Nonlinear Modeling of the Dynamic Effects of Infused Insulin on Glucose: Comparison of Compartmental with Volterra Models

Transcription

1 TBME-59-8 Nonlinear Modeling of he Dynamic Effecs of Infused Insulin on Glucose: Comparison of Comparmenal wih Volerra Models Georgios D. Misis, Member, IEEE, Mihalis G. Markakis, and Vasilis Z. Marmarelis, Fellow, IEEE Absrac This paper presens he resuls of a compuaional sudy ha compares simulaed comparmenal (differenial equaion) and Volerra models of he dynamic effecs of insulin on blood glucose concenraion in humans. In he second modeling approach, we employ he general class of Volerra-ype models ha are esimaed from inpu-oupu daa, and in he firs approach we employ he widely acceped minimal model and an augmened form of i, which incorporaes he effec of insulin secreion by he pancreas, in order o represen he acual closedloop operaing condiions of he sysem. We demonsrae boh he equivalence beween he wo approaches analyically and he feasibiliy of obaining accurae Volerra models from insulinglucose daa generaed from he comparmenal models. The resuls corroborae he proposiion ha i may be preferable o obain daa-driven (i.e. inducive) models in a more general and realisic operaing conex, wihou resoring o he resricive prior assumpions and simplificaions regarding model srucure and/or experimenal proocols (e.g. glucose olerance ess) ha are necessary for he comparmenal models proposed previously. These prior assumpions may lead o resuls ha are improperly consrained or biased by preconceived (and possibly erroneous) noions a risk ha is avoided when we le he daa guide he inducive selecion of he appropriae model wihin he general class of Volerra-ype models.. Index Terms Physiological sysems; Volerra-Wiener models; Laguerre-Volerra neworks. D I. INTRODUCTION iabees mellius represens an alarming hrea o public healh wih rising rends and severiy in recen years worldwide and is characerized by muliple and ofen no Manuscrip received 8. This work was suppored in par by he European Social Fund (75%) and Naional Resources (5%) Operaional Program Compeiiveness General Secrearia for Research and Developmen, Program ENTER, by he NIH/NIBIB Cener Gran No P4- EB978 o he Biomedical Simulaions Resource a he Universiy of Souhern California and by he Myronis Foundaion (Graduae Research Scholarship). Georgios D. Misis was wih he Insiue of Communicaions and Compuer Sysems, Naional Technical Universiy of Ahens, Ahens 578 Greece. He is now wih he Deparmen of Elecrical and Compuer Engineering, Universiy of Cyprus, Nicosia 678, Cyprus (phone: ; fax: ; gmisis@ucy.ac.cy). Mihalis G. Markakis is wih he Deparmen of Deparmen of Elecrical Engineering and Compuer Science, Massachuses Insiue of Technology, Cambridge MA 39 ( mihalis@mi.edu). Vasilis Z. Marmarelis is wih he Deparmen of Biomedical Engineering, Universiy of Souhern California, Los Angeles, CA 989 USA ( vzm@bmsr.usc.edu). readily observable clinical effecs []. There is, herefore, urgen need for improved diagnosic mehods ha provide more precise clinical assessmens and sensiive deecion of sympoms a earlier sages of he disease. This criical ask may be faciliaed (or enabled) by he uilizaion of advanced mahemaical models ha reliably describe he dynamic inerrelaionships among key physiological variables implicaed in he underlying physiology (i.e. blood glucose concenraion and various hormones such as insulin, glucagon, epinephrine, norepinephrine, corisol ec.) under a variey of meabolic and behavioral condiions (e.g. pre-/pos-prandial, exercise/res, sress/relaxaion). Such models would no only provide a powerful diagnosic ool, bu may also enable longerm glucose regulaion in diabeics hrough closed-loop model-reference conrol using frequen insulin micro-infusions adminisered by implaned programmable micro-pumps. This will preven he onse of he pahologies caused by elevaed blood glucose over prolonged periods in diabeic paiens []. Blood glucose concenraion flucuaes considerably in response o food inake, hormonal cycles or behavioral facors. These flucuaions may range from 7 o 8 mg/dl in mos normal subjecs, alhough blood glucose concenraion remains wihin he normoglycemic zone (7- mg/dl []) for mos of he ime. The inernal physiological regulaion of hese wide flucuaions is a complex, muli-facorial process. The mos criical regulaory role is played by he pancreas which, upon sensing an elevaion in blood glucose concenraion, secrees insulin hrough is bea cells, while an opposie change in glucose causes secreion of glucagon hrough is alpha cells. The secreed insulin assiss he upake of glucose by he cells and he sorage of excess glucose in he liver in he form of glycogen. Secreed glucagon assiss he caabolism of glycogen ino glucose ha is released from he liver ino he bloodsream, while insulin inhibis glycogen synhase [3]. Furhermore, free fay acids in he blood poeniae he shor-erm responsiveness of pancreaic bea cells o glucose oscillaions, bu may inhibi long-erm responsiveness [4, 5]. Finally, blood glucose concenraion and is relaion o insulin concenraion depend on he acion of several oher hormones (e.g. epinephrine, norepinephrine, corisol [6-8]) making he dauning complexiy of his mulifacorial regulaory mechanism eviden. The primary effec on blood glucose is exercised by insulin and mos effors o dae have focused on he sudy of his causal relaionship. Prolonged hyperglycemia is usually caused by defecs in insulin secreion by he pancreaic bea

2 TBME-59-8 cells or in he efficiency of insulin-faciliaed glucose upake by he cells. The exac quaniaive naure of he dependence beween blood glucose concenraion and he acion of he oher hormones menioned above, or facors such as die, endocrine cycles, exercise, sress ec., remains largely unknown (primarily because of lack of appropriae daa), alhough he qualiaive effec has been esablished. Thus, he aggregae effec of all hese oher facors for modeling purposes is viewed as random disurbances, addiive o he blood glucose level. Saring from he iniial work of Bolie [9] and Ackerman [], mos modeling sudies of he causal relaionship beween insulin and glucose (as he inpu and oupu of a sysem represening his relaionship) have relied on he concep of comparmenal modeling []. In his conex, he minimal model (MM) of glucose disappearance, combined wih he inravenous glucose olerance es (IVGTT), has been he mos widely used mehod o sudy whole body glucose meabolism in vivo [, 3]. The MM posulaes ha insulin acs from a remoe comparmen and affecs glucose uilizaion, in addiion o he insulin-independen uilizaion ha depends on he glucose level per se. These insulin-dependen and insulinindependen effecs on glucose uilizaion/ kineics are combined in a single comparmen. Cerain parameers of he MM (i.e., insulin resisance S I and glucose sensiiviy S G ) have been shown o be of clinical imporance and can be esimaed from IVGTT daa, using nonlinear leas-squares mehods [4, 5] or, more recenly, Bayesian esimaion echniques [6-8]. However, he accuracy of he esimaes obained from he MM has been quesioned because of he single-comparmen assumpion [5, 9, ], and wo-comparmen models for glucose kineics [-3], as well as muli-comparmenal models for glucose and insulin kineics, have been proposed [4, 5]. Oher modeling approaches ha have been recenly explored in he conex