Optimal Estimation Applications to Continuous Glucose Monitoring

Transcription

1 Optiml Estimtion Applictions to Continos Glcose Monitoring. Wyne eqette Isermnn Deprtment of Chemicl nd iologicl Engineering, Rensseler Polytechnic Institte, Troy, NY Abstrct Type dibetics mst freqently monitor their glcose to void hypoglycemi nd hyperglycemi. In this pper Klmn Filter is developed to estimte glcose concentrtion from noisy continos sbctneos sensor signls. The Klmn Filter formltion leds to ntrl glcose predictor tht cn be sed to lrm ptients to te corrective ction to void hypoglycemi. I. MOTIATION AND ACKGROUND Dibetes is disese where the pncres of n individl hs impired bility to prodce inslin to reglte glcose. A type dibetic hs completely lost the bility to prodce inslin nd mst receive dily injections, or continos infsions, of inslin. Dibetics mst closely monitor their glcose levels by pricing their fingers severl times ech dy to obtin cpillry glcose vles from glcose test meters. This informtion, long with expected mel consmption, is sed to djst the mont of inslin delivered. A significnt nmber of ptients re on inslin pmp therpy, where inslin is delivered in nerly continos fshion. While intensive glcose mngement, bsed on obtining 3-4 fingerstic glcose mesrements dy, cn reslt in redced micro- nd mcro-vsclr complictions (with some incresed hypoglycemi ris; DCCT, 993, few dibetics monitor themselves t this rte. A nmber of continos glcose monitoring devices re nder development, bsed on sbctneos or interstitil flid sensing rther thn direct glcose mesrement. The vilbility of continos glcose signl hs the potentil to rdiclly improve glcose mintennce; this is lso necessry for the development of closed-loop rtificil pncres (Prer et l., ; elzzi et l.,. Objectives of continos glcose monitoring inclde the redction of ris of hypoglycemi (low glcose nd the improvement of glcose mngement in individls with dibetes. Chllenges inclde compenstion for the dynmic lg between the glcose nd the sensed sbctneos vle, nd the mesrement noise ssocited with the sbctneos sensor. A review of nd sbctneos glcose dynmics is provided in section II, while sections III nd I develop techniqes for glcose estimtion nd hypoglycemi detection. II. LOOD AND UCUTANEOU GLUCOE DYNAMIC Crrent clinicl prctice is to me decisions on cpillry glcose mesrements, while most continos monitors nder development re bsed on sbctneos sensing. A nmber of stdies describe the -sbctneos glcose dynmics s first-order lg. Rebrin et l. (999 se comprtmentl pproch to model glcose dynmics, s shown in Fig.. Cpillry C 2 Interstitil ensor C Ft/ Mscle Cell Fig.. Comprtmentl Representtion of Cpillry lood nd bctneos (Interstitil Glcose Dynmics (Rebrin et l., 999. The mthemticl model is ( + C C ( where 2 represents rte constnt for the pte of glcose in the sbctneos tisses, 2 nd 2 re rte constnts for diffsion between the nd sbctneos comprtments, nd 2 re the volmes of the nd sbctneos glcose comprtments, respectively nd C nd C represent the nd sbctneos glcose concentrtions. Rebrin et l. (999 note tht diffsion coefficient cn be defined s D chmie et l. (998 consider sensing volme (, with glcose diffsion from the to the sensing

