EPSAC Predictive Control of Blood Glucose Level in Type I Diabetic Patients

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1 Proceedings of the 44th IEEE Conference on Decision and Control, and the Eropean Control Conference 005 Seville, Spain, December 1-15, 005 WeIA0.4 Predictive Control of Blood Glcose Level in ype I Diabetic Patients Clara Ionesc and Robin De Keyser Abstract An in-hose predictive control algorithm has been implemented to control blood glcose level in type I diabetic patients, by controlling the inslin infsion rate to a mechanical pmp. he role of the distrbance filter in model-d predictive control is nderlined and its possibility to improve control performance is exploited. For comparison prposes, a classic controller has been designed via CAD tools. Controller performance was assessed in terms of its ability to track a normoglycemic setpoint (81mg/dL) from initial states of hypo- and hyperglycemia, as well as in response to a meal distrbance. Unconstrained control gave satisfactory reslts, within imposed constraints. Both strategies are implemented in discrete manner, ths sitable for ftre implementation in wearable devices. I. INRODUCION IABEES Mellits is an incrable disease affecting Dmillions of people worldwide. Belgim is one of the leading contries in the astringent fight against the increasing nmber of diabetic patients so that this contemporary disease does not reach its crrent prediction of 50 million victims in Erope by 05 [1]. In parallel, scientists are focsing on developing a manifold of new techniqes and feasible instrmentation to offer wearable soltions and improve the life of type I diabetic patients. ype I, or inslin-dependent diabetes mellits (IDDM), is characterized by absent inslin secretion in the pancreas, reslting in plasma glcose concentrations elevated above the normoglycemic range (70-10mg/dL) []. he treatment methods for inslin-dependent diabetes, sbctaneos inslin injection or continos infsion of inslin, sally reslt in freqent variations of glcose concentration in blood. herefore, a significant effort has been pt toward the development of a device to control glycemia [3] [6] and the methods are continosly improved. A device of this type wold contain three major components: i) a mechanical pmp (MP); ii) an in vivo glcose sensor; and iii) a mathematical algorithm (R) to reglate the pmp given a sensor measrement (Fig.1). echnological advances allowed a wide variety of programmable and variable-rate infsion pmps to become C. Ionesc is with the Electrical energy, Systems and Atomation Department at Ghent University, echnologiepark 913, B905, Gent, Belgim (phone: ; fax: ; clara@atoctrl.ugent.be). R. De Keyser is with the Electrical energy, Systems and Atomation Department at Ghent University, echnologiepark 913, B905, Gent, Belgim ( rdk@atoctrl.ugent.be). available [7],[8]. Model-d predictive control (MBPC) algorithms have been recently reported in literatre to tackle sccessflly constraints posed by several biomedical control problems, not only in blood glcose control in diabetic patients [9], [10] bt also mean arterial pressre, and cardiac otpt control dring anesthesia [10], [11]. he prpose of this contribtion is to present reslts of an in-hose extended prediction self adaptive control () algorithm to blood glcose control of IDDM patients, and investigate the effect of a defalt and a designed distrbance filter in strategy. A comparison between a classical controller and is made in this report. Correction of hypo- and hyperglycemia states towards normoglycemia is performed and rejection of a meal distrbance is tested. As a first step, reslts are compared and discssed with respect to controller performance and possible improvements. reqired blood error infsion meal glcose rate blood glcose R Patient level Pmp level inslin - measred blood glcose level Fig. 1. Schematic overview of a blood glcose control device II. PAIEN MODEL In [1] it has been conclded that an injected glcose load in normal (glcose tolerant) individals immediately elevated the glcose concentration in plasma. his initiates the secretion of inslin from the pancreatic ß-cells. he provoked hyperglycemia indces an immediate peak in the inslin concentration in plasma, and the glcose ptake in mscles, liver and adipose tisse is raised by the remote inslin action. his lowers the glcose concentration in plasma, affecting the ß-cells to secrete less inslin (biological feedback). By 1 hor the glcose concentration is normalized and in the following hors a moderate ndershoot is observed. After -3 hors, it is sally fond that the pertrbed inslin and glcose concentrations have retrned to normal. A mathematical representation of glcose and inslin kinetics is necessary before implementing the model-d control scheme. Althogh it sffers from glcose effectiveness overestimation and inslin sensitivity nderestimation [13], the Bergman minimal model [14] proved to be sitable for exemplifying this stdy: G =pg 1 X( G GB) Gmeal / VI (1a) /05/$ IEEE 4845

