Glucose control by subcutaneous insulin administration: a DDE modelling approach

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1 Glucose control by subcutaneous insulin administration: a DDE modelling approach P. Palumbo P. Pepe S. Panunzi A. De Gaetano Istituto di Analisi dei Sistemi ed Informatica A. Ruberti, Consiglio Nazionale delle Ricerche IASI-CNR, BioMatLab - UCSC - Largo A. Gemelli 8, 0068 Roma, Italy, s: pasquale.palumbo@iasi.cnr.it, simona.panunzi@biomatematica.it, andrea.degaetano@biomatematica.it Dipartimento di Ingegneria Elettrica e dell Informazione, Poggio di Roio, L Aquila, Italy, pierdomenico.pepe@univaq.it Abstract: This note investigates the problem of plasma glucose regulation by means of subcutaneous insulin administration. A discrete delay differential equation model of the glucoseinsulin regulatory system has been considered, which properly takes into account also the pancreatic insulin release, in such a way to allow insulin therapies for both Type I and Type II diabetes in this latter case the endogenous insulin delivery is not negligible. The method of exact input/output feedback linearization and stabilization is used, in order to ensure the local convergence of the tracking error to zero. Simulations are performed in a virtual environment, and numerical results show the effectiveness of the proposed approach. IFAC Keywords: Glucose-Insulin System. Nonlinear Control. Time-Delay systems.. INTRODUCTION Diabetes is a major chronic disease affecting around 5 to 6% of the world population, with a heavy impact on national public health budgets. The term diabetes comprises a group of metabolic disorders characterized by hyperglycemia resulting from defects in insulin secretion, insulin action, or both. In one category Type diabetes, there is an absolute deficiency of insulin secretion caused by an autoimmune pathologic process occurring in the pancreatic islets. Individuals with this extensive beta-cell destruction, and therefore no residual insulin secretion, require insulin for survival. In the other, much more prevalent category Type 2 diabetes, the cause is a combination of resistance to insulin action and inadequate compensatory insulin secretory response. These individuals have therefore insulin resistance and usually have relative rather than absolute insulin deficiency, in the face of increased levels of circulating insulin. Exogenous insulin administration is a basic procedure to cope with any malfunctioning of the endogenous insulin feedback action in Type diabetes only exogenous insulin is available, while in Type 2 exogenous insulin complements pancreatic production. Glucose control strategies are mainly actuated by subcutaneous or intravenous injections or infusions. Control of glycemia by means of subcutaneous insulin injections, with the dose adjusted on the basis of capillary plasma glucose concentration measurements, is by far more widespread than control by means of intravenous insulin, since the dose is habitually administered by the patients themselves see Bellazzi et al. 200 and references therein. However, in order to design closed-loop control strategies, the insulin absorption from the subcutaneous depot needs also to be considered. A closed loop control strategy may be implemented according to a model-less or to a model-based approach. The first approach does not use a mathematical model of the glucose-insulin system, and provides an arbitrary while possibly very effective control rule for insulin infusion rate, based on experimental data: recent papers on this topic are mainly devoted to the application of PID controllers aiming to mimic the pancreatic glucose response see e.g. Chee et al. 2003; Steil et al. 2004; Gopakumaran et al. 2005; Marchetti et al On the other hand a model-based approach presupposes sufficiently detailed knowledge of the physiology of the system under investigation. The advantages of a model-based approach are evident since, by using a glucose/insulin model, the control problem may be treated mathematically and optimal strategies may be determined. Clearly, the more accurate the model, the more effective is the