of glucose conrol - include arificial neural neworks [6], probabilisic models [7] and linear/ nonlinear impulse response and Volerra models [8, 9]. In addiion o hese insulin-glucose models, aemps have been made o ake ino accoun he influence of some relevan physiological signals, such as glucagon [4] and free fay acids [3]. The aforemenioned comparmenal models rely on a priori assumpions and simplificaions regarding he underlying physiological mechanisms and heir primary aim is ofen o exrac clinically imporan parameers in conjuncion wih specific experimenal proocols (e.g., he IVGTT). Therefore, heir abiliy o quanify glucose meabolism under acual, more general operaing condiions remains limied. On he oher hand, recen echnological advances in he developmen of reliable coninuous glucose sensors and insulin micro-pumps [3, 3] have provided ime-series daa ha enable he applicaion of daa-rue modeling approaches [33]. These approaches offer new opporuniies owards he goal of obaining reliable models of he insulin-glucose inerrelaionships in a more general conex. Using sponaneous or exernally infused insulin and glucose daa, one can obain daa-driven models ha are no consrained by a priori assumpions regarding heir srucure. The presen paper examines he relaion beween exising comparmenal (differenial equaion) and Volerra-ype models, boh analyically and compuaionally. The resuls demonsrae he feasibiliy of obaining Volerra models of insulin-glucose dynamics ha are equivalen o widely acceped comparmenal models, using daa-records ha are pracically obainable. They also illusrae he physiological inerpreaion of nonlinear Volerra models by providing direc links o a well-known parameric model wih parameers of clinical significance. Since he Volerra approach does no require prior assumpions abou model srucure, i can provide he effecive means for obaining accurae daa-rue, paienspecific and ime-adapive models in a clinical conex. II. METHODS The presen sudy concerns comparmenal and Volerraype nonlinear dynamic models; among comparmenal models, we selec he minimal model of glucose disappearance (MM), as well as an augmened version of i (AMM), which incorporaes an insulin secreion equaion. The srucure and parameer values of hese models are aken from he lieraure [, 4, 34-37]. The equivalen Volerra models [38] are esimaed using simulaed inpu-oupu daa from he comparmenal models. A. The minimal model of glucose disappearance The MM of glucose disappearance is described by he following wo differenial equaions [], which describe he nonlinear dynamics of he insulin-o-glucose relaionship during an IVGTT: dg( dx( p g( x( [ g( + g = () b p x( + p i( ) = () 3 where g( is he deviaion of glucose plasma concenraion from is basal value g b (in mg/dl), x( is he inernal variable of insulin acion (in min - ), i( is he deviaion of insulin plasma concenraion from is basal value i b (in μu/ml), p and p are parameers describing he kineics of glucose and insulin acion respecively (in min - ) and p 3 is a parameer (in min - ml/μu) ha affecs insulin sensiiviy (see below). The iniial condiions for he simulaions are: g() = and x() = (i.e. we assume ha we sar a basal condiions which is a reasonable assumpion in he conex of simulaing he model for siuaions where he iniial ransien phase can be ignored). Noe ha he MM is nonlinear, due o he presence of he bilinear erm beween he inernal variable x( represening insulin acion and he variable [g(+g b ] represening he plasma glucose concenraion in he firs equaion. This bilinear erm describes he modulaion of he effecive kineic consan of he glucose uilizaion by insulin acion (i.e. insulin concenraion increases cause faser disappearance of blood glucose). The physiological inerpreaion of he MM parameers can be made in erms of insulin-dependen and insulinindependen processes ha enhance glucose upake and suppress ne glucose oupu [3]. The parameer p, ermed glucose effeciveness S G, represens he insulin-independen effec, while he insulin-dependen effec is represened by he raio p 3 /p (in min - /μu ml - ) and is ermed insulin sensiiviy ]

3 TBME S I. The values of S G and S I are ypically esimaed from IVGTT daa and he MM has proven o be successful in a clinical conex, requiring a relaively simple es procedure [3]. Noneheless, he accuracy and physiological inerpreaion of he MM parameer esimaes has been quesioned because of he use of a single comparmen for glucose kineics [9, ]. The MM, as formulaed in Eqs. ()-(), does no include an equaion describing he secreion of insulin from pancreaic bea cells in response o an elevaion in blood glucose concenraion, i.e., i is an open-loop model, which may be used along wih properly designed experimenal proocols (IVGTT) for parameer esimaion. However, he acual glucose meabolism process is a closed-loop sysem, excep in condiions of severe Type I diabees where he pancreaic bea cells are considered oally inacive. In order o accoun for his, an insulin-secreion equaion may be included, as described below (closed loop MM or AMM). Limiaions of he MM (and he AMM) include he absence of an explici glucogenic componen reflecing producion of new glucose by he liver in response o elevaed plasma insulin and/or glucose (such as he model presened in [39]) and he associaed glucagon secreion process (from he alpha cells of he pancreas) among ohers. The aggregae effec of hese processes, as well as he effec of oher facors (free fay acids, epinephrine ec.), can be incorporaed by disurbance erms ha are added o he glucose rae and insulin acion equaions. B. Closed-loop parameric model: The Augmened Minimal Model The closed-loop naure of insulin-glucose ineracions requires he incorporaion of an addiional equaion describing he insulin secreion dynamics by he pancreaic bea cells. Of several equaions ha have been proposed [4, 36, 37, 4, 4], we selec one ha uilizes a hreshold funcion see Equaions (5)-(6) below [4, 36, 37]. The resuling closed-loop model becomes: dg( + pg( = x( [ g( + gb ] (3) dx( = px( + p3[ i( + r( ] (4) dr() = ar() + βth[ g()] (5) where r( is he secreed insulin by he pancreaic bea cells in response o an elevaion in plasma glucose concenraion. The secreion is riggered by elevaed plasma glucose concenraions according o he hreshold funcion T h [g(] defined as: g( θ, g( θ T h [ g( ] =, oherwise where θ corresponds o he glucose concenraion value above which insulin is secreed. The dynamics of his riggered secreion process and he kineics of he secreed insulin are described (in firs approximaion) by he kineic consan a (in min - ) in Equaion (5). The parameer β (in μu min - /ml per (6) mg/dl) deermines he rae of insulin secreion (i.e. he srengh of he feedback pahway). C. Volerra-ype modeling The Volerra-Wiener framework has been employed exensively for modeling nonlinear physiological sysems [38]. In his conex, he inpu-oupu dynamic relaionship of a causal, nonlinear sysem of order Q and memory M is described by he Volerra funcional expansion: Q M... g( = k ( τ,..., τ ) i( τ )... i( τ ) dτ... dτ (7) n= M n n where i( and g( are he inpu and oupu of he sysem a ime (deviaions of plasma insulin and glucose concenraions from heir basal values, respecively). The unknown quaniies of he Volerra model ha are esimaed from he inpu-oupu daa are he Volerra kernels k n (τ,,τ n ). The firs-order kernel (n=) is