2 X = px p3i (1b) I = ni ( IB) t ( )/ V (1c) I he plasma inslin at time t, denoted by I(t), fills a remote inslin compartment with a constant rate p 3. he active inslin in the remote compartment accelerates glcose tilization by the peripheral tisses and liver. Conseqently, p 3 is referred to as the inslin-dependent increase in glcose ptake ability in tisse. he intra-celllar metabolism of the remote inslin effect is constant and determined by p. he dynamic inslin response denoted by X(t) is proportional to the active inslin in the remote compartment and describes the time dependent effect of the inslin on the net glcose disappearance. he plasma glcose concentration G(t) is governed by the balance between the glcose prodction/ptake by the liver and the tilization of glcose by the peripheral tisses. he glcose ptake in mscles, liver and tisses is assmed to be of constant rate p 1. Basal levels of glcose and inslin are denoted by G B and I B respectively. V I is the inslin distribtion volme, n is the fractional disappearance of inslin, and the inslin infsion rate delivered by the pmp is denoted as (t). Defining the state, inpt, and otpt variables as: x G x = X = y = G 1 1 U UB ; ; Gmeal 0 x 3 I one can derive a linear state space model where the first inpt is maniplated (inslin infsion rate) and the second inpt represents a meal glcose distrbance: p1 GB 0 0 1/ VI A= 0 p p 3 ; B 0 0 = ; 0 0 n 1/ VI 0 C = [1 0 0]; D = 0 0. [ ] Diabetic (glcose intolerant) sbjects can be modeled by the following set of parameters [15]: G B =81mg/dL, I B =15mU/L, U B =16.667mU/min, V I =1L, n=5/54min -1, p 1 =0min -1, p =0.05min -1, p 3 =13*10-6 mu/min. Assming the meal glcose G Meal and glcose otpt in mg/dl the reslting state space model is: A= B = 0 0 5/ 54 1/1 0 C = [18 0 0]; D = *10 ; 0 0 ; [ ] o avoid that the patient enters hypo- or hyperglycemia state, the glcose level is imposed between () (3) (4) 65 G G B 180 for an inslin infsion rate within 0 U U B 100. Notice that a negative/positive limit denotes the deviation from the basal vale. III. PREDICIVE CONROL strategy [16],[17] is d on a generic process model: yt () = xt () nt () (5) he distrbance n(t) incldes the effects in the measred otpt y(t) which do not come from the model inpt (t) via the available model. hese non-measrable distrbances have a stochastic character with non-zero average vale, which can be modelled by a colored noise process: 1 1 nt () = Cq ( )/ Dq ( ) et () (6) with: e(t) - ncorrelated (white) noise with zero mean vale; C(q -1 ) and D(q -1 ) - monic polynomials in the backward shift operator q -1 of orders n c and n d. he filter C(q -1 )/D(q -1 ) is considered to be a design filter. It plays an important role in MBPC and will be discssed later in this Section. A. Prediction Algorithm he model otpt x(t) represents the effect of the control inpt (t) on the process otpt y(t) and is also a nonmeasrable signal, and the relationship between (t) and x(t) is given by the generic dynamic system model: [ ] xt ( ) = f xt ( 1), xt ( ),, t ( 1), t ( ), (7). he fndamental step in MBPC methodology consists in prediction of the process otpt y(tk) at time instant t, indicated by { y( t k t), k = 1 N }, over the prediction horizon N, and d on: - the measrements available at sampling time instant t: { yt (), yt ( 1),, t ( 1), t ( ), } ; - he ftre vales of the inpt signal (postlated at time t): { t ( t), t ( 1 t), }. Using the generic process model (5), the predicted vales of the otpt are: y( t k t) = x( t k t) n( t k t) (8) Prediction of x(tk t) and of n(tk t) can be done respectively by recrsion of the process model (7) and by sing filtering techniqes on the noise model (6) [17]. B. Control Algorithm In for linear models, the ftre response is then considered as being the cmlative reslt of two effects: 4846

