Glucose-Insulin System based on Minimal Model: a Realistic Approach

2015 17th UKSIM-AMSS International Conference on Modelling and Simulation Glucose-Insulin System based on Minimal Model: a Realistic Approach Adriana Aguilera González, Holger Voos Interdisciplinary Centre for Security, Reliability and Trust (SnT) - University of Luxembourg L-1359, Luxembourg-Kirchberg, Luxembourg Email: adriana.aguilera/holger.voos@uni.lu Mohamed Darouach CRAN-CNRS UMR 7039 Université de Lorraine, IUT de Longwy 54400, Cosnes et Romain, France Email: mohamed.darouach@univ-lorraine.fr Abstract This paper presents a new approach to represent the glucose and insulin levels on patients with diabetes type 1 based on the well-known minimal model. In this work a new mathematical model is proposed which attempts modeling properly the malfunction of the pancreas. The model takes into account both: the description of the glucose level in the subcutaneous layer and a meal disturbance term. Further, also has an additional term that represents the insulin in a more realistic approach. In silico simulations are used for first evaluations of the proposed model effectiveness, considering variations on quantity of meals and of insulin supplied. Keywords glucose-insulin system; diabetes type 1; minimal model augmented I. INTRODUCTION Type 1 diabetes is a chronic and autoimmune disease. According to the World Health Organization in 2012, diabetes was the direct cause of 1.5 million deaths worldwide, and additionally in 2014, 9% of adults 18 years and older had diabetes. Type 1 diabetes is characterized by deficient or null insulin natural production, which causes deviations of the glucose concentration level in the bloodstream from normal values. Significant and prolonged deviations from the normal level can lead to numerous and serious pathologies as: adult blindness, end-stage renal failure, retinopathy, neuropathy, and lower-limb amputations. Therefore, diabetes represents a major threat to public health with alarmingly rising trends of incidence and severity in recent years; which motivates communities of control engineering, bio-engineering and medicine to continue efforts to improve engineering quality of life of patients. The current standard treatment procedure of diabetes primarily involves insulin medication coupled with strict dietary control. The insulin hormone is administered through discrete insulin injections, or through continuous subcutaneous infusion via an insulin pump. Discrete insulin injections is an open-loop process that is not an ideal treatment of diabetes, because several unknown factors exists which directly influence the level of glucose (meals, sport, mood, etc.). Continuous insulin infusion through a programmable insulin pump, on the other hand, offers a potential for a closed-loop control to regulate glucose. These new means offer the possibility to regulate the infusion rate in an individual form for each person. However, interpersonal variations are a big problem to regulate the level of glucose and make it difficult to determine an insulin dose for each particular patient. Some of these difficulties are: the variability of overnight insulin requirements, episodes of hyperglycemia after nocturnal hypoglycemia, variability of subcutaneous absorption of insulin or a defective counter-regulation of glucose. Therefore, closed loop control techniques are developed in order to maintain a normal physiological glucose level [1]. Human bodies need to maintain a glucose concentration level in a narrow range (70 110mg/dl or 3.9 6.04mmol/l after overnight fast). If the glucose concentration level is significantly out of the normal range, the person is considered to have a plasma glucose problem: hyperglycemia or hypoglycemia. The modelling of the glucose-insulin system has become an interesting topic and several models have been proposed towards a better understanding of the metabolic system, investigating possible pathways to control the diabetes as well as providing better insulin administration practices. In order to predict blood glucose concentrations, a model must incorporate such information as: 1) the effect of the blood glucose concentration and changes in the blood glucose concentration on beta cell insulin release; 2) the effect of exogenous insulin delivery into the subcutaneous or intravenous compartments on blood insulin levels; 3) the effect of food intake on glucose appearance into the circulation; and 4) the effect of a change in glucose appearance into the bloodstream combined with specified insulin levels on blood glucose concentrations [2]. In general, mathematical models have been used to estimate the glucose disappearance and insulin sensitivity which are used to study relative dependencies of insulin infusions. Different types of models which have been used in the literature can be classified mathematically as: ordinary differential equations [3], [4], [5], delay differential equations [6], [7], partial differential equations [8], [9], stochastic differential 978-1-4799-8713-9/15 $31.00 2015 IEEE DOI 10.1109/UKSim.2015.65 55