he linear componen of he sysem dynamics, while he higher order kernels (n>) form a hierarchy of he nonlinear dynamics of he sysem. The highes order Q defines he nonlinear order of he sysem. Many physiological sysems can be described adequaely by Volerra models of second or hird order [38]. The Volerra- Wiener approach is well-suied o he complexiy of physiological sysems since i yields daa-rue models, wihou requiring a priori assumpions abou sysem srucure. Among various mehods ha have been developed for he esimaion of he discreized Volerra kernels, a Volerraequivalen nework in he form of he Laguerre-Volerra Nework (LVN) is seleced because i has been proven o be an efficien approach ha yields accurae represenaions of high-order sysems in he presence of noise using shor inpuoupu records [4, 43]. The LVN model consiss an inpu layer of a Laguerre filer-bank and a hidden layer of K hidden unis wih polynomial acivaion funcions (Figure ) [4, 43]. A each discree ime, he inpu signal i( (insulin) is convolved wih he Laguerre filer-bank and weighed sums of he filer-bank oupus v j (where v j =i*b j and b j is he j-h order discree-ime Laguerre funcion) are ransformed by he hidden unis hrough polynomial ransformaions. Fig.. The Laguerre-Volerra nework. The sysem inpu i( is convolved wih a Laguerre filer bank wih impulse responses b j, he oupus of which (v j (n)) are fed ino a layer of K hidden unis wih polynomial acivaion funcions f K ha produce he sysem oupu g(. n n

4 TBME The model oupu g( (glucose) is formed as he summaion of he hidden uni oupus z k and a consan corresponding o he glucose basal value g b : L u () = w v () (8) k k, j j j= K K Q n k b n, k k b k= k= n= (9) g () = z() + g = c u() + g where L is he number of funcions in he filer bank and w k,j and c q,k are he weighing and polynomial coefficiens respecively. The insulin and glucose ime-series are used o rain he LVN model parameers (w k,j, c q,k and he Laguerre parameer which deermines he Laguerre funcions dynamic properies) wih a gradien-descen algorihm as follows [4]: ( r+ ) ( r) n ( r) '( r) ( r) β k k k= L δ = δ + γ ε ( n) f ( u ( n)) w [ v ( n ) + v ( n)] j= k, j j j () ( r+ ) ( r) ( r) '( r) ( r) w = w + γ ε ( n) f ( u ( n)) v ( n) () k, j k, j w ( r+ ) ( r) ( r) ( r) m c = c + γ ε ( n)( u ( n)) () mk, mk, c where δ is he square roo of he Laguerre parameer, γ β, γ w, γ c are posiive learning consans, r denoes ieraion and ε ( r ) ( n ) '( r and f ) k ( uk ) are he oupu error and derivaive of he polynomial acivaion funcion of he k-h hidden uni, evaluaed a he r-h ieraion, respecively. The equivalen Volerra kernels are hen obained in erms of he LVN parameers as: k n n n, k k, j k, jn j jn n k= j = jn = k K L L k ( τ,..., τ ) c... w... w b ( τ )... b ( τ ) = (3) The srucural parameers of he LVN model (L,K,Q) are seleced on he basis of he normalized mean-square error (NMSE) of he oupu predicion achieved by he model, defined as he sum of squares of he model residuals divided by he sum of squares of he de-meaned rue oupu. The saisical significance of he NMSE reducion achieved for model srucures of increased order/complexiy is assessed by comparing he percenage NMSE reducion wih he alphapercenile value of a chi-square disribuion wih p degrees of freedom (p is he increase of he number of free parameers in he more complex model) a a significance level alpha, ypically se a.5 [44]. The LVN represenaion is equivalen o a varian of he general Wiener-Bose model ermed he Principal Dynamic Mode (PDM) model. The PDM model consiss of a se of parallel branches, each one of which is he cascade of a linear dynamic filer (PDM) followed by a saic nonlineariy [38, 45]. Each of he K hidden unis of he LVN corresponds o a separae branch and defines he respecive PDM p K ( and polynomial nonlineariy. This leads o model represenaions ha allow physiological inerpreaion, since he resuling number of branches is ypically low in pracice. According o he PDM model form, he insulin inpu signal is convolved wih each of he PDMs p k (, where k=,,k and k j L k k, j j j= p () = w b (), and he PDM oupus u k are subsequenly ransformed by he respecive polynomial nonlineariies f k (.) o produce he model-prediced blood glucose oupu (he aserisk denoes convoluion): g () = gb + f[ u()] fk[ uk()] = = g + f [ p ( * i( ] f [ p ( * i( ] b K K (4) D. Equivalence beween comparmenal and Volerra models In order o examine he mahemaical relaionship beween he aforemenioned comparmenal and Volerra models, we employ he generalized harmonic balance mehod o derive analyical relaions beween he wo model forms, as oulined below for he second-order case of he nonparameric model [46]. This procedure can be exended o any order of ineres. By seing he inpu i( equal o, e s s and s e + e in he general Volerra model of Eq. (7) successively, he oupu g( s s becomes equal o k, k + e K( s) + e K( s, s) +... and k + s s ( s+ s) e K( s ) + e K( s ) + e K( s, s) +..., where K (s) and K (s,s ) are he Laplace ransforms of k (τ) and k (τ,τ ) respecively. If we subsiue hese hree inpu-oupu pairs ino he differenial equaions of he comparmenal models (Eqs. ()-() for he open-loop model and (3)-(5) for he closed-loop model) and equae he coefficiens of he resuling exponenials of he same kind, we can obain analyical expressions for k, K (s) and K (s,s ), in erms of he parameers of he respecive comparmenal model. To define he compuaional equivalence beween he wo model forms, we simulae he comparmenal models wih broadband inpu (insulin) daa and we hen esimae he kernels of he equivalen Volerra model, from he simulaed inpu-oupu daa. The accuracy of he esimaed firs and second-order Volerra kernels is assessed by comparison wih he exac kernels of he equivalen Volerra model ha is derived in analyical form from he differenial equaions of he comparmenal models. The accuracy and robusness of he kernel esimaes is evaluaed under measuremen noise condiions, in order o assess he performance of he Volerra approach. III. RESULTS A. Analyical expressions of he Volerra kernels of he comparmenal model: Open-loop case The bilinear erm beween insulin acion and glucose concenraion in Eq. () of he MM gives rise o an equivalen Volerra model of infinie order. However, for parameer values wihin he physiological range, a second-order Volerra model offers an adequae approximaion for all pracical purposes. Considering he insulin and glucose deviaions from he respecive basal values i( and g( as he inpu and he oupu respecively, we can derive analyically he Volerra kernels of he open-loop MM by applying he procedure oulined in Mehods o he inegro-differenial equaion:

