In this paper, the extension of the Bergman model (minimal model) is
proposed with an internal insulin control (IIC) part, representing the own
insulin control of the human body. The model has been verified with clinical
experiments, by oral glucose intake tests. Employing parameter estimation,
for inverse problem solution technique (SOSI - `single output single
input´) was developed using Chebysev shifted polynomials, and linear
identification in time domain based on measured glucose and insulin
concentration values was applied. The glucose and insulin input functions
have been approximated and the model parameters of IIC were estimated. This
extended Bergman model suits considerably better to the practical clinical
situation, and it can improve the effectivity of the external control design
for glucose-insulin process. The IIC part has been identified via dynamical
neural network using the proposed SOSI technique. The symbolic and numerical
computations were carried out with Mathematica 5.1, and with its application Neural
Networks 2.0

Kovács, Levente

English

Periodica Polytechnica (Budapest University of Technology and Economics)

PERIODICA POLYTECHNICA SER. EL. ENG. VOL. 50, NO. 1–2, PP. 23–32 (2006)EXTENSION OF THE BERGMAN MINIMAL MODEL FOR THEGLUCOSE-INSULIN INTERACTIONLevente KOVÁCSDepartment of Control Engineering and Information TechnologyBudapest University of Technology and EconomicsH–1117 Budapest, Magyar Tudósok krt. 2, Hungarye-mail: lkovacs@seeger.iit.bme.huReceived: April 28, 2005AbstractIn this paper, the extension of the Bergman model (minimal model) is proposed with an internal insulincontrol (IIC) part, representing the own insulin control of the human body. The model has been verifiedwith clinical experiments, by oral glucose intake tests. Employing parameter estimation, for inverseproblem solution technique (SOSI – ‘single output single input’) was developed using Chebysevshifted polynomials, and linear identification in time domain based on measured glucose and insulinconcentration values was applied. The glucose and insulin input functions have been approximatedand the model parameters of IIC were estimated. This extended Bergman model suits considerablybetter to the practical clinical situation, and it can improve the effectivity of the external controldesign for glucose-insulin process. The IIC part has been identified via dynamical neural networkusing the proposed SOSI technique. The symbolic and numerical computations were carried out withMathematica 5.1, and with its application Neural Networks 2.0.Keywords: Glucose–insulin interaction, internal insulin control, identification, dynamic neural net-work, symbolic computation, Mathematica.1. IntroductionNowadays health experts refer to diabetes mellitus as the ‘disease of the future‘. TheWorld Health Organization’s (WHO) newest statistics show that 4% of the adultsociety of the world suffer from diabetes mellitus, and this value could increaseto 5.4% by 2025. Van den Berghe [15, 16] has shown that tight glucose controlcan reduce the Intensive Care Unit (ICU) patient mortality by 45% if the glucoselevel is kept less than 6.1 mmol/L for a cardiac care population. It was shownthat automated control algorithms capable of tight regulation for glucose intolerantICU patients would reduce mortality, as well as the current burden on ICU medicalresources and time.To design an appropriate control, an adequate model is necessary. Duringthe last 50 years, various models for the interaction between glucose and insulinhave been suggested [1, 6, 9, 13, 14], but the benchmark model was the minimalmodel of Bergman [3]. Bergman’s two compartment minimal model explains andevaluates in the simplest way the results of the intravenous glucose tolerance test(IVGTT).24 L. KOVÁCSHowever, the shortcoming is its big sensitive to variance in the parametersand that the plasma insulin concentration must be known as a function of time.Therefore, extensions of this minimal model have been proposed, [4, 10, 11] usingeven control algorithms [5], trying to capture the changes in patient dynamics ofthe glucose-insulin interaction, particularly with respect