2 volme nd pte of glcose in the sbctneos tisse. The reltionship between (C nd sbctneos glcose (C concentrtion is glcose concentrtion, vs. s.c. sbctneos A m m ( C C rc (2 where A m is the mss trnsfer re nd m is the mss trnsfer coefficient. They define the reciprocl of the mss trnsfer time constnt s β A, nd rewrite (2 s m m ( r + β C + βc (3 chmie et l. fond tht β generlly rnged between.4 nd. min -, nd r in their rt stdies. Note tht ( nd (3 re identicl, with β 2, nd r 2. Recll tht the stndrd form for first-order model is dy y + b y + (4 τ τ where nd y represent pertrbtions of the inpt nd otpt from stedy-stte, nd nd τ represent the gin nd time constnt. For ( nd (3, the gin nd time constnt re then β β + r τ + β + Notice tht if there is no pte of glcose in the sbctneos comprtment, the gin, for both ( nd (3. A given chnge in glcose then reslts in the sme long-term chnge in sbctneos glcose. The rt stdies of chmie ssme, while Freelnd nd onnecze (999 show some simltion reslts for <. chmie et l. indicted tht τ rnged from 9 to 25 mintes in rts, while the dog stdies of Rebrin et l. (999 fond tht τ rnged from 5 to 2 mintes. teil et l. (3 stdied hmns nd fond τ 3 mintes. Consider now the simple simltion model stdied by Rebrin et l. (999. The time constnt is 2 mintes nd the gin is. The glcose decreses from to mg/dl, with first-order decy time constnt of 75 mintes. The corresponding sbctneos vles, with mesrement noise (stndrd devition mg/dl, re shown in Fig. 2. This exmple will be sed in the demonstrtion of estimtion techniqes in section III. While the mesrement noise in Fig. 2 does not pper tht drmtic, it does hve strong effect on the bility to estimte glcose from sbctneos mesrements. We shold lso note tht the glcose monitoring litertre sometimes erroneosly refers to dely between nd sbctneos glcose. The reslts in Fig. 2 re not chrcteristic of time-dely, bt rther first-order lg (time constnt. 2 2 r (5 5 5 Fig. 2. lood Glcose nd bctneos Glcose Responses. bctneos Mesrement Noise with tndrd Devition of mg/dl. III. ETIMATION A. Intitive Finite Differences Approch Rebrin et l. (999 develop n intitive estimtion pproch for by rerrnging (4 to solve for the glcose ( from the sbctneos glcose (y. dy y (6 b where ^ is sed to indicte n estimted vle. A finitedifferences (FD pproximtion for the derivtive yields y y y t b (7 y + y b t b b t Rebrin et l. (999 note tht this nmericl derivtive bsed pproch is very sensitive to mesrement noise, nd lso pply three-point moving verge filter on the mesred otpt.. Klmn Filtering ystems nd control engineers, of corse, immeditely nderstnd the limittions to the intitive finite-differences pproch presented bove, nd wold generlly pply optiml estimtion-bsed techniqes, sch s the Klmn Filter. We se the following conceptl model x + y Φx + Γ + Γ Cx + v w w (8

3 where w is n inpt noise vector (with covrince mtrix Q, nd v is mesrement noise vector (with covrince R. The noises re white Gssin noise with zero men. In prctice, the noises re not nown so Q nd R re tning prmeters tht ffect the estimtor performnce. Weighting Q high reltive to R indictes tht the mesred otpts re to be trsted more thn the model predicted otpts. Conversely, weighting R high reltive to Q indictes tht ny individl mesrement hs high degree of ncertinty nd the model predicted otpt shold be trsted more thn the mesrement. The mximm lielihood stte estimtes cn be fond recrsively (tengel, 994, sing the Klmn filter. The predictor-corrector eqtions re Φ + L + Γ ( y C where represents n estimte of the sttes nd the nottion for the sbscripts - mens the estimte t step is bsed on mesrements p (nd inclding step -. Note tht model is sed to propgte the stte estimte from the previos time step (- to the crrent time step (. The mesrement t the crrent time step is then sed to pdte the stte estimte, bsed on the Klmn Gin (L... tep Distrbnces The formltion sed here is slightly different thn in (9 bove; since the inpt ( glcose is nnown, it mst be estimted. Defining the glcose ( s second stte, we find (sing mtrix-vector nottion tht (9 cn be written in the form ( ( y (3 Here we hve sed stedy-stte Klmn gin, L. The estimtes sing the two techniqes re shown in Fig. 3. The KF-bsed estimtes re clerly less noisy thn the intitive FD pproch. The KF estimtes exhibit