Preclinically assessed optimal control of postprandial glucose excursions for type 1 patients with diabetes

Transcription

1 Preclinically assessed opimal conrol of posprandial glucose excursions for ype paiens wih diabees Thierry Prud homme, Alain Bock, Grégory François and Denis Gille Absrac Type Paiens wih Diabees (Type PwDs) have o frequenly adjus heir insulin dosage o keep heir Blood Glucose concenraion (BG) wihin normal bounds. Meal inakes represen he mos imporan disurbance ha has o be accouned for. Is effec differs for every individual as well as for every meal. These specificiies are auomaically aken ino accoun in he approach proposed in his paper. Model parameers are idenified for every couple (Paien, Meal) of ineres and opimal conrol is applied o generae individualized meal specific insulin profiles. The mehod does no require he use of a coninuous BG meer, he profiles being infused in an open-loop manner. Resuls from a preliminary clinical sudy are presened. The concep is shown o be effecive, despie limiaions due o he aggressive execuion chosen. Improvemens are proposed and a possible praical implemenaion is described. I. INTRODUCTION I has been shown ha a high average Blood Glucose concenraion (BG) over several years increases he risk of severe long erm complicaions, among hem nephropahy, reinopahy, cardiovascular diseases or renal failure []. On he oher hand, oo much insulin can lead o a hypoglycemic even, which is a poenially lehal condiion. Type Paiens wih Diabees (Type PwDs) have o compensae for evens ha can be perceived as disurbances from a conrol poin of view. Meal inakes, physical aciviy [], sress [] or mensrual cycles belong o he se of disurbances ha have o be accouned for by Type PwDs in heir insulin dosage. Opimized BG conrol afer meal inake has a high poenial for improving he reamen of Type PwDs. During he day, he BG depends mainly on he qualiy of he meal compensaions. However, his ask is also very challenging because he paien as well as he meal specificiies have o be accouned for. Today, mos of he Type PwDs esimae heir insulin boluses based on he carbohydrae conen of he meal and he measured BG a meal ime. They ypically infuse his bolus jus before he meal inake. One of he main limiaions of his sraegy lies in he fac ha only carbohydrae conen is considered, while each meal has is own specificiy and can lead o very differen BG profiles. Typical BG excursions are shown in Fig.. These measuremens were aken for he same paien a wo consecuive days, a he same T. Prud homme is wih he Lucerne Universiy of Applied Sciences and Ars, CH 8 Horw, Swizerland hierry.prudhomme@hslu.ch A. Bock and D. Gille are wih he School of Engineering and G. François is wih he Laboraoire d Auomaique, Ecole Polyechnique Fédérale de Lausanne (EPFL), CH Lausanne, Swizerland alain.bock@epfl.ch Fig.. Pospandrial glucose excursions afer a fas and a slow meal for he same paien, meal carbohydrae conen, and insulin bolus dayime, wih he same insulin bolus, and wih equal meal carbohydrae conen bu differen meal composiions. For he so-called fas meals (mainly carbohydrae conen), a high glucose peak is measured approximaely minues afer he meal inake. Afer his peak, he BG decreases very abruply. Very ofen, an inervenion is needed o compensae for he resuling low BGs. The BG excursion measured for he slow meal (imporan fa and proein conen) is compleely differen: no peak bu a drop followed by a slow and large increase, is observed. BG remains high several hours afer he meal inake. This clearly shows ha he sandard one-sho carbohydrae based insulin bolus canno handle variaions of meal conens, which limis he efficiency of mos bolus calculaors []. The delivery of exogenous insulin has undergone significan changes wih he inroducion of he insulin pump, a device ha allows coninuous infusion of fas acing insulin. Nowadays, mos pumps are equipped wih smar meers. These meers have increased compuaional capaciies and can exchange daa wih insulin pumps. Wih such devices, he one-sho meal insulin bolus can be replaced by a disribued insulin infusion profile. I makes i possible o shape he profiles and adap hem o he specificiy of each meal. This possibiliy has barely been used so far, and is he opic of his paper. Today, he research focus is on closed-loop conrol aiming a a so-called Arificial Pancreas (AP). Two differen ypes of conrollers are considered, eiher a pure feedback conroller [], or model predicive conrollers (MPC) [], []. These closed-loop approaches require coninuous BG measuremens, which are provided by a Coninuous Subcuaneous Glucose Monior (CGM). Such sensors suffer from severe limiaions, among hem reduced accuracy, poenial sensor drop-ous or ime delays of approximaely minues.