equations [10], [11] and integro-differential equations [12], [13]. Also, different software packages can be used for different types of models for numerical analysis and simulations. The so-called minimal model, is the model currently mostly used in physiological research on the metabolism of glucose and the insulin regulation. This model was proposed in the early eighties by [4] for the interpretation of the glucose and insulin plasma concentrations following the intravenous glucose tolerance test (IVGTT). The minimal model is commonly used to analyze the results of glucose tolerance tests in humans and laboratory animals. For instance, a model predictive controller (MPC) using the minimal model was implemented by authors in [14], in order to return the blood glucose to normoglycemic ranges when subjected to a meal disturbance. In the same way, [15] propose a mathematical model which takes into account all plasma glucose concentration, generalized insulin and plasma insulin concentration. This model shows a long-term diabetes progression but not considers the meals disturbance. Similarly, in [16] a mathematical model for describing the glucose infusion rate was introduced based on the minimal model. A glucose-insulin system was generated to obtain the real optimized model parameters, from the experimental data using the modified minimal model. Equality important is the parameter estimation, for this, the authors in [17] present a novel approach for guaranteed the parameter estimation of the minimal model based on the Set Inversion via Interval Analysis (SIVIA) algorithm. In this work, the proposed model takes into account three main conditions that must be considered for successful treatment of the disease. These three conditions are: i). The description of the glucose level in the subcutaneous layer, which it is very important because the measurements of glucose level and the insulin injections via a insulin pump, are made subcutaneously. ii). A meal disturbance term it is considered into the model because is a variable that have an important influence on the system, causing deviations from the normal glucose level. iii). A term that allows to represent the insulin dose considering the type, which it is very important factor due that the insulin is classified by how fast they start to work and how long their effects last. Consequently, to consider these aspects our model becomes in an approach more realistic. The paper is organized as follows. The minimal model and its basic properties are described in Section II. In Section III some extensions are introduced that are necessary in order to adapt the model in a better way to reality. In the Section IV simulation results are presented and discussed. Section V provides the conclusions and comments. II. MINIMAL MODEL: PRELIMINARS The origin of the minimal model began with the idea to construct a mathematical model of the pancreatic islets, which was a particularly difficult problem, due to the fact that beta-cell have a complex behavior, with a response that is altered by previous history of stimulation and a non-linear dose response curve. Bergman et. al. in [4], via intravenous glucose tolerance test (IVGTT) modified, achieved revealing the complex dynamic relationship between plasma glucose and insulin and they concluded that the resulting data can be described by a stimulus-response (input-output) model of the extra pancreatic tissues that utilize glucose. Such model it is considered as being simple enough to account totally for the measured glucose (given the insulin input), however allows, using mathematical techniques, to estimate all the characteristic parameters of the model from a single data set (thus avoiding unverifiable assumptions) [18]. Figure 1. Minimal model of glucose disappearance [4] Figure 2. Minimal model of insulin kinetics [4] Glucose minimal model involves two physiological compartments: a glucose compartment, whereas plasma insulin is assumed to act through a remote compartment to influence net glucose uptake (see Figs. 1-2). The minimal model attempts to describe the dynamics of the system as simply as possible, dividing the model in two parts: the Eqs. (1-56

Figure 3. Blood glucose level using the minimal model for a normal patient. Figure 4. Blood insulin level using the minimal model for a normal patient. 2) which describe the glucose plasma concentration timecourse treating insulin plasma concentration as a known forcing function (it is represented by Fig. 1). These equations are: dg dt = p 1G X(G + G b ) (1) dx dt = p 2X + p 3 I (2) where G is the blood glucose concentration, X is the effect of active insulin, I is the plasma insulin concentration, G b and I b are the basal values of glucose and insulin concentration, p 1 is the glucose clearance rate independent of insulin, p 2 is the rate of clearance of active insulin (decrease of uptake) and p 3 increase in uptake ability caused by insulin. Eq. (3) consists of a single equation, which describes the time course of plasma insulin concentration treating glucose plasma concentration as a known forcing function (it is represented by Fig. 2). di dt = n[i Ib]+γ(G h)+ t (3) where n is the fractional disappearance rate of insulin, γ is the rate of pancreatic release after glucose bolus, h is the target glucose level and t represents the time interval from the glucose injection, (values and units of this variables can be seen in the Appendix, Table I). This function was firstly presented by [4] and adjusted by [12]. Only the positive part of the term (G(t) h) is taken, otherwise the value is zero. Therefore h is considered as a threshold level to decide when the pancreas should produce more insulin