3 y( t k t) = y ( t k t) y ( t k t) (9) optimize he two contribtions have the following origins: y ( t k t) : effect of past control {(t-1), (t-),...} (initial conditions at time t); effect of a ftre control scenario, called ( t k t), k 0, which is defined a priori [17]; for linear systems the choice is irrelevant, a simple choice being { ( t k t) 0, k 0 } ; effect of ftre (predicted) distrbances n(tk t). he component y ( t k t) can be easily obtained sing (6)(7)(8) taking ( t k t) as the model inpt for (7). y ( t k t) : optimize effect of the optimizing ftre control actions δt ( ), t δt ( 1 ), t δt ( N 1 ) t with { } δ t ( k t) = t ( k t) ( t k t). Refer to Fig. for the concepts of and optimizing controls. Notice that (tk t) is constrained to be constant from k=n on (and this is realized by selecting (tk t) constant from k=n on and by imposing that δ(tk t) shold be constant from k=n on). he design parameter N is called the control horizon (a well-known concept in MBPCliteratre). N =4 (tk t) yoptimize ( t k t) = hk δ( t t) hk 1δ( t 1 t) (10)... g k N 1 δ ( t N 1 ) t he parameters g1, g,... gk,... gn are the coefficients of the nit step response of the system, i.e. the response of the system for a stepwise change of the inpt (with amplitde 1). he parameters h1, h,... hk,... h N are the coefficients of the nit implse response of the system and can be easily calclated from the step response coefficients and vice versa: hk = gk gk 1 (and h0 = h-1 =... = g0 = g-1 =... 0 ). Using (9) and (10), the key -MBPC eqation: is obtained, where: Y= Y GU (11) [ yt ( N1 t) yt ( N t) ] [ y ( t N1 t) y ( t N t) ] [ δt ( ) t δt ( N 1 ) t] Y = Y = U = hn h 1 N11 hn 1... gn 1 N 1 hn 1 1 hn h N G = hn h N1 hn... g N N 1 (1) he controller otpt is then the reslt of minimizing the cost fnction: δ(tk t) (tk t) past crrent time t ftre Fig.. he concept of /optimizing controls time From Fig. it is obvios that the component y ( t k t) optimize is the cmlative effect of a series of implse inpts and a step inpt: an implse with amplitde δ t ( t) occrring at time t, reslting in a contribtion hkδ ( t t) to the process otpt at time tk (k sampling periods later); an implse with amplitde δ t ( 1 t) occrring at time t1, reslting in a contribtion hk 1 δ ( t 1 t) to the predicted process otpt at time tk (k-1 sampling periods later); etc; finally a step δ t ( N 1 t) at time t N 1, reslting in a contribtion gk N 1 t ( N 1 ) δ t to the predicted process otpt at time tk. he cmlative effect of all implses and the step is: U N = (13) k= N1 V( ) [ r( t k t) y( t k t)] with rt ( k t) the desired reference trajectory and the horizons N 1, N being design parameters. It is now straightforward to derive the soltion. he cost fnction (13) is a qadratic form in U, having the following strctre sing the matrix notation from (1) and with R defined similarly to Y: 1 V ( ) = [ ] [ ] U R Y GU R Y GU (14) which leads after minimization w.r.t. U to the optimal soltion: * 1 U = [ G G] G ( RY ) (15) he matrix G G which has to be inverted has dimension N x N. For the defalt case N =1, this reslts in a simple scalar control law. Only the first element δ( t / t) in U * is reqired in order to compte the actal control inpt applied to the process: 4847

4 () t = ( ) t t δ ( ) t t = ( ) t t U (1) (16) At the next sampling instant t1, the whole procedre is repeated taking into accont the new measrement information y(t1). his is called the principle of receding horizon control, another well-known MBPC-concept. C. Distrbance Filter Design he distrbance n(t) incldes all effects in the measred otpt y(t) which do not come from the model otpt x(t) (effects of process distrbances, effects of n-modeled process inpts, measrement noise, model errors, etc). he net effect of all these nknown distrbances has a stochastic character with non-zero average vale and can be modeled by a colored noise process as in (6) where the filter C/D is the distrbance model. In the MBPC approach this filter is considered as a design filter and can be sed - in order to improve the qality of the control performance - to spply information to the controller abot the type of distrbances that can be expected. A basic choice for the distrbance model might be: Cq ( )/ Dq ( ) = 1/1 q (17) -zeros in the transfer fnction K( sz1)( s z)/ s, so that the controller proportional gain K is as large as possible, to redce the settling time (s<10min), bt still satisfying the reqired specification for overshoot (OS<30%) and a relatively high robstness (Ro>0.65 on a scale 0 1). he zeros are located at z 1, = -0.0 and the gain is K=-0.5, as depicted in Fig. 3. OS%<30% Ro>0.65 