control law. Different approaches have been proposed, recently, based on nonlinear models such as the Minimal Model of Bergman et al. 979 and Toffolo et al. 980, or more exhaustive compartmental models like the ones of Cobelli et al. 982, of Sorensen et al. 982 and of Hovorka et al see, e.g., papers on Model Predictive Control, Parker et al. 999, on nonlinear Model Predictive Control, Hovorka et al. 2004, on Parametric Programming, Dua et al. 2006, on Neural Predictive Control, Trajanoski et al. 998, on H control, Parker et al. 2000, on non-standard H control, Chee et al and Ruiz-Velázquez et al It has to be stressed that most of these approaches are based on the Copyright by the International Federation of Automatic Control IFAC 47

2 approximation of the original nonlinear model, provided by linearization, discretization and model reduction balanced truncation. An excellent review of the available models presently adopted for blood glucose regulation as well as the closed loop control methodologies and technical devices blood glucose sensors and insulin pumps may be found in Chee et al In the present work, a model-based closed-loop control scheme is proposed. Differently from previously mentioned model-based approaches, which use nonlinear Ordinary Differential Equation ODE models, the one presented here uses a nonlinear Delay Differential Equation DDE model to describe the glucose/insulin regulatory system, reference to Panunzi et al and Palumbo et al Despite the great spread of DDE models in the last decade, which allow a better representation of pancreatic Insulin Delivery Rate IDR see e.g. Makroglou et al and references therein, their use is still lacking in the field of glucose control, according to the authors knowledge. First attempts have been proposed in Palumbo et al. 2009a,b, where an intra-venous insulin administration was designed to track a desired plasma glycemia, by means of a DDE model-based approach DDE model taken from Panunzi et al In this paper we achieve the same goal of tracking a desired glucose reference, this time by means of subcutaneous instead of intra-venous infusions. To this aim, the above mentioned model of the glucose-insulin regulatory system has been coupled to a linear model of the subcutaneous insulin absorption see references of Nucci et al. 2000; Wilinska et al. 2005; Li et al to have a comprehensive review of the many different models of insulin absorption. Note that when attempting to design a model-based glucose control, the works published so far have concentrated on Type diabetic patients who have essentially no endogenous insulin production and are very well described by suitably exploiting ODE models, avoiding in this way the need to take pancreatic IDR into account. In the present work we do take into account spontaneous pancreatic IDR, thereby treating healthy, Type 2 diabetic and Type diabetic patients in a unified fashion. As far as the proposed control law, it is based on recent results on differential geometry for time-delay systems see Germani et al. 2000, Germani et al. 2003, Marquez- Martinez et al. 2004, and Oguchi et al An exactly linearized input/output map is first obtained by a nonlinear inner feedback which makes use of the state variables at present and delayed time. Then, the tracking of the output the blood glucose concentration is achieved by means of an outer feedback on the linear input/output map. The control law is obtained without linearizing, by first order approximations, the system equations: this way, the control law here provided is meant to work also in case of large deviations from the desired final level, and not only for small deviations. The paper is organized as follows: next section deals with some preliminaries, including a short description of the adopted DDE model; section III is devoted to develop the main results, detailing on the theoretical properties concerning the proposed control algorithm. Simulations are reported in section IV. Conclusions follow. 