5 TBME Fig.. Top panel: The firs-order (lef and second-order (righ Volerra kernels of he minimal model for ypical values of is parameers wihin he physiological range (S G =. min - and S I =.36 min - /μu ml - ). Boom panel: Effec of he wo key parameers p and p of he open-loop MM on he form of he equivalen firs-order kernel. Noe ha he glucose effeciveness S G is equal o p and he insulin sensiiviy S I is inversely proporional o p (and proporional o p 3 ). These plos offer a visual undersanding of he effecs of changes in hese parameers (p beween. and.4 min -, p beween. and.5 min - ) on he firs-order insulinglucose dynamics (see ex. g () + pg () + p exp( pτ )( i τ) gd () τ 3 = gp exp( pτ) i ( τ) dτ b 3 (5) The above equaion is derived from he MM by subsiuing he convoluional soluion of Equaion (): 3 exp( pτ ) i( τ ) x ( = p dτ (6) ino Equaion (). Upon applicaion of his mehod, we derive he following analyical expressions in he Laplace domain for he firs- and second-order Volerra kernels of he MM (k = ): K( s) = p3 g b (7) ( s + p )( s + p ) gp b 3 K( s, s) = ( s+ p)( s+ p) (8) p + ( s + p)( s + p) s+ s + p The MM has, in principle, Volerra kernels of any order. However, i can be shown ha he magniude of he nh-order kernel is proporional o he n h power of p 3 and, subsequenly, an adequae Volerra model may only include he firs wo kernels (since he value of p 3 is on he order of -5 o -4 ). The resuling expressions for he firs and second order kernels in he ime domain are given in Equaions (9) and () (nex page) respecively: p3 k( τ ) = g b [ exp( pτ ) exp( pτ )] (9) p p These firs and second-order Volerra kernels are ploed in Figure (op panel) for ypical MM parameer values wihin he physiological range [5, 35]: g b =8 mg/dl, p =S G =. min -, p =.8 min - and p 3 = -4 min - ml/μu, which yield S I =.36 min - /μu ml -. Since he specific parameer values define he MM descripion of insulin-glucose dynamics, hey also define he form of he equivalen Volerra kernels. The form of he firs-order kernel in Figure (op lef panel) indicaes ha an μu/ ml insulin concenraion increase will cause a firs-order drop in plasma glucose concenraion ha will reach a minimum of abou -. mg/dl abou 36 min laer, rising afer ha o half he drop in abou hour and relaxing back o he basal value abou 4 hours afer he minimum. The posiive values of he second-order Volerra kernel indicae ha he acual glucose drop caused by he insulin infusion will be slighly less han he firs-order predicion (sublinear response). For insance, an insulin concenraion increase of μu/ml will no cause a maximum glucose drop of mg/dl (as prediced by is equivalen firs-order kernel) bu a drop of abou.5 mg/dl due o he anagonisic second-order kernel conribuion. Changes in hese parameer values affec he form and he values of he kernels in he precise manner described by Equaions (9) and (). The effecs of changes in he wo MM parameers p and p on he equivalen firs-order kernel are illusraed in Figure (boom panels) for a range of physiological values (p beween. and.4 min - and p beween. and.5 min - [5], keeping p 3 = -4 min - ml/μu consan. Noe ha changes in p 3 simply scale he firs-order kernel according o Equaion (9) and do no aler is form (proporional dependence) nor do hey aler he form of he second-order kernel (hey scale i quadraically). A direc sense of he effecs of parameer changes is obained by he waveforms of Figure : for insance, as p (S G ) increases, he maximum drop of he firs-order kernel becomes smaller and is dynamics (i.e. he drop o he minimum and he reurn o basal value) become faser. Similar effecs are observed when p increases (or S I decreases). B. Analyical expressions of he Volerra kernels of he comparmenal model: Closed-loop case To derive he analyical expressions of he kernels in he closed-loop case, we approximae he hreshold funcion of Equaion (6) wih a polynomial as indicaed below, assuming ha θ is equal o zero (i.e. insulin secreion is riggered when he glucose concenraion rises above is basal value): β T h [ g( ] βg( + β g ( + () where g( is he deviaion of glucose plasma concenraion from is basal value. Then Equaion (5) can be rewrien as: dr() = ar() + βg() + βg () + () The soluion of Equaion () is given by: r( = β f ( * g( + β f ( * g ( ) + (3)

6 TBME g a.5 μu min - /ml per mg/dl (lef panel) and β varying b p 3 k( τ, τ) = {[ exp( pτ) exp( pτ) ] [ exp( pτ) exp( beween pτ) ] + p. exp[ and p(. τ+ τμu min )](exp[ p - /ml min( per τ, τ)] mg/dl ) ( p wih a p) p () exp[ p( τ+ τ)] [exp( pτ pτ ) + exp( pτ pτ )](exp[ p min( τ, τ )] ) + (exp[( p p) min( τ, τ)] ) p p p H( s+ s) F( s+ s) K( s) K( s) H( s) K( s) + H( s) K( s) K( s, s) = p3 ( β + β) gb + (9) ( s+ s + p) + p3gbβh( s+ s) F( s+ s) ( s+ s + p) + p3gbβh( s+ s) F( s+ s) where he aserisk denoes convoluion and: a f ( = e u( (4) Also, from Equaion (4) we have: dx( = p3h( *[ i( + r( ] (5) where: p h( = e u( (6) Then Equaion (3) becomes: dg() + pg () = pg 3 ()[ h ()*() i + + β h ()* f()* g () + β h ()* f()* g () + ] (7) The above equaion can be used o obain he equivalen Volerra kernels of he closed-loop model, following he procedure oulined before for he open-loop model. The resuling expressions for he firs-order and he second-order kernels in he Laplace domain are given by Equaions (8) and (9) (op of page) respecively (k =): H ( s) K( s) = p3g b (8) s + p + p g β H ( s) F ( s) 3 b remaining consan a.3 min - (righ panel). The nominal value of a (.3 min - ) was aken from [37], while he value of β was se a.5 μu min - /ml per mg/dl, since he value repored in [37] (.54) resuled in negligible effecs of endogenous insulin secreion for he simuli used in his sudy. The decrease of a (slower insulin secreion dynamics) and increase of β (sronger feedback) affec he AMM firs-order kernel waveform similarly - i.e., hey resul in faser dynamics wih a small decrease of he negaive peak value and he appearance of an overshoo which is characerisic of closedloop sysems. C. Simulaion resuls: open-loop model In order o demonsrae he feasibiliy of esimaing he Volerra kernels of he open-loop MM direcly from inpuoupu measuremens, we simulae i by numerical inegraion of Equaions ()-() for he following values of MM parameers: p =. min -, p =.8 min -, p 3 = -4 min - ml/μu, g b =8 mg/dl ha are around he middle of he physiological ranges repored in he lieraure [4, 5]. The inpu signal for his simulaion is a zero-mean Gaussian whie noise (GWN) sequence of insulin ime-series (i.e. independen samples every 5 min), wih a sandard deviaion of 4 μu/ml, which may be viewed as sponaneous flucuaions around is basal value or arising from sep-wise coninuous infusions of insulin a random levels, changed every 5 min, superimposed on a consan (posiive) baseline infusion. Due o he low-pass dynamic characerisics of he model, one sample every 5 min is sufficien for represening he inpu-oupu daa. An inpuoupu record of 44 sample poins (i.e., hr long) is used o perform he raining of he LVN and he esimaion of he kernels of he equivalen Volerra model. Fig. 3. The firs-order kernels of he AMM for a varying beween. and.3 min - wih consan β=.5 (lef panel) and for β varying beween. and. μu min - /ml per mg/dl wih consan α=.3 min - (righ panel). where F(s), H(s) are he Laplace ransforms of f(, h( respecively, i.e.: F s) = s + a H ( s) s + p ( (3) = (3) The above relaions were invered numerically o yield he ime-domain expressions for he firs-order kernel, which are shown in Figure 3 for he following parameer values: a varying beween. and.3 min - wih β remaining consan TABLE I OUTPUT PREDICTION NMSES FOR VARIOUS LVN MODEL STRUCTURES AND VALUES OF P 3, GWN INPUT (OPEN-LOOP CASE). L p 3 =5-5 min - ml/μu p 3 = -4 min - ml/μu p 3 =5-4 min - ml/μu Linear NMSE Nonlinear NMSE Linear NMSE Nonlinear NMSE Linear NMSE Nonlinear NMSE The value of p 3 deermines he relaive conribuion of he nonlinear erms: noe ha for p 3 =5-5 min - ml/μu he NMSE reducion achieved by nonlinear models is marginal, while for 5-4 min - ml/μu i is over %. Using L>5 does no improve model performance furher.