s<150 Design: K=-0.5; p=0; Z1=Z=-0.0 ProcessController reslting in a distrbance signal n(t) with non-zero average vale. In this case the MBPC-controller will intrinsically take action to remove steady-state errors, similar to the effect of the integrator in a -type controller. Notice that (17) is the defalt distrbance model (dd) that is sally applied in practice. Generally, the distrbance signal n(t) can be reconstrcted sing the generic model (5), by measring the process otpt y and calclating the model otpt x with the system model (7). As n(t) has sally the character of a correlated (colored) random signal, a sefl and simple approach is then to calclate its power spectra density sing a spectral analysis software. his allows then to detect arond which freqency the main distrbance energy is sitated. Once the main freqency is known, a more intelligent design of the filter C/D can be made so that it has a band-pass characteristic arond this freqency (id). Some gidelines on designing an effective distrbance filter are detailed in [18]. IV. CLASSICAL CONROLLER DESIGN Controller design is d on the CAD (Compter Aided Design) methodology, reqiring an available dynamic model of the system. he CAD-software d on Freqency Response techniqes (FR-tool) has been sed. his in-hose developed methodology is d on specifications sch as: robstness, settling time, overshoot, and is a very interactive and visal design tool. he principle is the following: play with the -zeros to reshape the Nichols crve, so that it fits into the reqirements of the given specifications. Notice that it is necessary to provide a dynamic model of the system. he design procedre consists of playing with the Fig. 3. Sample figre from CAD freqency response toolbox for design Frther on, the parameters are sed in a discrete-time control scheme, with a software implemented controller: t () = t ( 1) cet 0 () cet 1 ( 1) cet ( ) (18) with the error: et ( ) = wt ( ) yt ( ) (19). Denoting the shift-operator: q 1 e() t = e( t1) reslts in: 1 1 Cq ( ) c0 c1q cq t () = et () = et () (0) 1 1 1q 1q and the control loop is depicted below. he sampling period of the controller was s =3min [19]. R w(t) e(t) C(q -1 1 (t) y(t) ) 1 MP Patient 1 q - Sensor Fig. 4. control loop scheme (R denotes the reglator from Fig.1) V. PERFORMANCE EVALUAION A. Hypoglycemia and Hyperglycemia Correction If inslin is applied in excess, the blood glcose level goes below normal (<65mg/dL) casing hypoglycemia. On a short term, hypoglycemia can indce fainting, mscle convlsions, 4848

5 deeper state of consciosness-loss sch as coma. On the other hand, if inslin is spplied insfficiently, the blood glcose level elevates above the normoglycemia vales (>10mg/dL) casing hyperglycemia. Most of the long-term complications associated with diabetes, sch as nephropathy (a kidney disease) and retinopathy (diseased condition of the eye-retina, esp. non-inflammatory) reslt from sstained hyperglycemia [10], ths an immediate retrn to the normoglycemic vales is important. Glcose (mg/dl) Inslin (mu/min) (detail) Fig. 5. Correction of a hypoglycemic state he controllers are tested from the performance standpoint for correcting a state of hypo- and hyperglycemia, respectively. Correcting a state of hypoglycemia implies bringing the glcose level in the blood as soon as possible from the hypoglycemic vale (<65mg/dL) to the basal vale (81mg/dL). Respectively, correcting hyperglycemia implies bringing the glcose level in blood from the hyperglycemic vale (>10mg/dL; in this case 180mg/dL). Fig.5 and Fig.6 depict the and controller (N =1, N 1 =1, N =5) performance by means of inpt (inslin infsion rate) and otpt (glcose concentration) variables for hypo- and hyperglycemia state corrections, respectively. Control is evalated in the absence of exogenos glcose infsion (zero meal distrbance). B. Glcose Level Control Following a Meal Distrbance In this test the effect of infsing (exogenos) glcose is examined and controllers are compared. It is expected that the extra information given in the distrbance filter of the id strategy leads to improved reslts compared to dd. It has been reported that a realistic meal glcose distrbance is arond 0.3g glcose per kg bodyweight [1]. In this paper, a 3g glcose infsion distrbance has been considered. In the oral glcose tests, the glcose does not enter directly the blood stream and is more realistic to consider a lag in the gt of abot 0 mintes. his information is then provided to the distrbance