2. THE DDE GLUCOSE-INSULIN MODEL Denote Gt, [mm], It, [pm], plasma glycemia and insulinemia, respectively, and S [pmol], S 2 [pmol] the insulin mass in the accessible and not-accessible subcutaneous depot, respectively. The model considered consists of a single discrete-delay differential equation system: dg dt = K xgigtit + T gh, di dt = K xiit f Gt τ g +, t ds 2 = S t, dt t t ds dt = t S t + ut where K xgi, [min pm ], is the rate of glucose uptake by tissues insulin-dependent per pm of plasma insulin concentration; T gh, [min mmol/kgbw], is the net balance between hepatic glucose output and insulin-independent zero-order glucose tissue uptake mainly by the brain;, [L/kgBW], is the apparent distribution volume for glucose; K xi, [min ], is the apparent first-order disappearance rate constant for insulin; T igmax, [min pmol/kgbw], is the maximal rate of second-phase insulin release;, [L/kgBW], is the apparent distribution volume for insulin; τ g, [min], is the apparent delay with which the pancreas varies secondary insulin release in response to varying plasma glucose concentrations; t, [min], is the timeto-maximum insulin absorption; ut, [pm/min], is the subcutaneous insulin delivery rate, i.e. the control input. The nonlinear function f models the pancreas Insulin Delivery Rate as: G γ G fg = + G γ, 2 G where γ is the progressivity with which the pancreas reacts to circulating glucose concentrations and G [mm] is the glycemia at which the insulin release is half of its maximal rate. In case of no exogenous insulin input, by neglecting the insulin dynamics in the subcutaneous depot, model -2 reduces to a basic glucose-insulin regulatory system, and belongs to the family of DDE models described in Palumbo et al see the reference for more details on the qualitative behavior. It represents equally well healthy subjects and insulin-resistant or severe diabetic patients, changing the parameter values as appropriate. Moreover, it does belong to the class of minimal models, in the sense that according to a minimal set of independent parameters, it allows to very well resemble the physiology of the glucose/insulin kinetics, and is perfectly identifiable from data according to standard perturbation experiments IVGTT see Panunzi et al The model of insulin absorption here adopted third and fourth eq.s in refers to Puckett et al. 995 with no insulin degradation at the injection site. It has been recently analyzed in the papers of Wilinska et al and Clausen et al. 2006, and it has been exploited 472

3 with the aim of glucose control in Hovorka et al. 2004, according to which, here we assume the same notation. 3. NONLINEAR CONTROL We make use here of the elementary theory of nonlinear feedback for time-delay systems see Germani et al. 2000; Germani et al. 2003; Marquez-Martinez et al. 2004; Oguchi et al Let G ref t be the desired glucose reference signal to be tracked, which we assume to be smooth and bounded. Let x t Gt x xt = 2 t It x 3 t =, yt = Gt G ref t, x 4 t S t Gt G ref t d z t yt z zt = 2 t z 3 t = y dt Gt G reft t y 2 t = d 2 z 4 t y 3 t dt 2 Gt G reft d 3 dt 3 Gt G reft Lemma. The state variable zt defined in 3 obeys the following equation: żt = A b zt + B b α K xgi t 2 Gtut 4 with A b and B b the Brunowski pair: A b = 0 0 0, B b = and function α given by: α = K xgi K xgi GtIt + T gh KxgiI 2 3 t + 3K xgi K xi I 2 t 3K xgit igmax ItfGt τ g 3K xgi t It +KxiIt 2 K xit igmax fgt τ g K xi + t t f Gt τ g Ġt τ g + t 2 K xgi K xi It f Gt τ g + t S t 3KxgiGtI 2 2 t 2K xgit gh It +6K xgi K xi GtIt 3K xgit igmax GtfGt τ g 3K xgi Gt + K 2 t xigt K xgi f Gt τ g Ġt τ g 3K xgit igmax GtIt + 2T ght igmax K xit igmax Gt K xgi S t t t 3K xgi GtIt K xi + Gt t t t K xgi TiGmax Gtf Gt τ g Ġ 2 t τ g Gtf Gt τ g Gt τg t 3 GtS t G 4 ref t 6 Proof. Consider the derivatives of the components of zt. By using equations, we have the following results. ż t = z 2 t = K xgi GtIt + T gh Ġreft 7 ż 2 t = z 3 t = K xgi K xgi GtI 2 t + T gh It K xi GtIt GtfGt τ g V I + Gt t G ref t 8 ż 3 t = z 4 t = K xgi KxgiGtI 2 3 t K xgit gh I 2 t +3K xgi K xi GtI 2 t 3K xgit igmax GtItfGt τ g 3K xgi GtIt t +2 T ght igmax 2T gh fgt τ g + t +KxiGtIt 2 K xit igmax GtfGt τ g K xi + Gt t t Gtf Gt τ g Ġt τ g + t 2 GtS t G 3 ref t 9 Finally, by computing the derivative of z 4 t it results: K xgi ż 4 t = α t 2 Gtut. 0 from which, the Lemma is proven. 473