to insulin sensitivity.Our researches have been focused on the optimization of the amount of insulin.The proposed modified Bergman model has kept the minimal model’s properties[2, 7], and good results have been achieved [8, 12], but the shortcomings of theBergman model remained.This paper proposes an extension of the minimal model, but in a differentway, than the mentioned works have done. We are adding an internal insulincontrol (IIC) part, representing the own insulin control of the human body by usingmostly control theory and less physiological aspects. However, the obtained modelis built by clinical experiments made on oral glucose intake tests, and measuredat the Heim Pál Hospital for Sick Children from Budapest, proving the long-termapplicability of our model.The symbolic and numerical computations were carried out and presented as alive worksheet with Mathematica 5.1, and with its application Neural Networks 2.0.2. Materials and Results2.1. Model EquationsThe original Bergman model consists of two well-known equations, [3]:deq1 = X'[t℄ == p1X[t℄ + p2h[t℄;deq2 = Y'[t℄ == (p3 - X[t℄)Y[t℄ + i[t℄ + p4; (1)with the following parameters, [6]:parameters = {p1 → -0.021151, p2 → 0.092551,p3 → -0.014188, p4 → 0.077947}; (2)The terms h(t) and i(t) stand for exogenous insulin and glucose as inputs of thesystem. X(t) and Y (t) are the concentration of glucose in the plasma and that of theinsulin remote from plasma, respectively. In our caseX(t) and Y (t) represent boththe states and the output of the system, because the dynamic of the measurement andactuator devices are considerably faster than that of the system itself. The constantspi(i = 1, 2,…, 4) are the model parameters.We should like to extend this model in order to take into consideration animportant phenomenon, the internal insulin control (IIC), namely the own insulincontrol of the body. We neither want to develop a new model nor estimate new valuesfor parameters pi of Bergman’s model, just to add a new part, which considerablyimproves its applicability for real clinical situations.Therefore, we tried to extend the minimal model with a third equation:EXTENSION OF THE BERGMAN MINIMAL MODEL FOR THE GLUCOSE-INSULIN INTERACTION 25iint = F(iint,X(t),Y(t),h(t)) (3)where the insulin inlet i(t) in the second equation of (1) is divided into an internaliint (t) and an external part iext(t) (see Fig. 1):i(t) = iint(t) + iext(t) (4)In order to identify our suggested extension, the input and output signals of themodel are needed. The clinical experiments were taken using oral glucose intake(OGI), instead of intravenous injection and only the values of X(t) and Y (t) weremeasured. Therefore, first the signals iint (t) and h(t) should be computed fromthese measurements.Fig. 1. The block diagram of the extended Bergman model.2.2. Theoretical Approach of Oral Glucose IntakeIn case of oral glucose intake – in contrast to intravenous glucose injection – it takestime while the glucose is absorbed in the stomach and the glucose concentrationin the blood increases. The simplest model for this process is the perfectly mixedtank model with first order reaction,V dXdt = -kX. (5)Solving this equation with the initial condition X(0) = X0,X(t) = X0e- kV t (6)Then glucose inlet flux into the blood circuit is proportional with,- dXdt = X0 kVe- kV t (7)26 L. KOVÁCSConsequently, the glucose influx in the Bergman model h(t) can be expressed ash[t_℄ := αe-βt (8)2.3. Estimation of the Parameters α and βNow, the first equation of the Bergman model is:deq1X′TtU == αe−βtαp2 + p1XTtU (9)which can be solved with the initial condition:i1=X[0℄==X0 (10)X[t_℄=X[t℄/.(DSolve[{deq1,i1},X[t℄,t℄//Simplify)[[1℄℄etp1(βX0 + p1X0 + (α − e−t (β+p1)α)p2)β + p1We have determined α and β parameters by clinical experiments using theoral glucose intake test as measurement method. At the Heim Pál Hospital for SickChildren from Budapest we have made tests on 14 different patients.Oral glucose intake test known as oral glucose tolerance test (OGTT) is usuallyperformed to verify diagnosis when diabetes is suspected but symptoms are absent.Data from an OGTT can be used to determine if an individual has an impairedglucose tolerance, which is a prerequisite for diabetes.Patients should be drink in five minutes 2-3 dl water with maximum 75gglucose. After this, glucose level is measured at adults in two hours duration ateach 30 minute. In children measurements are taken also 150 and 180 minutes afterdrinking the sugared water.The estimation is presented over an oral glucose measurement set of a child(see Fig. 2):gluosedata={0.,0.0432,0.0378,0.024,0.0234,0.009,0.