slight lg from the ctl vles, prticlrly immeditely fter the initil decrese in glcose. glcose ctl nd estimtes ctl KF estimte ctl FD filtered estimte Fig. 3. Comprison of Klmn Filter-bsed Approch (top with Finite-Differences Filtered Estimte (bottom..2. Rmp Distrbnce x A limit to the KF performnce shown in Fig. 3 is the + Φ Γ x w ssmption of rndom step chnge. Now, ssme tht the ( + + rte-of-chnge of glcose is constnt. Define the x y [ ] chnge in glcose t step s d - + v x + Φ Γ x where it is ssmed tht the glcose chnges + + w rndomly in step fshion. Notice tht, in this cse, the (4 d d + noise vectors re sclrs, so Q nd R re both sclrs nd x only their rtio is importnt. Using gmented stte y [ ] + v nottion d x Φ Γ, w x,, Γ, [ ( Φ C ] At ech time step the Klmn filter is estimting the sbctneos nd glcose, nd the rte-of-chnge of The prediction-correction eqtions for this gmented -glcose, from noisy sbctneos mesrements. stte formltion re The reslts re shown in Fig. 4, where Q/R.5 ws sed. Notice tht there is less of lg in the estimtes bsed on rmp chnge vs. step chnge. Estimting the Φ (2 rte-of-chnge of glcose will lso qite ntrlly L ( y C + led to formltion for predicting when hypoglycemic threshold might be reched, s shown in section I. For time constnt of 2 min. nd smple time of min., the prediction nd correction eqtions re (for Q/R 5

4 ctl KF estimte ssming rmp 5 5 ctl KF estimte ssming step 5 5 Fig. 4. Comprison of Klmn Filter-bsed Approch Assming Rmp Glcose Chnge Estimte (top with tep Chnge Estimte (bottom. There hs some debte bot the reltionship between nd sbctneos glcose, with some sorces indicting tht there is glcose pte in the sbctneos comprtment nd, therefore, lower stedy-stte glcose concentrtion. This reslts in gin < for the first-order model. A Klmn Filter pproch hndles this cse qite nicely, s shown in Fig. 5 (Q/R.5. s.c. mesred min., llowing n individl to consme glcose to rise sgr. They se liner regression lgorithm tht, by ntre, dds time dely to the prediction. We sggest better pproch bsed on optiml estimtion theory. In section III we showed how Klmn Filter cn be sed to estimte the rte-of-chnge of glcose, even with significnt mesrement noise. This rte-of-chnge estimte cn obviosly be sed to predict when criticl vle of glcose will be reched. The prediction of glcose N steps into the ftre is N + + N d (5 defining crit s criticl glcose concentrtion (7 mg/dl in or stdies, we cn solve for the criticl time, t crit, by t crit N t crit t (6 d A chrcteristic decrese in glcose stdied by Chole et l. (2 is shown in Fig. 6. Note the sbstntil lg between nd sbctneos glcose, with over min. difference in the times to rech the threshold of 7 mg/dl. Here gin we ssme mesrement noise with stndrd devition of mg/dl. Althogh there is cler lg in the s.c. mesrements, the glcose estimtes trc the ctl vles very well, s shown in Fig. 7. Also shown is t crit, the predicted time before the hypoglycemic threshold of 7 mg/dl is reched. Note tht the criticl time of 2 min. occrs t pproximtely 8 min. The hypoglycemi prediction lrm cold be set to wrn the ptient to consme glcose to void reching hypoglycemic stte. 5 5 glcose concentrtion, vs. s.c. ctl sbctneos ctl KF est. of 5 5 Fig. 5. lood vs. Mesred bctneos ( mg/dl std. dev. noise Glcose (top; lood Glcose vs. Estimte (bottom. I. HYPOGLYCEMIA DETECTION One of the min objectives of continos glcose monitoring is to redce the ris of hypoglycemi. Chole et l. (2 estimte the rte of chnge of glcose concentrtion to predict when hypoglycemic threshold (7 mg/dl will be reched, nd sggest tht n lrm cold be set to fnction when this prediction time drops to Fig. 6. Decrese in lood nd bctneos (Noise-free Glocse Concentrtions to elow the Hypoglycemic Threshold of 7 mg/dl.