2 Imporan safey issues relaed o he auomaically changing insulin infusion generaed by an AP are anoher drawback. All hese limiaions preclude he use of closed-loop conrollers, for he ime being. For his reason, an open-loop opimal conrol sraegy is proposed. I does no require he use of a CGM, bu i could be exended o use coninuous measuremens. The sraegy provides meal specific disribued insulin profiles. These may be inspeced for heir safey before he infusion. Paien and meal specific model parameers are idenified ha are subsequenly used o compue opimal insulin infusion profiles. The effeciveness of his mehod is evaluaed in a preliminary clinical sudy, where BG is moniored and compared o BG excursions shown in Fig., for he same wo-meal scenario. This paper is organized as follows: In secion II, he proposed approach is deailed and he model, he parameer idenificaion sraegy and he opimal conrol problem are explained. A possible implemenaion using exising devices is also skeched in his secion. In secion III, clinical resuls are presened and discussed. Finally, in secion IV, conclusions are drawn and possible improvemens are skeched. A. Modeling II. METHODS The opimal conrol approach chosen in his work requires a dynamical model of he sysem. The minimal model, see [8], has been exended o accoun for a subcuaneous insulin infusion and an oral meal inake. The insulin acion and absorpion submodel have hen been simplified o improve he idenifiabiliy of he parameers. The resuling model is given in () o (). du g,gu () = U g,gu () () d U g,gu () = a g U g,gu () a gu g,gu () + K g a gu cho () () dq() = X()Q() S g,zero Q() + U endo + C g mmol M U g,gu() () dx() = a x X() + a x X () () dx () U i,sq () = a x X () + K x a x M () G() = Q() V ga, () where he saes are he gu glucose absorpion U g,gu in g/min, he ime derivaive of he gu glucose absorpion U g,gu in g/min/min, he glucose amoun Q in mmol/kg, he insulin acion X in min, and he inermediae insulin acion X in min. Model parameers are he paien s body weigh M in kg, he volume of he accessible comparmen per body mass V ga in L/kg, he uniless bioavailabiliy of he meal of ineres K g, he inverse of he ime consan of he meal of ineres a g in min, he insulin sensiiviy K x in kg/mu, he inverse of he ime consan of he insulin absorpion/acion a x in min, he glucose effeciveness a zero insulin S g,zero in min, and he insulin independen endogenous glucose producion U endo in mmol/kg/min. C g mmol convers g of glucose ino mmol. The only manipulaed variable is he subcuaneous insulin infusion, U i,sq () in mu/min, and he oupu is he BG G() in mg/dl. The carbohydrae inake rae U cho () in g/min can be viewed as a predicable disurbance. The model equaions () o () can be expressed wih he compac equaion ẋ = f(x, u, v, θ) where x is he vecor of saes, u he manipulaed inpu U i,sq, v he predicable disurbance U cho, and θ represens he vecor of model parameers. Values for C g mmol and M are known or can be measured direcly, while K g, a g, K x, a x, S g,zero, and U endo have o be idenified. V ga can be assumed consan and equal o he populaion mean [9]. Therefore, he vecor of parameers o idenify is θ id = [K g a g K x a x S g,zero U endo ] T. B. Parameer idenificaion The model parameers have o be compued for each couple (paien = P, meal = A) as individualized meal specific insulin recommendaions are looked for. Therefore, for each meal o be conrolled wih he proposed approach, corresponding paien BG measuremens, wih a maximum sampling ime of abou hour (for example Fig. ), are needed around he meal ime. These measuremens can be performed wih a classical srip-based meers and do no require he use of a CGM. These daa are exploied o compue he parameer values for he couple (P, A) ha minimize a Maximum A Poseriori (MAP) crierion. The necessary a priori knowledge can be compued using experimenal daa hrough a populaion-based approach, like for example he Ieraive Two Sage (ITS) mehod described by [], or i can be deduced from available knowledge on he physiology (meal absorpion, insulin kineics). The simple model srucure ogeher wih he use of a priori knowledge lead o an idenifiabiliy wih parameers coefficiens of variaion below % (cf []). For a MAP idenificaion, illusraed in Fig., he following opimizaion problem is solved. θ id, P,A M = arg min J id (M) () ) N g (Ĝ(i, θ id P,A, v, u) G( i ) s.. J id (M) = i= N θ + j= M j θ id P,A = exp(am + B), where G( i ) are he N g glucose measuremens used for he idenificaion. Measuremen errors are supposed o be σ i

