Mathematical Beta Cell Model for Insulin Secretion following IVGTT and OGTT

Transcription

1 Annals of Biomeical Engineering, Vol. 3, No. 8, August 2006 ( C 2006) pp DOI: 0.007/s Mathematical Beta Cell Moel for Insulin Secretion following IVGTT an OGTT RUNE V. OVERGAARD,, 2, 3 KATARINA JELIC, 2 MATS KARLSSON, 3 JAN ERIK HENRIKSEN, an HENRIK MADSEN Informatics an Mathematical Moelling, Technical University of Denmark; 2 Novo Norisk A/S, Måløv, Denmark; 3 Department of Pharmaceutical Biosciences, Uppsala University, Sween; an The Diabetes Research Centre, Department of Enocrinology M, Oense University Hospital, Denmark (Receive 30 November 2005; accepte 8 June 2006; publishe online: 3 July 2006) Abstract Evaluation of beta cell function is conucte by a variety of glucose tolerance tests an evaluate by a number of ifferent moels with less than perfect consistency among results obtaine from ifferent tests. We formulate a new approximation of the istribute threshol moel for insulin secretion in orer to approach a moel for quantifying beta cell function, not only for one, but for several ifferent experiments. Data was obtaine from 0 subjects that ha both an oral glucose tolerance test (OGTT) an an intravenous tolerance test (IVGTT) performe. Parameter estimates from the two experimental protocols emonstrate similarity, reproucibility, an inications of prognostic relevance. Useful first phase inexes comprise the steay state amount of reay releasable insulin A 0 an the rate of reistribution k r, where both yiel a consierable correlation (both r = 0.67) between IVGTT an OGTT estimates. For the IVGTT, A 0 correlates well (r = 0.96) with the 0 min area uner the curve of insulin above baseline, whereas k r represents a new an possibly more funamental first phase inex. For the useful secon phase inex γ, a correlation of 0.75 was foun between IVGTT an OGTT estimates. Keywors Parameter estimation, Pancreatic beta cell, Mixeeffects, Physiological moels. AUC BG BOV BSV CV FDR HGC IVGTT MTT OGTT RRI ABBREVIATIONS Area uner the curve Baseline glucose Between occasion variability Between subject variability Coefficient of variation First egree relatives to patients with iabetes Hyperglycaemic clamp Intravenous tolerance test Meal tolerance test Oral glucose tolerance test Reay releasable insulin Aress corresponence to Rune V. Overgaar, Informatics an Mathematical Moelling, Technical University of Denmark. Electronic mail: rvo@imm.tu.k 33 RRP SE Reay releasable pool Stanar error INTRODUCTION Type 2 iabetes is a heterogeneous isorer characterize by a combination of impaire insulin secretion an insulin resistance, 7 in which either factor can be ominant. Of these interrelate subjects, the present work eals with the assessment of beta cell function, which relative to insulin resistance must be impaire for evelopment of type 2 iabetes, an may even represent the primary factor preisposing iniviuals to type 2 iabetes. 9 Insulin secretion in response to an abrupt increase in plasma glucose is known to be biphasic with a rapi peak at 2 min (first-phase), ecrease to nair at 0 5 min, an then graually increase within the next couple of hours (secon-phase). Early insulin release after glucose ingestion is a key etermining factor for the subsequent glucose concentration, 3 inicating that a reuce first-phase may be responsible for the evelopment of impaire glucose tolerance. 6 Eviently, iagnostic tests for the assessment of insulin secretion as well as insulin resistance have great value for epiemiological an clinical stuies. The most common oral aministration tests are the oral glucose tolerance test (OGTT) an the meal tolerance test (MTT), whereas the most common intravenous (IV) tests are the intravenous glucose tolerance test (IVGTT) an the hyperglycemic clamp (HGC). Several escriptive mathematical moels an moel base methos have been propose to calculate inexes for characterization of beta cell function from the various tests. 2,7,27,28 Although useful for the analysis of a specific experiment type, the moels can rarely be use across ifferent tests, an similar inexes obtaine from ifferent experiments are not necessarily in agreement, leaing to the conclusion that further work is neee for these inexes to be routinely use in clinical an epiemiological stuies /06/ /0 C 2006 Biomeical Engineering Society