and when to stop, finally the difference between G(t)h determines how much it should be produced. The reason for the multiplication with t is because the pancreas response is proportional not only to the hyperglycemia attained but also to the time elapsed from the glucose stimulus. Figs. 3-4 shown the glucose and insulin levels of a normal patient. It is possible to see that the glucose level reaches its basal level at 60 minutes approximately, and so does insulin, too. This model is useful to represent a normal patient, since it describes the pancreas as the source of insulin. In a healthy person a small amount of insulin is always created or cleared. This helps to keep the insulin basal concentration I b. If the insulin level is above basal concentration the clearance increases, if the insulin level is below basal concentration the basal production increases. When the glucose level gets high the pancreas reacts by releasing more insulin at a certain rate. The minimal model is still classified as a very simple model of the dynamics of the interaction of blood glucose and insulin, it retains some compatibility with known physiological facts and has been validated in a number of clinical studies [19]. However, to use this model for a diabetic patient Type 1, it is necessary to introduce some changes into the minimal model. Some variables and functions will be added in order to have a more approximate representation of reality and daily conditions of the patient. III. MINIMAL MODEL ADAPTED TO TYPE 1DIABETIC PATIENT It is possible to increase the functionality of the glucose minimal model, thus it could be used to simulate more than an IVGTT, for which some additions could be done. The first of these additions corresponds to a function describing what a meal would do to the glucose level. This is done by adding a meal disturbance term called D(t) to Eq. (1). To represent a person in a diabetic state, the original minimal model is described by the following differential equations: dg dt = p 1G X(G G b )+D(t) (4) dx dt = p 2X + p 3 I (5) where the parameters p 1, p 2, p 3, and V G should be estimated via mathematical methods. D(t) represents the 57

meal disturbance that can be normalized to the subject s body weight m BW and the glucose distribution volume V G for do it explicit the appearance of exogenous glucose into the minimal model [20], [21], [22]. D(t) can be obtained as follow: D(t) = R abs (6) m BW V G where R abs is the rate of absorption (mg/min) and is simply scaled by the glucose distribution volume V G and for the body weight m BW, in order to determine the glucose infusion rate per volume. This model can be adapted to change basal blood glucose concentrations and to body mass, where the parameters are previously known, i.e., can be adapted to each patient [23]. This process of meal absorption needs a description, which was suggested by [19] as follows: R abs (t) =B e (drate t) (7) The author suggested that the meal absorption description should be a function which rapidly increases after the meal, and then decays to 0 in 2-3 hours. Like the glucose model, the insulin model is used to interpret the IVGTT. That simplifies any interpretation of the model, however complicates the description of the insulin kinetics for a type 1 diabetic person with no endogenous insulin production. Therefore, it is possible to modify this condition by exchanging the incoming part representing the pancreas by a function U(t) that describes exogenous or endogenous insulin infusion [14]. This modification is presented as follow: di dt = n(i I b)+ U I(t) (8) V I where endogenous insulin secretion (i.e. the term σ(g(t)h)t) in Eq. (3) was removed, and the terms of exogenous infusion of insulin U I (t) was added. V I is the distribution volume of insulin in blood that should be estimated. The exogenous infusion of insulin (U I (t)) can be modeled as the rate of insulin absorption after a subcutaneous insulin injection according to authors in [24], as follow: U I (t) = s it si T si 50 D I t[t si 50 + ] tsi 2 (9) where t is the time elapsed from the injection (t s means t raised to the power s i ), T 50 is the time at which 50% of the insulin dose D I has been absorbed and s i is a parameter which defines the insulin absorption pattern depending of types of insulin (regular, intermediate, slow, etc.). The linear dependency of T 50 on dose is defined as: T 50 = a i D I + b i (10) where a i and b i are preparation-specific parameters the values of which are given in [24] along with values for s i. In order to make the proposed model more realistic, another addition which could be done is a description of the glucose level in the subcutaneous layer. A patient with type 1 diabetes, usually test their blood sugar frequently (3 to 10 times per day), both to assess the effectiveness of their prior insulin dose and to help determine their next insulin dose. For this reason, it is more easy to do the blood glucose concentration monitoring through the subcutaneous layer measurements and not via intravenous, in order to avoid invasive techniques. For the purpose of simplifying any later comparison with measured data, the function G sc (t) is introduced, which describes the glucose concentration in the subcutaneous layer. A differential equation describing this behavior is introduced with the following function: dg sc dt = G(t) G sc 5R utl G sc (0) = G(0) 5R utl (11) This equation models a first-order delay (5 minutes) between the blood glucose concentration and the subcutaneous glucose concentration. The R utl, is the rate of utilization, which is the difference between the two concentrations in the steady state [25]. One of the major problems concerning creating a artificial pancreas, is the delay between these two concentrations. IV. RESULTS AND DISCUSSION In Fig. 5 the result of three meals of different sizes are given, using the meal disturbance function given by Eq. (7) and values of B =3,B =5and B =12. These figures show that the glucose level rises and then decays inside a 1.5 hours period. This fast clearance of glucose is most likely due to the value parameters estimated by [14]. So p 1 should have a lower value, which would slow down the decay. But this example was just to show that the model reacts to the meal disturbance Eqs. (6-7). Figure 5. Subcutaneous and blood glucose level at different meal disturbance values and zero insulin. In Fig. 6 a day s scenario (24 hours) is presented including three meals: breakfast at 7:00a.m., lunch at 12:30p.m. and dinner at 7:00p.m.. In this figure it can be seen the increase of glucose levels in each meal and return to basal glucose 58

level between two or three hours after. This figure shows that the meal disturbance is an external variable that causes important deviations from the normal glucose level, which will be considered a completely unknown and individual disturbance. Figure 8. Subcutaneous and blood glucose level in 24h, three meals and three insulin injections response. Figure 6. Subcutaneous and blood glucose level in 24h, three meals and zero insulin. Fig. 7 presents 3 injections of size D I = 250mU of regular insulin at the meal times. The larger the dose of regular insulin, the faster the onset of action, but the longer the time to peak effect and the longer the duration of the effect. This figure shows the rate of insulin absorption after three subcutaneous insulin injections for a type 1 diabetic patient, without endogenous insulin production. Figure 7. Subcutaneous insulin level. In Fig. 8 it is shown a comparison between a profile without insulin therapy (as shown in Fig. 6) and a profile with insulin injections. The insulin was supplied at the same time as the meals, and in the three cases the quantity was the same (D I = 250mU). It is possible to see that the glucose level is less in the case of insulin therapy and also, the time to come back a basal glucose level was less. This shows how the model describe an insulin injection effect on the glucose concentration. V. CONCLUSION An analysis about the minimal model of Bergman was presented. This model is considered the simplest and has been useful to identify important issues in systems biology as they relate to carbohydrate metabolism. In this paper, new modifications of the minimal model are provided in order to allow the exploration of new strategies to maintain blood sugar level in normal values. On the glucose-insulin system, usually multiple conditions are found as: insulin delays, physical activity, inaccessible/unmeasurable meal disturbances, etc., which hindering an accurate representation. The proposition presented in this work, is an alternative that seeks to obtain an more approximate model which allows to take into account these conditions. These auxiliary variables and functions result in a more realistic model, useful to develop control algorithms. Evaluating the performance, the proposed model has the possibility to represent the reaction to a meal disturbance and the reaction to a insulin injection or a change in basal insulin delivery. It could be used to give approximations how the glucose-insulin system would react in a certain situation and one of the advantages is that it is very easy to use. The most interesting function of the modified model is the possibility to attach a controller, which controls the glucose level by manipulating the insulin delivery U I. Which at same time allows represent the type of insulin used in the treatment. This feature makes it interesting, because one of the great issues in the diabetes treatment is the search of the best controller acting as an artificial pancreas. ACKNOWLEDGMENT The authors would like to thank the Fonds National de la Recherche (FNR) of Luxembourg for the financial support in the developing of the project I2R-DIR-AFR-090000. REFERENCES [1] D. Radomski, M. Lawrynczuk, M. Marusak and P. Tatjewski. Modeling of glucose concentration dynamics for predictive control of insulin 59

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Parameter Value G b, basal value of glucose (mg/dl) 200 I b, basal value of insulin (mu/l) 15 V G, glucose distribution volume per kg 0.22 body weight (L/kg) V I, insulin distribution volume per kg body 0.1421 weight (L/kg) m BW, body weight (kg) 102.32 p 1, glucose clearance rate independent of 0.028735 insulin (min 1 ) p 2, decreasing level of insulin action with 0.028344 time (min 1 ) p 3, increase in uptake ability caused by 5.035e 5 insulin ((μu/ml) 1 min 2 ) n, decay rate of blood insulin (mu/l) 0.22 γ, rate of pancreatic release after glucose mu dl bolus ( L mg min ) 0.0041 h, pancreatic target glycemia level (mg/dl) 83.7 a i, regular insulin parameter (h/u 1 ) 0.05 b i, regular insulin parameter (h) 1.7 s i, regular insulin parameter 2 R utl, rate of utilization 0.74 60