filter: Cq ( )/ Dq ( ) = 1/ (1 q )(10.9 q ) (1) and the corresponding reslts (id) are in Fig.7. Glcose (mg/dl) Inslin (mu/min) (detail) Fig. 6. Correction of a hyperglycemic state Glcose (mg/dl) Inslin (mu/min) id dd dd id 5 Fig. 7. Correction after an exogenos glcose infsion C. Discssion he benefit of sing predictive control instead of classical control algorithm is the estimation of ftre glcose behavior d on the past inslin inpts. By definition, the predictive controller takes action for a predicted hypo- or hyperglycemic state of the patient before it occrs, while a classical feedback controller reacts afterwards. In the present contribtion, constrained control has not been applied, althogh the design of N, N 1 and N parameters had been made sch that it flfills the reqested constraints. Mixing types of inslin is expected to diminish controller performance, since it is desirable that the reslts are obtained fast (ths a fast-active inslin type is applied). For hypo- and hyperglycemia state correction, the designed controller performed smoothly, converging to the setpoint vale in approximately 3 hors. Notice that the design of the 4849

6 controller d on the specifications does not necessarily give optimal controller parameters. he controller performed better from the settling time standpoint (approx. 1 hor), bt it applies a more aggressive control. For distrbance rejection, as expected from theory of distrbance filter design, the id had the best performance. his improved reslt is de to the fact that the controller knows abot the type of distrbance it can expect and takes the appropriate control action. he settling times for both are similar, arond 1.5 hors, and they otperform the controller (3 hors). An important aspect is the comparison between control efforts of the controllers in the distrbance rejection test. For the, the control effort range is within 5mU/min, while the controller reqires more energy. Notice that the maximm control effort range for dd is abot (4-16)mU/min, while for id is abot (9-16)mU/min; respectively 9.5% and 15.5% (considering (100-16)mU/min as 100% and relating to basal vales in Section II). he remarkable observation is that the otpt of the id is significantly better than the one of dd, reqiring only a 6% increase in its control effort. his observation enhances the importance of the role of distrbance filter in - MPC methodology. VI. CONCLUSION As a starting point, the reslts presented in this paper comprise the application of the in-hose algorithm to blood glcose control in IDDM patients. Althogh constrained control has not been yet applied, sefl reslts within imposed constraints have been obtained. A common advantage of the and -MBPC strategies stands in their discrete natre, sitable for DSP implementation in recent circit technology. he nonlinear applied on a generalized (nonlinear) patient model is an ndergoing research. herefore, inter- and intra-patient variability is sbject to frther investigation, in robstness tests. Using a designed distrbance filter, the n-modeled effects, modeling errors and other distrbances are expected to be well tackled in this -MBPC control strategy. Moreover, if sed in constrained control, the predictive algorithms provide optimal reslts in coping with the imposed constraints on inpt-otpt variables. Clinical trials are the final aim of this research, whereas a modified Biostator (an device that monitors blood glcose and gives incrementally inslin to mimic normal inslin secretion) is considered to be sed on sbjects. Since stdies employing fast acting inslin and the sbctaneos delivery rotes with a controller reported good reslts [19], some frther investigation is reqired for parameter tning via the FR-tool toolbox. REFERENCES [1] IDF-International Diabetes Federation, Annal Report 004, 5p. Available: [] A.C. Gyton, extbook of Medical Physiology. 7 th Edition. Philadelphia, USA: Sanders W.B., 1986 [3] E.M. Fisher, A semi-closed loop algorithm for the control of blood glcose levels in diabetics IEEE rans. on Biomedical Engineering, vol. 38, pp.57-61, Jan [4] B. Candas, J. Radzik, An adaptive plasma glcose controller d on a nonlinear inslin/glcose model, IEEE rans. on Biomedical Engineering, vol. 41, pp , Feb [5] R. Belazzi, G. Ncci, C.Cobelli, he sbctaneos rote to inslindependent diabetes therapy, IEEE Engineering in Medicine and Biology, vol. 0, pp , Jan-Feb 001. [6] R.S. Parker, F.J. Doyle III, N.A. 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