3 Daa collecion for Paien P Meal A Meal inake hisory Parameer Idenificaion Model Forecass Meal inake forecas Opimal Profile Generaion Insulin Infusion Hisory Objecive Funcion Pre-Meal Colleced Daa Pre-Meal insulin infusion hisory Model Objecive funcion Measured glucose Glucose Measuremen a : Idenified Model Parameers Opimized Insulin Profile A priori Knowledge on Idenified Model Parameers Fig.. MAP parameer idenificaion Fig.. Overview of he opimizaion process normally disribued and σ i are he corresponding sandard deviaions. Ĝ( i, θ id P,A, v, U i,sq ) are model predicions, N θ is he number of parameers o idenify, and M j are he N θ normalized Gaussian variables described in he Appendix. C. Opimal profile generaion Once he parameers have been idenified for a couple (P, A), a model-based opimal conrol problem is solved. The manipulaed inpu, i.e. he subcuaneously infused insulin U i,sq, is parameerized wih a piecewise consan funcion wih consan sampling ime T. The conrol horizon is defined by he sar ime h,s and he end ime h,e. Wih such an inpu parameerizaion, U i,sq () = U(π), [ h,s, h,e ], i.e. U i,sq () can be generaed using a vecor of scalar parameers π = [ π... π Nπ ] T, which corresponds o a sequence of infusion raes. N π is he number of sampling inervals for he period [ h,s, h,e ]. Thus, he opimizaion problem reads: π = arg min J op (π) (8) h,e s.. J op (π) = (Ĝ(, θ id, P,A g,a()), v, π) G h,s π π max, As seen, he square of he difference beween a meal dependen arge profile G g,a () for meal A and he prediced blood glucose concenraion Ĝ() is minimzed. π max is he insulin pump s maximum infusion rae. The arge profiles for he wo meals of ineres in he preliminary clinical sudy are ploed in Fig.. Paien s BG measuremen a ime h,s as well as informaion on infused insulin before h,s are used o compue he values of he model saes a h,s. Therefore, he compued opimized insulin profile will auomaically correc for an iniially oo low or oo high BG. I will also accoun for he pre-meal insulin infusion hisory (so called insulin on board), as illusraed in Fig.. D. Pracical implemenaion A possible implemenaion sraegy of his mehod is skeched in his secion. Today, some insulin pumps are coupled o a smar meer. The Accu-Chek R Combo sysem from Roche is a good example. The glucose meer has been exended and offers addiional funcionaliies. I communicaes wih he pump and can be used o program a bolus or change he basal rae. On he clinician side, a daa managemen sofware like Accu- Chek R can be used o collec and process he paien daa from he pump and meer. The Accu-Chek R Combo ogeher wih he sofware Accu-Chek R is used as an example for he implemenaion of he mehod proposed in his aricle. The implemenaion is divided in main seps. They are described in wha follows and are illusraed in Fig.. ) Based upon his BG hisory, he PwD and his Healh Care Provider (HCP) idenify he meals ha have been badly conrolled since he las visi. Among hese problemaic meals, hose who can be poenially improved by he mehod presened in his paper are seleced. The HCP daa managemen sofware is used o assis he HCP in he selecion process. A BG measuremen schedule as well as he lis of problemaic meals are uploaded o he PwD s smar meer. ) The PwD will follow his BG measuremen schedule (e.g. in Fig. ) for he seleced meals. A reminder is implemened in he smar meer o ensure compliance wih his schedule. The daa colleced during heses sequences are sored in he smar meer. ) During he PwD s nex visi o his HCP, model parameers values are idenified as shown in II-B for he meals for which he BG measuremen schedule has been followed. The compued model parameers are uploaded o he smar meer which is able o perform insulin profile opimizaion. Typically, he idenificaion sraegy is implemened as a module of he HCP daa managemen sofware. In Fig., i is assumed ha he model parameers have been compued for he wo