5 tcrit, min time to hypoglycemic concentrtion ctl KF estimte s.c. mesrement Fig. 7. Hypoglycemi Prediction. Top: Time predicted to rech the criticl glcose concentrtion of 7 mg/dl. ottom: Estimted nd ctl glcose concentrtions, nd sbctneos mesrement. There re mny importnt chllenges in hypoglycemi detection nd prediction, inclding the selection of thresholds nd prediction horizons to provide resonble rtes of flse positive lrms. In Plerm nd eqette (4, Klmn Filter is developed bsed solely on the sbctneos glcose concentrtion. The rte-of-chnge (first derivtive nd the second derivtive of glcose re sed s estimted vribles, nd sed to predict ftre concentrtions of sbctneos glcose. The effect of mesrement smpling freqency, hypoglycemic threshold, nd prediction horizon on the sensitivity nd specificity re stdied. We propose mlti-tiered lrm system sing different wrning levels depending on the prediction horizon; i.e. predicting hypoglycemi over shorter prediction horizon leds to higher level wrning thn over longer prediction horizon.. UMMARY A Klmn Filtering pproch hs been sed to estimte glcose from noisy sbctneos mesrements. y ppending stte relted to the rte-of-chnge of glcose ntrl formltion for predicting possible hypoglycemi rises. This predictor cn then be sed in hypoglycemi wreness monitor, to provide dibetics with enogh time to te corrective ction to prevent hypoglycemi. The estimtion strtegy presented cn lso be sed in model-bsed rtificil pncres to reglte glcose bsed on sbctneos mesrements nd the djstment of inslin infsion rtes to mintin desired glcose setpoint. REFERENCE elzzi, R., G. Ncci nd C. Cobelli The bctneos Rote to Inslin-Dependent Dibetes Therpy, IEEE Engineering in Medicine nd iology Mgzine, 2(, (Jn/Feb. Chole, C., P. Doldl, J.-C. Klein, W.K. Wrd, G.. Wilson nd G. Rech Prevention of Hypoglycemi Using Ris Assessment With Continos Glcose Monitoring ystem, Dibetes, 54, (2. The Dibetes Control nd Complictions Tril Reserch Grop The Effect of Intensive Tretment of Dibetes on the Development of Long-Term Complictions in Inslin-Dependent Dibetes Mellits, N. Eng. J. Med., 329, (993. Freelnd, A. C. nd R. onnecze Inference of lood Glcose Concentrtions from bctneos Glcose Concentrtions: Applictions to Glcose, Annls iomed. Eng., 27: (999. Plerm, C.C. nd.w. eqette Isses in Hypoglycemi Prediction, in Proceedings of the 3 th Northest ioengineering Conference, pringfield, MA, April 6-7 (4. Prer, R.., F.J. Doyle III nd N.A. Pepps The Intrvenos Rote to lood Glcose Control, IEEE Engineering in Medicine nd iology Mgzine, 2(, (Jn/Feb. Rebrin, K., G.M. teil, W.P. n Antwerp nd J.J. Mstrototro bctneos Glcose Predicts Plsm Glcose Independent of Inslin: Implictions for Continos Monitoring, Am. J. Physiol. 277 (Endocrinol. Metb. 4: E56-E57 (999. Rebrin, K. nd G.M. teil Cn Interstitil Glcose Assessment Replce lood Glcose Mesrements? Dibetes Technology & Therpetics, 2(3, (. chmie, D. W., A.C. Freelnd, A. Heller, nd R. onnecze Mesrement nd Modeling of the Trnsient Difference etween lood nd bctneos Glcose Concentrtions in the Rt fter Injection of Inslin, Proc. Ntl. Acd. ci. UA Medicl ciences, 95: (998. teil, G.M., K. Rebrin, J. Mstrototro,. ernb nd M.F. d Determintion of Plsm Glcose Dring Rpid Glcose Excrsions with bctneos Glcose ensor, Dibetes Technology & Therpetics, 5(, 27-3 (3. tengel, R.F. Optiml Control nd Estimtion, Dover, New Yor (994. This is nbridged, corrected repbliction of tochstic Optiml Control: Theory nd Appliction, Wiley, New Yor (986.