}; (11)times={0.,30.,60.,90.,120.,150.,180.}; (12)Employing our solution, the parameters α and β can be estimated on the bases ofthese measured values. Substituting the parameter values p1 and p2, we get:model=X[t℄/.parameters/.X0 → 0 (13)0.092551e−0.021151t(α - e−t (−0.021151+β)α)β − 0.021151<< Statistis`NonlinearFit` (14)αβ =BestFitParameters/.NonlinearRegress[Transpose[{times,gluosedata}℄,model,t,{α, β}℄{α → 0.0340007, β → 0.0322824}EXTENSION OF THE BERGMAN MINIMAL MODEL FOR THE GLUCOSE-INSULIN INTERACTION 27model=X[t℄/.αβ8.31443e−0.021151t (0.0340007 − −0.0340007e−0.0111314t )Checking statistical significance it can be seen that the resulted model fitswell the measured values (11) and (12):NonlinearRegress[Transpose[{times,gluosedata}℄, model,t,{α, β}℄(15){BestF itP arameters → {α → 0.0340007, β → 0.0322824}2.4. Approximation of the Insulin inlet of Internal Insulin ControlIn order to identify a model for IIC, first one needs to estimate the insulin influxproduced by the internal insulin control. Therefore an inverse problem shouldbe solved, namely, we know the model (now represented by the second equationof the Bergman model (1)) and the output (the measured values of the insulinconcentration in time) exemplified by the same patient as in (11), but now for hisinsulin parameters:insulindata={25.3, 91.2, 64.2, 73.5, 51.3, 37.4, 24.1}; (16)Fig. 2. The measured glucose values and the estimated model.We are looking for the input, the insulin i(t) function. In these experiments,no external insulin control took place, therefore i(t) ≡ iint (t). To solve this inverseproblem for non-autonomous SISO system in formy'(t) = a(t)y(t) + b(t)u(t) + (t) (17)a numerical method has been developed (SOSI), which approximate the unknownu(t) function (representing iint (t)) with the linear combination of shifted Chebyshevpolynomials (Ti), namely:28 L. KOVÁCSu(t)= n∑i=0γ i Ti (2 tθ − 1) (18)where these polynomials are valid in [0, θ] interval. The derivative of y(t)is ap-proximated by a three point finite difference formula.2.5. Implementation of Function SOSIIn the followings the realization of the SOSI module is presented. This is basicallythe implementation of the Chebyshev polynomials:<<LinearAlgebra`MatrixManipulation`SOSI[a_, b_, _, tm_, xm_, n_℄ := Module[{m, 3, i, t, x, u, eqs, A,B, P, p},m = Length[xm℄;3= tm[[2℄℄-tm[[1℄℄;Do[ti= i3, {i, 0, m-1}℄;Do[xi= xm[[i+1℄℄, {i, 0, m-1}℄;p = Table [zi, {i, 0, n}℄u=Apply[Plus,Table[zi Chebysev[i, 2ηtm-1 -1℄,{i,0,n}℄℄;eqs = {t22 (x0 -x1 )+ 2to (t2 (-xo +x1) +t1 (xo -x2 ))++t21 (-xo +x2)+t2o (-x1 +x2)}/((to -t1)(to -t2)(t1 -t2) ====a[to℄xo +b[to ℄(u/.η → to)+;eqs=Join[{eqs},Table[(t21+i(x-1+i -xi ) --2ti (t1+i (x-1+i -xi)+t-1+i (xi -x1+i))++t2i (x-1+i -x1+i)+t2-1+i(xi -x1+i))//((t-1+i -ti )(t-1+i -t1+i )(ti -t1+i))==== a[ti℄xi +b[ti℄(u/.η → ti)+,{i,1,m-2}℄℄;eqs=Append[eqs,(t2-1+m (-x-3+m +x-2+m) --2t-2+mt-1+m (x-3+m-x-1+m)+t2-3+m (x-2+m -x-1+m)++t2-2+m(-x-3+m +x-1+m)+2t-3+mt-1+m(-x-2+m +x-1+m))//((t-3+m-t-2+m)(t-3+m-t-1+m)(t-2+m-t-1+m ))====a[tm-1℄xm-1+b[tm-1℄(u/.η → tm-1)+℄;{A, B}= LinearEquationsToMatries[eqs,p℄;P=PseudoInverse[A℄;EXTENSION OF THE BERGMAN MINIMAL MODEL FOR THE GLUCOSE-INSULIN INTERACTION 29p=P.Bn+1∑i=1p[[i℄℄ChebysevT[i-1, 2ηtm-1 -1℄Applying the SOSI algorithm on the glucose-insulin model we have got:a[t_℄ = (p3 -- X[t℄) /. parameters /.