4 Paien wih Diabees Healh Care Provider Meal Inake Glucose measuremens Accu-Chek Combo Insulin Pump Accu-Chek Download ) Problemaic meals Idenificaion Proocol Upload - Which meals are badly conrolled? - BG measuremen schedule? Accu-Chek combo smar Meer BG [h] [h] Fig.. BG measuremen schedule ) Daa Collecion Proocol Reminder BG BG Meal B Meal A ) Parameer Idenificaion I Download Parameer idenificaion Model Parameers Fig.. ) Insulin Profile Opimizaion Glucose arge profiles Opimal profile for Meal A Meal A Pre-meal BG Fig.. Possible implemenaion overview meals A and B. ) The nex ime one of hese wo meals is eaen, he PwD s smar meer compues an opimal insulin profile, on he basis of he pre-meal BG, he insulin infusion hisory, and he idenified model parameers for his meal as described in. The resuling profile is auomaically injeced by he insulin pump. I should be noed ha he PwD s safey is of highes imporance and needs o be guaraneed before any ou-paien implemenaion. III. P RELIMINARY CLINICAL STUDY exhibi a sandard deviaion equal o % of he measured glucose value. This value is of paricular imporance when he number of measuremens is low and he weigh of he Bayesian erm in he objecive funcion is no negligible. The second in-paien period, referred o as second block, ook place approximaely monhs afer he firs one. During his period, he opimizaion rouine described in subsecion II-C was used o compue opimized insulin profiles. h,s was se o hour before meal inake and h,e o hours afer meal inake. A sampling period T of min was fixed. This led o Nπ = decision variables o opimize. The wo arge profiles depiced in Fig. were used for he fas and slow meals. I can be noed ha hese arge profiles exhibi similar shapes, bu he peak of he profile used for he slow meal is shifed by an hour. The resuling opimized insulin profiles were infused o he paiens, while heir BG was measured every min from inravenous blood draws o assess he performances of he proposed approach. A. Sudy design B. Sudy resuls A clinical sudy was performed o es he proposed mehod on Type PwDs. Boh fas and slow meals, as depiced in Fig., were considered. For conciseness reasons, only resuls for paiens are discussed. During he firs in-paien period, referred o as firs block, he paiens had o ea he wo meals on wo consecuive days a 9: a.m. and o conrol heir BG using he sandard herapy. As illusraed in Fig., eigh differen BG measuremens were made around he meal inake wih a sampling ime of h. The firs one was made hour before he meal inake and he las one hours afer. The daase was used o idenify he parameers for each couple (P, A) using he sraegy discussed in subsecion II-B. The a priori knowledge was generaed using ITS (cf II-B). σi has been chosen independen of he model and equal o.g(i ). This describes he assumpion ha glucose measuremens The resuls of he sudy for he wo paiens are shown in Fig. o. On he lef hand side, he infused insulin profiles for he wo blocks are ploed. The green and he blue curves represen he one-sho boluses and he opimal insulin profiles, respecively. On he righ hand side, he BG excursions measured in he wo blocks are ploed. The green and blue lines represen he measured BGs in he firs and second block, respecively. The doed blue line gives he expeced BG excursion for he second block. The predicion capabiliies of he model can be sudied by comparing he plain blue line wih he doed blue line. Similar conclusions can be drawn for he wo paiens. A he beginning of he opimizaion horizon (before meal inake), he opimizaion compued large insulin amouns. I can be seen in he four figures ha hese correcions were efficien. For he fas meals, his large iniial amoun