7 TBME Fig. 4. Top panel: The esimaed firs and second order Volerra kernels of he MM using a GWN inpu of 44 poins ( hrs) when differen realizaions of independen GWN signals are added o he oupu for an SNR of 6.5 db. The obained firs-order (lef panel solid: mean value, dashed: +/- one sandard deviaion, doed: noise-free esimae) and second-order kernel esimaes (righ panel - mean value) are no affeced significanly relaive o heir exac counerpars (Fig. op panel), demonsraing he robusness of his approach. Boom panel: The esimaed firs and second order Volerra kernels of he MM for an insulin inpu composed of 8 insulin infusions over hrs. The iming and ampliude of each infusion are random (see ex. Noe he similariy of hese esimaes o he esimaes obained from GWN inpus. In order o illusrae model srucure selecion, we show he obained NMSEs for various values of L, as well as for linear (Q=) and nonlinear (Q=) models for hree differen values of p 3, which deermines he srengh of he MM nonlineariy, in Table I. For p 3 =5-5 min - ml/μu he model is weakly nonlinear, whereas for p 3 =5-4 min - ml/μu he NMSE reducion achieved for Q= is over %. The conribuion of he n-h order Volerra erm is proporional o he n-h power of he produc of parameer p 3 wih he power level of he inpu (i.e., his conribuion increases for larger insulin variaions); however, for he range of values examined, a second-order model is found o be sufficien. Also, using L>5 reduces he NMSE minimally in all cases. Therefore, we selec a second-order LVN wih one hidden uni and five Laguerre funcions (i.e., L=5, K=, Q=) for he esimaion of he equivalen Volerra model, wih he resuling oupu predicion NMSE being.9% (p 3 = -4 min - ml/ μu). The esimaed kernels of firs (Fig. 4 doed) and second order for he noise free case are almos idenical o he rue kernels given by Equaions (9)-() (Fig op panel). In order o examine he effec of measuremen noise on he kernel esimaes, we repea he kernel esimaion wih he aforemenioned inpu-oupu daa afer he addiion of independen whie-noise signals wih maximum ampliude equal o approximaely % of he basal glucose value (i.e., error range of ±6 mg/dl) o he oupu [47]. This corresponds o an SNR of around 6.5 db relaive o he de-meaned glucose deviaions oupu. The resuling kernel esimaes are also shown in Figure 4 (op panels) and demonsrae he robusness of his modeling approach in he presence of measuremen noise. The corresponding linear and nonlinear NMSEs are equal o 4.±.7% and 3.6±.7% respecively (mean± sandard deviaion), i.e., he oupu addiive noise is no accouned by he model. Also in Figure 4 (boom panels), we presen he kernel esimaes obained wih an insulin inpu of he same lengh (44 poins) composed of a random sequence of impulses (represening insulin concenraion increases ha could be due o insulin infusions), wih a mean frequency of impulse every hrs and a normally disribued random ampliude wih sandard deviaion μu/ml. The resuling kernel esimaes are almos idenical o heir GWN-inpu counerpars, demonsraing he feasibiliy of esimaing accurae Volerra models using sparser, infusion-like simuli. D. Simulaion resuls: Closed-loop model The closed-loop AMM was simulaed wih he same GWN inpu used for he open-loop MM by numerical inegraion of Equaions (3)-(5), for p =. min -, p =.8 min -, p 3 = -4 min - ml/μu and parameer values of a=.3 min -, β=.5 μu min - /ml per mg/dl, θ=8 mg/dl for he addiional insulinsecreion equaion. Represenaive ime-series daa of he resuling insulin inpu, insulin secreion, insulin acion and glucose, used for raining he equivalen LVN model, are shown in Figure 5, where he effec of insulin secreion, relaive o he open-loop case, can be seen in he boom righ panel (solid: closed-loop oupu, dashed: open-loop oupu. Fig. 5. Represenaive realizaion of he closed-loop AMM ime-series daa for a GWN insulin inpu used for LVN raining (lengh: hrs). The insulin ime series represen deviaions from is basal value. The effec of he secreion equaion is seen by comparing he wo oupu waveforms of glucose deviaions shown in he boom righ panel (dashed: open-loop, solid: closed-loop for β=.5). An LVN wih L=5, K= and Q=3 was employed in his case - i.e., a more complex srucure of higher order is required relaive o he open-loop case. In he noise-free case, he obained nonlinear model reduces he predicion NMSE considerably, from.4% - yielded by he linear model - o.8% (Figure 6, op lef panel). As before, we repea he kernel esimaion afer adding independen whie noise sample signals (wih he same variance as above) o he oupu. Noe ha he resuling SNR is now around 4.5 db, i.e. lower han he open-loop case, since he noise-free oupu (glucose deviaions) has a smaller mean-square value in he closed-loop case, due o he effec of he endogenous insulin secreion. Therefore, he corresponding NMSEs are larger -

8 TBME i.e. 48.% for he linear model and 34.±4.% for he nonlinear model and correspond, for he nonlinear model, o he noise conen. This demonsraes he predicive capabiliy of he obained models in he presence of considerable oupuaddiive noise ha emulaes he observed errors in he measuremens of curren coninuous glucose moniors [47]. The kernel esimaes for boh cases are shown in Figure 6, illusraing he robusness of his approach. The firs and second order kernels of he closed-loop AMM exhibi biphasic characerisics (i.e. regions of posiive and negaive response o a posiive change in he inpu, and vice versa). The firsorder kernel conribuion o he oupu remains dominan over he second-order kernel conribuion for impulsive inpus up o abou μu/ml. lower branch exhibis faser dynamics (shorer laency of he firs peak of abou 3 min) and has a nonlineariy ha resembles a sigmoidal (sof sauraing) characerisic. Fig. 6. Represenaive model predicions (noise-free oupu, op lef and esimaed firs and second order Volerra kernels of he closed-loop AMM for a GWN inpu of 44 poins ( hrs) for noise-free oupu (op righ doed and boom lef and when differen realizaions of independen GWN measuremen noise are added o he oupu for an SNR of 4.5 db (op righ solid black: mean, dashed black: +/- one sandard deviaion and boom righ - mean). Nonlinear models achieve beer predicions (over % NMSE reducion). The obained kernel esimaes are no affeced significanly relaive o heir noise-free counerpars despie he low SNR. The obained equivalen PDM models for boh he open-loop and closed-loop models are shown in Figure 7. In he openloop case (op panel), since we have used K= in he LVN model, he equivalen PDM model has one branch, wih he PDM dynamics exhibiing similar characerisics o he openloop firs-order kernel (Fig. ) and he saic nonlineariy being close o linear. In he closed-loop case (boom panel), we have used K=; herefore, he equivalen