αβ ) /. X0 →0 (19)−0.014188 − 8.31443e−0.021151t (0.0340007 − 0.0340007e−0.0111314t )b[t_℄=1; (20) = p4 /.parameters0.077947For SISO input we need to generate artificial measurements using splineinterpolation (see Fig. 4), because we need to estimate the γi parameters (i =0 . . . n) of the shifted Chebyshev polynomials, (6). Results are presented in Fig. 4illustrating the approximation of the input inlet insulin flux generated by IIC.By using the computed i(t) function, the second equation of the Bergmanmodel can be solved only by numerical integration with the initial condition Y[0] =25.3 µU/ (kg min) (see (16)). The obtained solution gave us the same interpolationas in Fig. 3 and it corresponds also to the open loop simulation test of the insulin.Fig. 3. The interpolated measured insulin values.2.6. Identification of IIC Model via Dynamic Neural NetworkNow having the output iint(t) and the inputs X(t), Y (t) and h(t) of IIC a dynamicneural network can be built to identify the IIC model. As input function we haveused the previously calculated α and β values:30 L. KOVÁCSFig. 4. The insulin inlet flux generated by IIC.h[t℄/.αβ (21)0.0340007e−0.0322824tTherefore, the equation for iint(t) is (see Fig. 1):iint[t℄ =F[iint[t-1℄,X[t℄,Y[t℄,h[t℄,X[t-1℄,Y[t-1℄,h[t-1℄℄ (22)Consequently, the dynamical neural network of IIC has been built as follows:<<NeuralNetworks`y=Table[{i[t℄},{t,0,180}℄;u=Table[{Xint[t℄,Yint[t℄,h[t℄/.αβ},{t,0,180}℄;{model1,fitreord}=NeuralARXFit[u,y,{{1},{2,2,2},{0,0,0}},FeedForwardNet,{1},CriterionPlot → False,RandomInitialization → LinearParameters,Regularization → 0.0001℄;As a result the estimated internal insulin regulator nonlinear model in discrete formis:model1[[1℄℄[{y[t-1℄,u1[t℄,u2[t℄,u3[t℄,u1[t-1℄,u2[t-1℄,u3[t-1℄}℄(23)−3.46165 + 10.671/(1 + e0.935576t+0.269209u1T−1+tU−3.71082u1TtU.e−3.8279u2T−1+t U−2.87889u2Tt U−3.46827u3T−1+t U−0.278068u3TtU.e−0.195113yT−1+t U)Results show (see Fig. 5) that in the obtained model the insulin variation coulddescribe not only critical care situation, but it fits well also the interstitial insulin(two peaks in the insulin response). Therefore, we improved the minimal modelproposed by Bergman.EXTENSION OF THE BERGMAN MINIMAL MODEL FOR THE GLUCOSE-INSULIN INTERACTION 31Fig. 5. Comparison between the insulin inlet flux generated by IIC and by Dynamic NeuralNetwork.3. ConclusionIn this paper we proposed an extension of the minimal model, based on controltheory aspects and we have validated it based on clinical experiments. We wouldlike to point out that we didn’t develop a new model or estimate new values forparameters pi of Bergman’s model, but just added a new part, the internal insulincontrol (IIC) part, representing the own insulin control of the human body.The advantage of this model is that insulin variation could describe not onlycritical care situation, but it fits well also the interstitial insulin and not only theinsulin in a remote compartment as proposed by Bergman.Statistical parameters gave good results, also demonstrated by the insulin inletflux generated by Dynamic Neural Network.As a result we believe that after further verifications the model could providea useful help to control the blood glucose level, and in the optimization process ofdiabetic administration.4. AcknowledgementsThis research has been supported by the Hungarian National Research Fund, Grants No.F046726 and T042990.I would like to express special thanks to Béla Paláncz for the help he gave me inusing Mathematica and to Piroska Simonné-Sebestyén who helped me in collecting themeasurements.32 L. KOVÁCSReferences[1] BENETT, D. L.–GOURLEY, S. A., Asymptotic Properties of a Delay Differential EquationModel for the Interaction of Glucose with Plasma and Interstitial Insulin, Elsevier, AppliedMathematics and Computation, 151/1 (2003), pp. 189–207.[2] BENYÓ, B.–BENYÓ, Z.–PALÁNCZ, B.–KOVÁCS, L.–SZILÁGYI, L., A Fully Sym-bolic Design and Modelling of Nonlinear Glucose Control with Control System Profes-sional Suite (CSPS) of Mathematica, Proc. of the World Congress on Medical Physicsand Biomedical Eng. Sydney, Australia, 2003, #2813, Available: http://library. wol-fram.com/infocenter/Mathsource/5043.[3] BERGMAN, B. N.–IDER, Y. Z.–BOWDEN C. 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Extension of the Bergman Minimal Model for the Glucose-insulin Interaction

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Extension of the Bergman Minimal Model for the Glucose-insulin Interaction

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