5 Insulin Infusion (mu/min) Learning:. (U) Opimized:. (U) Blood Glucose Concenraion (mg/dl) 8 8 Insulin Infusion (mu/min) 9 8 Learning:. (U) Opimized:. (U) Blood Glucose Concenraion (mg/dl) 8 8 Fig.. Clinical resuls, paien, fas meal Fig.. Clinical resuls, paien, slow meal Insulin Infusion (mu/min) 8 Learning:. (U) Opimized: 8. (U) Fig. 8. Blood Glucose Concenraion (mg/dl) 8 8 Clinical resuls, paien, slow meal TABLE I VALUES OF J op FOR THE s AND nd BLOCK AND THEIR RELATIVE VARIATION. BG MEASUREMENTS AFTER HYPOGLYCEMIA EVENTS WERE NOT TAKEN INTO ACCOUNT FOR THE COMPUTATION OF J op. Sandard Opimized Relaive s block nd block variaion Paien fas meal % Paien slow meal 9-9.9% Paien fas meal -.% Paien slow meal % also compensaes for he effec of he meal, as nearly no addiional insulin injecion was needed afer meal inake. For he slow meals, a second insulin injecion was necessary beween and hours afer he meal inakes. The shapes of he BG predicions and measuremens were similar, for he fas as well as for he slow meals. Table I gives he values of he objecive funcion J op for he wo blocks. Excep for he couple (paien, fas meal), he opimized insulin profiles led o significan reducions of J op. These reducions are paricularly imporan for he slow meals. The insulin infused beween hour and hours afer he inake of he slow meals prevens from he hyperglycemia ha was observed during he firs blocks. For he couple (paien, fas meal), he BG excursion measured a he firs block was close o he arge profile excep for a hypoglycemic even ha occurred - hours afer he meal inake. To avoid his, he opimizaion recommended a smaller insulin infusion. However, his correcion was oo imporan and led o a higher value for J op. As already menioned, he BG model predicions for he second block (plain blue line) exhibi shapes in accordance Insulin Infusion (mu/min) Learning:. (U) Opimized:. (U) Fig. 9. Blood Glucose Concenraion (mg/dl) Clinical resuls, paien, fas meal wih he expeced BG excursions (doed blue line). However, i should also be noed ha he opimizaion procedure led o oo aggressive correcions beween he blocks. The size of he differen boluses can be found in Table II. As seen, hey differ significanly. Obviously, he model is no able o predic BG wih a good confidence when he insulin profile used for he idenificaion differs oo much from he insulin profile used for he compuaion of he predicions. The limiaions of he predicions capabiliies of he model can have differen reasons, some of hem are lised below: The choice of he model srucure was mainly driven by he idenifiabiliy of he parameers and by he accuracy of he fis. I is known ha glucagon, free fay acids or growh hormones, as well as many oher molecules, influence he BG. They have been inenionally negleced, as i would have been impossible o quanify heir conribuion wih only a few BG measuremens. Therefore, he model used for his implemenaion is an oversimplificaion of he glucose meabolism. This is he main reason for he limied predicion capabiliies. The model parameers for a couple (P,A) were idenified on a single daase colleced during he firs block. As menioned earlier, several disurbances like sress, physical aciviy or mensrual cycles have no been considered. These disurbances should, a leas, be filered TABLE II TOTAL INFUSED INSULIN (U) Sandard Opimized Paien fas meal.. Paien slow meal. 8. Paien fas meal.. Paien slow meal..

6 o minimize heir influence on he model parameers idenified. This could be done, e.g., by using more han one daase o idenify he parameers for a couple (P,A). Insulin absorpion exhibis nonlineariies wih respec o he amoun of injeced insulin. However, he model used in his work is linear. Therefore, he idenified parameers should no be used o compue predicions for insulin infusion profiles ha differ oo much from he profiles used for he idenificaion. Physiological parameers of a PwD change over ime. As he second block was performed approximaely monhs afer he firs one, he idenified model parameers may no be accurae enough anymore. IV. CONCLUSION AND OUTLOOK The resuls