PDM model has wo branches. The lower PDM exhibis a clear biphasic response characerisic (corresponding o a glucose decrease and increase respecively, in response o an insulin increase) ha is no presen in he open-loop model. The upper PDM branch exhibis slower dynamics (peak laency of abou 8 min) han he open-loop PDM (peak laency a 4 min) and a sricly negaive nonlineariy (i.e., always leading o a reducion of glucose), while he nonlineariy of he open-loop model has boh posiive and negaive response regions. The PDM of he Fig. 7. The obained PDM model for he open- and closed-loop models, which consis of one and wo branches (op and boom panels respecively). The open-loop single PDM (op lef panel) exhibis a glucolepic characerisic (reduces he glucose oupu for posiive insulin inpus in a mildly sublinear manner. The closed-loop upper PDM branch exhibis a glucolepic characerisic for posiive or negaive insulin inpus in a mildly supralinear manner, unlike he single PDM branch of he open-loop MM. Noe ha he laency of he peak response (abou 8 min) is much longer for his closed-loop PDM han for he open-loop PDM (abou 4 min), and he slope of is oupu nonlineariy is differen for posiive/negaive inpu (abou 4 o ). The lower PDM is biphasic wih he firs glucolepic peak having a laency comparable o he open-loop PDM (abou 3 min) and he second glucogenic peak being much smaller (abou 5%) and having a laency of abou min. The nonlineariy of he lower PDM branch reains he biphasic response characerisic (increase of insulin leads o glucose decrease and vice versa) and is mildly sublinear (resembling a sof sauraing characerisic). IV. DISCUSSION In he presen paper, we have rigorously examined he relaion beween nonlinear comparmenal and Volerra models of glucose meabolism. Two widely used comparmenal models, he minimal model (MM) of glucose disappearance and is closed-loop exension (AMM), which includes he effecs of insulin secreion, were formulaed in he Volerra-Wiener framework and equivalen descripions, in he form of Volerra models, were derived analyically. The effec of parameric model parameers of clinical imporance on hese descripors (Volerra kernels) was examined. Using simulaed daa generaed from he aforemenioned comparmenal models, we have demonsraed he feasibiliy of obaining Volerra models ha describe hese daa accuraely, using boh random-like and impulsive insulin

9 TBME simuli. We have also shown ha hese esimaes are no affeced significanly by oupu-addiive noise corresponding o measuremen noise. The resuls provide evidence ha Volerra models, free of a priori assumpions, may be esimaed reliably from paien-specific daa. These models may provide quaniaive descripions ha reflec he underlying physiological mechanisms under general operaing condiions and may prove useful in diagnosic or herapeuic (e.g., for glucose regulaion for an iniial repor, see [48]) applicaions. This should be furher verified using glucose disurbance paerns and experimenal daa from diabeic paiens, a ask ha is currenly underway. We should noe ha for model-based glucose conrol applicaions, addiional facors, such as he delay beween plasma glucose and he sensor signal, should be aken ino accoun. The parameric models examined herein are nonlinear due o he presence of a bilinear erm in Equaions () and (3), which modulaes he effecive ime consan of glucose disappearance and depends on he acion of plasma insulin (in he case of MM) and boh plasma and endogenous secreed insulin (in he case of AMM) respecively. An addiional nonlineariy is found in he endogenous insulin secreion Equaion (5) of he AMM in he form of a nonlinear hreshold operaor. The range of values for he MM and AMM parameers is aken from he lieraure [4, 5, 34-37]. The value of p 3 was seleced owards he upper limi of previously repored values in order o increase he conribuion of he bilinear erm, while he parameer β in Equaion (5), which deermines feedback srengh was seleced o be larger han he value repored in [37] since, for he simuli examined in he presen paper, he effec of endogenous insulin was almos negligible for his laer value (i corresponds o low olerance, obese paiens [37]). Noe ha in he more general case, he value of β could be viewed as being dependen on g, in order o accoun for he effec of blood glucose concenraion on insulin secreion. The value of he hreshold θ in he endogenous insulin secreion Equaion (5) was seleced equal o zero in order o simplify he analyical derivaions. This hreshold can be generally se o a larger value, paricularly when glucose disurbance erms ha are non-insulin dependen, are included. However, in he conex of he simulaions presened herein, his value yielded reasonable paerns for he insulin secreion profile (Fig. 5). Two ypes of inpus (variaions of insulin concenraion) were used in his compuaional sudy for he simulaion of he parameric models: Gaussian whie noise (GWN) flucuaions around a puaive basal value (corresponding o he GWN mean) and random sequences of sparse insulin increases (abou one every wo hours on he average), which may resul from insulin/glucose infusions. I was shown ha reliable and robus nonparameric models can be obained wih boh ypes of simuli in he presence of measuremen noise. The GWN insulin flucuaions may also be viewed as inernal sponaneous flucuaions and, herefore, he applicabiliy of his approach can be exended o he case of sponaneous glucose/insulin measuremens. The use of random sequences of larger sparse impulsive insulin increases, alhough unconvenional, was shown o be effecive in erms of model esimaion and may offer clinical advanages as i is likely o miigae he risk of induced hypoglycemia an issue ha mus be examined carefully in fuure sudies. The Volerra approach does no require specific prior posulaes of comparmenal model srucures (e.g. i is no commied o any paricular number of comparmens) and allows esimaion of he model (i.e. he Volerra kernels) direcly from arbirary inpu-oupu daa. Therefore, i offers he advanage of yielding models ha are rue-o-he-daa and valid under all inpu condiions wihin he range of he experimenal daa. Therefore, his fundamenally differen approach provides significan benefis relaive o exising approaches in erms of modeling flexibiliy and accuracy. The robusness of he Volerra modeling approach (i.e. he effec of oupu-addiive noise on he obained kernel esimaes) was sudied by selecing as noise sample signals from a Gaussian whie noise process wih variance consisen wih wha is known abou glucose measuremen errors (i.e. a sandard deviaion equal o 4-% of he glucose basal value [47]). However, we mus make he disincion beween noise (which is primarily relaed o measuremen errors) and sysemic disurbance (which is relaed o