4 Remark 2. Note that from, for t τ g, it is: Ġt τ g = K xgi Gt τ g It τ g + T gh, and Gt τ g = K xgi It τ g K xgi Gt τ g It τ g + T gh K xgi Gt τ g K xi It τ g f Gt 2τ g + S 2 t τ g 2 t On the other hand, it can be assumed without loss of generality that, for θ [ τ g, 0], it is Ġθ = 0 and, as well, Gθ = 0. Indeed, such assumption would mean a patient at rest steady state solution for plasma glycemia before the controller is applied. According to Lemma, the inner feedback control law α + vt ut = K, 3 xgi t 2 Gt where vt is a new outer input, yields the following linear equation żt = A b zt + B b vt. 4 Finally, by choosing the new input vt as the outer feedback vt = Γzt, 5 with Γ a suitable row vector in IR 4, the following equation is obtained, żt = A b + B b Γzt 6 Thus, by designing Γ such that A b + B b Γ is Hurwitz this is possible since A b, B b is a controllable pair, we get that zt goes to zero exponentially, which returns the glucose to converge to the desired reference signal exponentially. From a mathematical point of view, the control law 3, 5 can always be computed, since the variable Gt at the denominator of 3 never vanishes: indeed, as it is required from basic assumptions on the qualitative behavior of the solutions, the glucose dynamics is strictly positive whatever are chosen the initial conditions in the positive orthant see Palumbo et al It follows that, from a mathematical point of view, the control law 3, 5 as well as equation 6 can be used with any physically meaningful initial conditions. Remark 3. In general, in the application of the elementary theory of nonlinear feedback for systems with time-delays, further dynamics, given by continuous time difference equations, must be taken into account see Germani et al. 2000; Germani et al. 2003, even if the relative degree is full as in our case. In this case, no unstable zero dynamics occur, since the relationship between the variable xt and the variable zt does not involve any further dynamics. 4. SIMULATIONS Simulations have been carried out on a virtual patient on the basis of parameter estimates obtained from data related to an IVGTT experiment conducted on an obese patient Body Mass Index 50, studied at the Catholic University of Rome, Department of Metabolic Diseases, Panunzi et al Below the estimated values are reported in terms of their original scale they refer to the glucose-insulin regulatory system: G b = 5.6 γ = =0.87 = 0.25 I b = G = 9 K xi = K xgi = T igmax =.573 τ g = 24 T gh = Parameter t = 55 is taken from Hovorka et al These parameters show high-normal glycemia G b = 5.6 and a substantial degree of insulin resistance K xgi 0 4. This picture moderate hyperglycemia, obesity, insulin resistance is consistent with the picture of a pre-diabetic patient, whose long-standing obesity has induced such a state of insulin resistance for such a long time that pancreatic glucose toxicity is apparent and insulin delivery which should be above normal to compensate for increased insulin resistance is progressively failing. This subject would be expected to develop frank Type 2 Diabetes Mellitus within a relatively short time, unless therapeutic maneuvers first of all weight loss are vigorously employed. We allow for a certain length of time one or two years, say to have gone by without any effective therapy. In this case, the natural progression of disease has determined the failure of pancreatic insulin secretion and, in the face of unchanged insulin resistance, a dropping insulin concentration. This in turn determines the emergence of severe hyperglycemia and the establishment of a state of frank Type 2 Diabetes Mellitus. Therefore, the pancreatic glucose sensitivity T igmax is reduced to 5% of its normal value, consequently determining new values of the G b and I b parameters: T igmax = 0.236, G b = 0.66, I b = Parameters 7, with the substitution of the first line with 8, identify our virtual diabetic patient. In order to regulate the resulting hyperglycemia down to a safe level, we choose matrix Γ such that the closed loop matrix A + BΓ has eigenvalues 0.027, 0.028, 0.029, The reference signal is chosen such to obtain the plasma glycemia decreasing exponentially from the value of 0.66 to the new value 5.0: G ref t = exp 0.0t. 9 The subject is supposed to be at rest before the experiment begins, which means that the initial state is given by G 0 τ = G b, I 0 τ = I b for τ [ τ g, 0]. Initial conditions for the subcutaneous depot are S 0 = 0 and S 2 0 = 0. As it clearly appears from Fig., a reasonably low plasma glycemia i.e. < 6mM is reached within the first four hours of simulation. 474

