subcutaneous Route, Proceedings of the International Symposium on Intelligent Control, Houston, 3-5 October, 23. [4] E.R. Carson and T. Deutsch, A Spectrum of Approaches for Controlling Diabetes, IEEE Control System Magazine, Vol. 12, No. 6, pp. 25-31, December 1992. [5] L.M. Dickerson, Therapy in the Treatment of Diabetes Mellitus, http://www.musc.edu/dfm/pharmd/med%2resident/insulin.pdf, 1999. [6] F.J. Doyle III, B. Srinivasan and D. Bonvin, Run-to-Run Control Strategy for Diabetes Management, Proceedings of the 23rd Annual EMBS International Conference, Istanbul, Turkey, October 25-28, 21. [7] M.E. Fisher, A Semiclosed-Loop Algorithm for the Control of Blood Levels in Diabetics, IEEE Transactions on Biomedical Engineering, Vol. 38, No. 1, pp. 57-61, January 1991. [8] K.H. Kienitz and T. Yoneyama, A Robust Controller for Pumps Based on H Theory, IEEE Transactions on Biomedical Engineering, Vol. 4, No. 11, pp. 1133-1137, 1993. [9] E.D. Lehmann and T. Deutsch, A Physiological Model of - in Type I Diabetes Mellitus, Journal of Biomedical Engineering, Vol. 14, pp. 235-242, 1992. [1] S.M. Lynch and B.W. Bequette, Model Predictive Control of Blood in Type I Diabetics Using Subcutaneous Measurements, Proceedings of the American Control Conference, Anchorage, AK, May 8-1, 22. [11] R.S. Parker, F.J. Doyle III and N.A. Peppas, A Model-Based Algorithm for Blood Control in Type I Diabetes Patients, IEEE Transactions on Biomedical Engineering, Vol. 46, No. 2, pp. 148-156, 1999. [12] R.S. Parker, F.J. Doyle III, J.H. Ward and N.A. Peppas, Robust H Control in Diabetes Using a Physiological Model, AIChE Journal, Vol. 46, No. 12, 2537-2549, 2. [13] R.S. Parker, F.J. Doyle III and N.A. Peppas, The Intravenous Route to Blood Control, IEEE Engineering in Medicine and Biology, Vol. 2, pp. 65-73, January/February 21. [14] W.R. Puckett, Dynamic Modelling of Diabetes Mellitus, Ph.D. Dissertation, Chemical Engineering Department, University of Wisconsion- Madison, June 1992. [15] E. Ruiz-Velazquez, R. Femat and D.U. Campos-Delgado, Blood Control for Type I Diabetes Mellitus: A Robust Tracking H Problem, To Apper in Control Engineering Practice, 24. [16] T. Shimauchi, N. Kugai, N. Nagata, and O. Takatani, Microcomputer- Aided Dose Determination in Intesified Conventional Therapy, IEEE Transactions on Biomedical Engineering, Vol. 35, No. 2, pp. 167-171, February 1988. [17] J.T. Sorensen, A Physiologic Model of Metabolism in Man and its Use to Design and Assess Improved Therapies for Diabetes, Ph.D. Dissertation, Chemical Engineering Department, MIT, Cambridge, 1985. [18] J.A. Tamada, M. Lesho and M.J. Tierney, Keeping Watch on, IEEE Spectrum, Vol. 39, No. 4, pp. 52-57, April 22. [19] Z. Trajanoski and P. Wach, Neural Predictive Controller for Delivery Using the Subcutaneous Route, IEEE Transactions on Biomedical Engineering, Vol. 45, No. 9, pp. 1122-1134, 1998. [2] V. Tresp, T. Briegel and J. Moody, Neural-Network Models for the Blood Metabolism of a Diabetic, IEEE Transaction on Neural Networks, Vol. 1, No. 5, pp. 124-1213, 1999.