sysemic perurbaions ha are no explicily accouned for in he model). The sysemic disurbance signal may include he effec of meals [49], he effec of circadian and ulradian endocrine cycles [5] and he effec of randomly occurring evens of acceleraed meabolism (due o exercise or physical exerion) as well as neuro-hormonal excreions (due o sress or menal exerion). The ampliudes and he relaive phases of hese disurbance componens will generally vary among subjecs and over ime. Since he selecion of such disurbance componens is raher complex, he sudy of heir effec on he robusness of he model esimaion is deferred o fuure sudies. The MM approach is based on he noion ha esimaes of he hree model parameers (p, p and p 3 ), obained hrough a glucose olerance es, provide he necessary clinical informaion for diagnosic purposes in he form of he equivalen indices of glucose effeciveness (S G ) and insulin sensiiviy (S I ). Alhough his proposiion has meri and has proven o be useful so far, i is widely recognized ha i has serious limiaions [5, 9, ]. To overcome some of hese poenial limiaions, our approach advances he noion ha a Volerra-ype model (in he form of kernels or he PDM model) provides he requisie clinical informaion in a more complee manner (i.e., no model consrains). In order o compare he relaive uiliy of he Volerra approach wih he convenional MM approach in a clinical conex, we mus define clinically relevan aribues for he wo approaches ha are direcly comparable. For insance, if we are ineresed in deriving quaniaive descripions/measures of how insulin affecs he plasma glucose concenraion in specific subjecs (i.e. based on colleced daa), we may use cerain feaures of he esimaed firs-order kernels, such as he inegraed area, peak value and iniial slope, which deermine he linear componen of he overall effec of an insulin injecion, is maximum insananeous effec and how fas his effec occurs respecively, insead of he esimaed MM parameers. In his conex, he combined effec of errors in he esimaes of he hree parameers of he MM (p, p, p 3 ) may be compared o esimaion errors in he inegraed area of he

10 TBME-59-8 firs-order kernel, which is equal o he raio S I /S G (i.e. p 3 /(p p )), as a measure of how much a uniary insulin impulse will affec he plasma glucose concenraion. Also, since S G = p is he inverse of he long ime-consan of he kernel (providing a measure of he exen of he kernel), i follows ha "insulin sensiiviy" S I is akin o he average kernel value. Thus, one may sugges ha he clinical index of insulin sensiiviy may be defined alernaively by he average kernel value and "glucose effeciveness" by he exen of he kernel in he daa-driven modeling conex. I also sands o reason ha he peak value of his kernel is likely o have some clinical significance, since i quanifies he maximum effec of an insulin injecion on blood glucose in a given subjec. Finally, he slope of he firs-order kernel a he origin (a measure of how rapidly glucose drops in response o an insulin infusion) is equal o (g b p 3 ). Since he basal glucose value is known, a quick esimae of p 3 can be obained from he slope of he firs-order kernel. In he above conex, PDM models (Figure 7) may prove very beneficial, since hey faciliae meaningful physiological inerpreaions relaive o he general Volerra formulaion. Therefore, cerain characerisics of he PDM branches (e.g., he dynamics of he linear filers and he characerisics of he nonlineariies) may also be associaed o clinical indices ha describe insulin acion and is efficiency in specific subjecs. As a firs illusraion, we provide he esimaes of several firs-order kernel feaures in he presence of noise in Table, in he case of he open-loop MM (Fig. 4 op panel). I can be seen ha he effec of oupu addiive noise is smaller in he esimaed kernel feaure values (variaion coefficien beween 8 % and 5%) han in he oupu daa (variaion coefficien of 5 %). However, he relaive uiliy of hese differen measures in a clinical conex will also depend on he robusness of heir esimaion in he presence of sysemic disurbances; herefore i is an issue ha deserves furher aenion and mus be examined in fuure sudies. TABLE II THE MEAN AND STANDARD DEVIATION (SD) OF ESTIMATED FEATURES OF THE FIRST-ORDER KERNEL FOR THE SIMULATED MM DATA OVER RUNS IN THE CASE OF NOISY OUTPUT AT SNR= 6.5 DB Firs-order kernel feaures Noise-free Noisy (SNR=6.5 db) (Mean ± SD) Area ±.3 Peak value ±.7 Time o peak ± 6.8 Iniial slope ±.8 The RMS of he noise-free oupu is approximaely wice he noise SD). The values of hese kernel feaures have specific analyical relaions wih he MM parameers p, p and p 3 (see ex. For example, he values of p 3 ha correspond o he esimaed iniial slope are min - ml/μu (noise-free case) and (9.68 ±.3) -5 min - ml/μu (noisy oupu respecively. Finally, we presen resuls from fiing he MM and LVN models o simulaed daa obained from he model proposed by Sorensen [5], which has been used as a comprehensive represenaion of he meabolic sysem in several sudies (e.g., [9], [35]) for insulin inpu signals considered above (i.e., random insulin variaions around a puaive basal value). Noe ha we do no make claims abou he universal validiy of his paricular model, bu we use i as a hird-pary meabolic simulaor for comparaive purposes. We considered wo disinc cases of Sorensen model parameers: one ha corresponds o a healhy subjec and anoher ha corresponds o a Type- diabeic subjec, following he procedure described in [39]. The MM parameers were obained by using a nonlinear opimizaion mehod (Levenberg-Marquar mehod) in order o fi p, p and p 3 o he Sorensen modelgeneraed daa. We considered differen realizaions of he insulin inpu signal (of he same lengh considered above) and provide he resuls in Table III and Figure 8. The resuls show ha he oupu predicion performance of he LVN model is superior in boh cases, paricularly for he Type- diabeic case. We noe ha an LVN model srucure wih L=5, K= and Q= was deemed appropriae in his case. TABLE III COMPARATIVE RESULTS OBTAINED FROM FITTING MM AND LVN MODELS TO SIMULATED DATA OBTAINED FROM THE SORENSEN MODEL FOR DIFFERENT RANDOM INSULIN INPUTS (VARIATIONS AROUND A PUTATIVE BASAL VALUE). Healhy Type- diabeic LVN NMSE [%] 3.6± ±4.9 MM NMSE [%].