5 Fig.. Plasma glycemia [mm], compared with the desired glucose reference; time is in hours Fig.2 shows the plasma insulin concentration, compared to the reference plasma insulinemia obtained if Gt G ref t, that is: T gh V g Ġreft I ref t = 20 K xgi G ref t Fig. 2. Plasma insulinemia [pm] compared with the desired insulin reference; time is in hours Fig.3 reports the insulin infusion rate to be administered, according to the proposed control law. Fig. 3. Insulin infusion [pmol/min]; time is in hours Remark 4. While the control input cannot be negative, this paper does not consider, from a theoretical point of view, saturation problems for the control law. We have taken into account this fact in the simulations: whenever the designed control law becomes negative, a zero control input is given to the system. Note that we have chosen the control parameters in order not to allow such a drawback, as it appears from Fig.3. Remark 5. As a final remark, we point out that the present paper is a starting point for further, and more significant results. Indeed, the proposed control law is based on the complete knowledge of the state vector, which means available measurements from both glycemia and insulinemia, as well as from the subcutaneous depot. Of course, such assumptions are far to be reliable, especially for what concerns the subcutaneous depot. The next step, which is a work in progress of the authors, will be to design the feedback control by using only plasma glucose measurements, by suitably exploiting a state observer for time-delay systems, as it has already been presented in Palumbo et al. 2009b for the case of intra-venous insulin administration. 5. CONCLUSIONS In this paper we have found a state feedback nonlinear control law for the glucose-insulin system. Asymptotic tracking of a desired time evolution for the blood glucose concentration is achieved by means of this nonlinear control law. The control input consists of subcutaneous insulin administration. The method of exact input/output feedback linearization for nonlinear time-delay systems is used. No linear approximations are used. Simulations have been reported, showing the high performance of the proposed control law. REFERENCES R. Bellazzi, G. Nucci and C. Cobelli, The subcutaneous route to insulin-dependent diabetes therapy, IEEE Engineering in Medicine and Biology, 20,54 64, 200. R.N. Bergman, Y.Z. Ider, C.R. Bowden and C. Cobelli, Quantitative estimation of Insulin sensitivity, Am. Journal on Physiology, 236, , 979. F. Chee and T. Fernando, Closed-loop control of blood glucose. Berlin Heidelberg, Springer-Verlag, F. Chee, T.L. Fernando, A.V. Savkin and V. van Heeden, Expert PID control system for blood glucose control in critically ill patients, IEEE Trans. Inf. Tech. in Biomedicine, 7,49 425, F. Chee, A.V. Savkin, T.L. Fernando, S. Nahavandi, Optimal H insulin injection control for blood glucose regulation in diabetic patients, IEEE Trans. on Biomedical Engineering, 52,625 63, W.H.O. Clausen, A. De Gaetano and A. Vølund, Withinpatient variation of the pharmacokinetics of subcutaneously injected biphasic insulin aspart as assessed by compartmental modelling, Diabetologia, 49, , C. Cobelli, G. Federspil, G. Pacini, A. Salvan and C. Scandellari, An integrated mathematical model of the dynamics of blood glucose and its hormonal control, Math. Biosci., 58,27 60, 982. P. Dua, P, F.J. Doyle, E.N. Pistikopoulos, Model-based blood glucose control for Type diabetes via parametric programming, IEEE Transactions on Biomedical Engineering, ,

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