presened in his paper are promising. An opimized disribued insulin profile led o a reduced BG peak afer a fas meal inake. I also correced for he hyperglycemic evens observed during he firs block a few hours afer he slow meal inakes. However, correcions were shown o be poenially aggressive and fuure work should include he following improvemens: Small sep sraegy: The difference beween wo consecuive insulin profiles for he same meal should be penalized in J op, as proposed in []. Indeed, he preliminary clinical resuls have shown ha he exrapolaion capabiliies of he model are oo limied o allow big variaions beween wo consecuive insulin profiles. By inroducing his penaly erm in J op, several ieraions will be necessary o converge o an opimal insulin profile for a couple (P, A), bu safey will be improved. No meal-dependen arge profile: The proposed approach requires meal-dependen arge BG profiles as shown in Fig.. I assumes ha some a priori knowledge is available on he absorpion dynamics of each meal. In pracice, his is very difficul o conceive. Insead, a consan arge profile could be used for all of he meals and a new objecive funcion J op should be used. This new J op would penalize deviaion from arge in an asymmerical manner. BGs below arge would have a much sronger conribuion o J op han BGs above arge o avoid hypoglycemia. An objecive funcion ins red by he meric defined in [] could be used advanageously. Robus idenificaion: Some of he model parameers (K x, a x, S g,zero, and U endo ) do no depend on he meal bu only on he paien. In his paper, all he parameers are idenified for each couple (P, A). Resuls could be improved if he paien dependen parameers would be idenified using daases from differen meals. Learning horizon reducion: The learning horizon should be reduced o h for pracical reasons. Indeed,he ime beween wo consecuive meal inakes is frequenly smaller han h. However, he main limiaion of he proposed approach lies in he simpliciy of he model used for opimizaion. This is clearly an advanage for idenificaion, which is made possible on he basis of sole glucose measuremens, bu meanwhile i can be a drawback a he opimizaion sage. To be able o predic he real opimal insulin profiles for a given paien, he endency model we are using should fulfill some adequacy requiremens []. In pracice, hese requiremens are hard o mee, bu fuure research should compare he accuracy of his simplified model wih more complex models (for example [9], []). Also, o avoid hyper- or hypoglycemia, robus opimizaion (i.e. opimizaion considering uncerainy, e.g. deduced from confidence inervals) should be invesigaed. REFERENCES [] DCCT, The effec of inensive reamen of diabees on he developmen and progression of long-erm complicaions in insulin-dependen diabees mellius. New Engl J Med, vol. 9, pp. 9 98, 99. [] B. Zinman, N. Ruderman, B. N. Campaigne, J. T. Devlin, and S. H. Schneider, Physical Aciviy/Exercise and Diabees, Diabees Care, vol., no. supplemen, pp. S9 S,. [] D. A. Finan, H. Zisser, L. Jovanovic, W. C. Bevier, and D. E. Seborg, Auomaic Deecion of Sress Saes in Type Diabees Subjecs in Ambulaory Condiions, Ind Eng Chem Res, vol. 9, pp. 8 88,. [] O. K. Hejlesen, S. Andreassen, R. Hovorka, and D. A. Cavan, DIAShe diabees advisory sysem: an ouline of he sysem and he evaluaion resuls obained so far, Compuer mehods and programs in biomedicine, vol., no. -, pp. 9 8, Sep. 99. [] S. A. Weinzimer, G. M. Seil, K. L. Swan, J. Dziura, N. Kurz, and W. V. Tamborlane, Fully auomaed closed-loop insulin delivery versus semiauomaed hybrid conrol in pediaric paiens wih ype diabees using an arificial pancreas, Diabees Care, vol., no., pp. 9 99, 8. [] R. Hovorka, V. Canonico, L. J. Chassin, U. Haueer, M. Massi- Benedei, M. O. Federici, T. R. Pieber, H. C. Schaller, L. Schaupp, T. Vering, and M. E. Wilinska, Nonlinear model predicive conrol of glucose concenraion in subjecs wih ype diabees, Physiological Measuremen, vol., no., p. 9,. [] P. Dua, F. Doyle, and E. Pisikopoulos, Model-based blood glucose conrol for