± ±.73 p [min - ].6±.6.8±.5 p [min - ].58±.4.4±.9 p 3 [min - ml/μu] (.3±.4) -5 (.5±.53) -5 The obained NMSE values correspond o he de-meaned glucose oupu daa. The LVN models yielded beer predicion performance overall, paricularly in he Type- diabeic case, while he obained MM parameer esimaes were influenced considerably by he paricular inpu realizaion. Values are Mean ± SD. Fig. 8. The predicions of he MM and LVN models for a represenaive Sorensen-model simulaed daa se (healhy subjec; lef panel) and he average firs-order kernel esimae of he LVN model for differen insulin inpu realizaions (righ panel solid black: mean dashed line: +/- one sandard deviaion) The presened resuls demonsrae he relaive advanages and disadvanages of he Volerra modeling mehodology versus he comparmenal approach for hese paricular parameric models (MM and AMM). The Volerra approach is inducive (daa-driven) and yields models wih minimum prior assumpions [38]. The comparmenal approach is deducive (hypohesis-based) and yields models wih he desired level of complexiy ha are direcly inerpreable bu no necessarily inclusive of all funcional characerisics of he sysem. The recen availabiliy of coninuous measuremens of glucose (hrough coninuous glucose sensors) and he feasibiliy of frequen infusions of insulin (hrough implanable insulin micro-pumps) make possible for he firs ime he realisic applicaion of daa-driven modeling approaches in a subjec-specific and adapive conex, which does no require he prior posulaes of comparmenal models. The poenial benefis include he inheren compleeness of he

11 TBME-59-8 obained models (in he sense ha hey will include all funcional characerisics of he sysem conained wihin he daa), he robusness of is esimaion in a pracical conex, is subjec-specific cusomizaion and is ime-dependen adapabiliy when he sysem characerisics are changing slowly over ime allowing effecive racking of hese changes in each specific subjec. REFERENCES [] The Diabees Conrol and Complicaions Trial Research, "The Effec of Inensive Treamen of Diabees on he Developmen and Progression of Long-Term Complicaions in Insulin-Dependen Diabees Mellius," The New England Journal of Medicine, vol. 39, pp , 993. [] A. D. Associaion, "Sandards of Medical Care in Diabees--8," Diabees Care, vol. 3, pp. S-54, 8. [3] J. Radziuk and S. Pye, "Hepaic glucose upake, gluconeogenesis and he regulaion of glycogen synhesis," Diabees/ Meabolism Research and Reviews, vol. 7, pp. 5-7,. [4] J. D. McGarry and R. L. 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12 TBME-59-8 [4] G. D. Misis and V. Z. Marmarelis, "Modeling of nonlinear physiological sysems wih fas and slow dynamics. I. Mehodology," Ann Biomed Eng, vol. 3, pp. 7-8, Feb. [43] V. Z. Marmarelis and X. Zhao, "Volerra models and hree-layer perceprons," Neural Neworks, IEEE Transacions on, vol. 8, pp , 997. [44] J. Sjoberg, "Non-linear Sysem Idenificaion Wih Neural Neworks," Linkoping, Sweden: Linkoping Universiy, 995. [45] V. Z. Marmarelis, "Modeling mehodology for nonlinear physiological sysems," Ann Biomed Eng, vol. 5, pp. 39-5, 997. [46] V. Z. Marmarelis, "Wiener analysis of nonlinear feedback in sensory sysems," Ann Biomed Eng, vol. 9, pp , 99. [47] J. Ginsberg, "The Curren Environmen of CGM Technologies," J Diabees Sci Technol, vol., pp. -7, 7. [48] M. G. Markakis, G. D. Misis, G. P. Papavassilopoulos, and V. Z. Marmarelis, "Model Predicive Conrol of Blood Glucose in Type Diabeics: he Principal Dynamic Modes Approach," in Proc. 3h Annual IEEE-EMBS Conf., Vancouver, BC, Canada, 8, pp [49] M. E. Fisher, "A semiclosed-loop algorihm for he conrol of blood glucose levels in diabeics," IEEE Trans. Biomed. Eng., vol. 38, pp. 57 6, 99. [5] E. V. Van Cauer, E. T. Shapiro, H. Tillil, and K. S. Polonsky, "Circadian modulaion of glucose and insulin responses o meals relaionship o corisol rhyhm," Am. J. Physiol., vol. 6, pp. R467 R475, 99 he Universiy of Souhern California, Los Angeles, where he is currenly Professor and Direcor of he Biomedical Simulaions Resource, a research cener funded by he Naional Insiues of Healh since 985 and dedicaed o modeling/simulaion sudies of biomedical sysems. He served as Chairman of he Biomedical Engineering Deparmen from 99 o 996. His main research ineress are in he areas of nonlinear and nonsaionary sysem idenificaion and modeling, wih applicaions o biology, medicine, and engineering sysems. Oher ineress include spaioemporal and nonlinear/nonsaionary signal processing, and analysis of neural sysems and neworks wih regard o informaion processing. He is coauhor of he book Analysis of Physiological Sysems: The Whie-Noise Approach NewYork: Plenum, 978; Russian ranslaion: Moscow, Mir Press, 98; Chinese ranslaion: Academy of Sciences Press, Beijing, 99) and edior of hree volumes on Advanced Mehods of Physiological Sysem Modeling (987, 989, and 994). He has published more han papers and book chapers in he area of sysem and signal analysis. His mos recen book, is Nonlinear Dynamic Modeling of Physiological Sysems, (Piscaaway, NJ: Wiley/IEEE, 4). Georgios D. Misis (S 99, M ) was born in Ioannina, Greece in 975. He received he Diploma in Elecrical and Compuer Engineering from he Naional Technical Universiy of Ahens, Greece in 997, M.S. Degrees in Biomedical and Elecrical Engineering from he Universiy of Souhern California, Los Angeles, CA in and respecively and he Ph.D. Degree in Biomedical Engineering from he Universiy of Souhern California in. Afer posdocoral appoinmens a he Biomedical Simulaions Resource, Los Angeles CA and he fmrib Cenre, Universiy of Oxford, UK and an ENTER Research Fellowship a he Naional Technical Universiy of Ahens, Greece, he joined he Deparmen of Elecrical and Compuer Engineering, Universiy of Cyprus, Nicosia, Cyprus, where he is currenly a Lecurer. His research ineress include nonlinear and nonsaionary sysems idenificaion, wih applicaions o quaniaive/sysems biology and physiology, as well as funcional magneic resonance imaging of he brain. Dr. Misis is a member of he Technical Chamber of Greece. He is currenly serving as an Associae Edior for he Annual IEEE EMBS conference (Biosignal Processing Theme). Mihalis G. Markakis was born in Ahens, Greece, in 98. He received he B.S. degree from he Naional Technical Universiy of Ahens, in 5, and he M.S. degree from he Universiy of Souhern California, in 8, boh in elecrical engineering. He is currenly affiliaed wih he Laboraory for Informaion and Decision Sysems a he Massachuses Insiue of Technology, working owards his Ph.D. His research ineress are in he areas of modeling and conrol of dynamic and sochasic sysems, wih applicaions ranging from physiological sysems o communicaion neworks. Vasilis Z. Marmarelis (M 79 SM 94 F 97) was born in Myilini, Greece, on November 6, 949. He received he Diploma degree in elecrical and mechanical engineering from he Naional Technical Universiy of Ahens, Ahens, Greece, in 97 and he M.S. and Ph.D. degrees in engineering science (informaion science and bioinformaion sysems) from he California Insiue of Technology, Pasadena, in 973 and 976, respecively. Afer wo years of posdocoral work a he California Insiue of Technology, he joined he faculy of Biomedical and Elecrical Engineering a