ype diabees via parameric programming, IEEE Trans Biomed Eng, vol., no. 8, pp. 8 9, aug.. [8] R. N. Bergman, Y. Z. Ider, C. R. Bowden, and C. Cobelli, Quaniaive esimaion of insulin sensiiviy, Am J Physiol Gasroines Liver Physiol, vol., no., pp. G, 99. [9] R. Hovorka, F. Shojaee-Moradie, P. V. Carroll, L. J. Chassin, I. J. Gowrie, N. C. Jackson, R. S. Tudor, A. M. Umpleby, and R. H. Jones, Pariioning glucose disribuion/ranspor, disposal, and endogenous producion during IVGTT. American journal of physiology. Endocrinology and meabolism, vol. 8, no., pp. E99,. [] P. Vicini and C. Cobelli, The ieraive wo-sage populaion approach o ivg minimal modeling: improved precision wih reduced sampling, Am J Physiol-Endoc M, vol. 8, no., pp. E9 E8, January. [] M. E. Wilinska, L. J. Chassin, H. C. Schaller, L. Schaupp, T. R. Pieber, and R. Hovorka, Insulin kineics in ype- diabees: coninuous and bolus delivery of rapid acing insulin, IEEE Transacions on Biomedical Engineering, vol., no., pp.,. [] T. Prud homme and C. Cris, US/ A. Prandial blood glucose excursion opimizaion mehod via compuaion of imevarying opimal insulin profiles and sysem hereof, 8. [] B. P. Kovachev, W. L. Clarke, M. Breon, K. Brayman, and A. Mccall, Quanifying emporal glucose variabiliy in diabees via coninuous glucose monioring: Mahemaical mehods and clinical applicaion. Diabees Technol Ther, vol., no., pp. 89 8,. [] J. Forbes, T. Marlin, and J. MacGregor, Model adequacy requiremens for opimizing plan operaions, Compuers & Chemical Engineering, vol. 8, no., pp. 9, 99. [] C. Dalla Man, R. A. Rizza, and C. Cobelli, Meal simulaion model of he glucose-insulin sysem, IEEE Trans Biomed Eng, vol., no., pp. 9,.

7 APPENDIX A. Maximum A Poseriori idenificaion sraegy: Transformaion In he Maximum A Poseriori mehod, he vecor θ of model parameers is a random variable. In his paper, his random variable is assumed o have a log-normal disribuion. This disribuion is widely used in he conex of physiological model parameers as i ensures ha any realizaion of his random variable will have a posiive value. By definiion we have: θ i = exp(θ) i for each i {,, N θ }, (9) wih N θ = dim(θ), and Θ = [Θ,, Θ Nθ ] is a random vecor having a mulivariae normal disribuion. Θ is hus fully defined by a mean vecor µ Θ and a covariance marix Σ Θ wih: µ Θ = E[Θ] () Σ Θ = E[(Θ E[Θ])(Θ E[Θ]) T ], () wih he auocorrelaion coefficiens being: and he correlaion coefficiens: σ Θ (i) = Σ Θ (i, i), () σ Θ (i, j) = Σ Θ(i, j) σ Θi σ Θj. () A ransformaion ha defines a one-o-one relaionship beween Θ and a vecor M of uncorrelaed normalized (zero mean and ideniy covariance) Gaussian vecor is applied. M is also of dimension N θ. This ransformaion is defined as follows: Θ = AM + B, () where A is called he ransformaion marix. A and B should be uniquely defined by µ Θ and Σ Θ. To deermine he relaionship beween hese hree eniies, firs he mean of Θ is compued as follows: M has zero mean. Thus: E[Θ] = AE[M] + B. () The covariance of Θ is compued: E[Θ] = µ Θ = B. () Σ Θ = E[(Θ E[Θ])(Θ E[Θ]) T ]. () Θ = AM + B and E[Θ] = B, hus: Developing: Σ Θ = E[(AM)(AM) T ]. (8) Σ Θ = E[AMM T A T ] = AE[MM T ]A T. (9) However E[MM T ] is he ideniy marix, remembering ha M is a se of independen normalized Gaussian variables. Thus, we have: Σ Θ = AA T. () If A is chosen as a riangular marix, i can easily be proven ha he componens A(i, j) of he marix A are defined by he wo following equaions: A(j, j) = j σθ (j) A(j, k) () k= A(i, j) = σ Θ(j)σ Θ (i)σ Θ (j, i) j k= (A(j, k)a(i, k)). A(j, j) () A and B are now fully defined. As a reminder, we have: θ = exp(θ) = exp(am + B). () B. Maximum A Poseriori idenificaion sraegy: Objecive funcion The minimized objecive funcion for he parameer idenificaion akes he following form: N g (Ĝ(i, M) G( i )) N θ J id = + Mj, i= σ i j= where N g is he oal number of glucose measuremens used for he idenificaion and N θ is he number of parameers being idenified. M j are he corresponding N θ normalized Gaussian variables which ogeher are he previously menioned M vecor.