2 3 OVERGAARD et al. Besies escriptive moels for characterization, more comprehensive mathematical moels,0,20 have been use to communicate an increase the unerstaning of the physiological mechanisms behin insulin secretion. 9 Of these, the istribute threshol hypothesis 0 has been use to argue an erive many of the escriptive moels. 6,2 However, to our knowlege, escriptive insulin secretion moels up until now all fail to incorporate the funamental mechanisms that have enable the istribute threshol hypothesis to escribe insulin secretion in response to a long series of glucose challenges, an hereby increase unerstaning of the beta cell function. In the present work we formulate a moel that inclues threshol istribution, reistribution, an incretin effects, an investigate the applicability of this moel to ata from the IVGTT an the OGTT. At a longer perspective, the present work is a step towars a moel for characterizing beta cell function, not only for one, but for many of the glucose tolerance tests; with parameters that are closer relate to the physiology than those of more empirical moels. Moel evelopment an evaluation was performe on ata from 0 iniviuals,,2 where each subject ha both an IVGTT an an OGTT performe. Inexes obtaine from the OGTT were compare to those from the IVGTT in orer to emonstrate parameter reproucibility an similarity, giving creit to the applicability of inexes, e.g. in epiemiological stuies. Inication of prognostic relevance was emonstrate by the extreme parameter values foun for subjects that subsequent to the stuy have evelope type 2 iabetes, also other such parameters exist. 3 RESEARCH DESIGN AND METHODS A total of 0 healthy normoglycemic subjects ha both an IVGTT an an OGTT performe; 20 subjects with no family history of iabetes an 20 first egree relatives (FDR) to patients with type 2 iabetes of which four subjects have evelope type 2 iabetes themselves within 0 years of the initial investigation. The protocols were approve by the local Ethics Committee an informe written consents were obtaine from all participants before testing both at the initial testing an at 0 years follow up. Clinical characteristics of the stuy populations are given in Table. The OGTT was performe by ingestion of 75 g glucose in a liqui solution. Bloo samples were obtaine in the fasting state (three samples) an for a total of 3 h following the glucose loa (5 samples). The IVGTT was performe, with an infusion of a 25% solution of glucose (300 mg of glucose per kg boy weight (max 25 g)) being given over min, immeiately followe by a saline flush (50 ml). Time zero was taken as the start of the glucose bolus an samples were collecte at 30, 20, 0,, 2, 3,, 5, 6, 8, 0, 2,, 6, 9, 22, 25, 30, 0, 50, 60, 70, 80, 90, 00, 0, 20, 0, 60 an 80 min, for etermination of plasma glucose an insulin. Plasma glucose concentration was measure at the besie by the glucose oxiase TABLE. Physical characteristics of the stuy subjects at initial examination. Relatives Control subjects N (F/M) 20 (8/2) 20 (8/2) Age (yr) 29. ± ±.7 HbAc (%) 6.7 ± ± 0.08 BMI (kg/m 2 ) 25. ± ± 0.9 Weight (kg) 76.6 ± ±.0 Fasting glucose (mm) 5. ± ± 0.08 Fasting insulin (pmol/l) 5.6 ± 3.0. ± 3.0 Note. Values are mean ± SE. HbAc: glycate haemoglobin (normal range: 5. 7.%); BMI: boy mass inex. metho with a Glucose Analyzer (Beckman Instruments, Inc., Fullerton, Calif., USA). Bloo samples for plasma insulin were immeiately centrifuge at Catthetime of stuy an store at 20 C until analysis an concentrations measure by a ouble antiboy raioimmunoassay in uplicate (Kabi Pharmacia Diagnostics AB, Uppsala, Sween). Moel Description Backgroun The istribute threshol hypothesis 0 successfully explains the ose epenent first phase insulin release following IV glucose aministration with a pool of insulin store in packets. Accoring to this hypothesis, ifferent packets have ifferent threshols, secreting insulin only when glucose concentration has exceee this threshol. Changes in plasma glucose concentration will alter the istribution of insulin in the packets, while provision of new insulin an reistribution of insulin among packets ensures convergence towar steay state when glucose concentration is constant. In principle, each packet correspons to a compartment, yieling a complicate numerical problem with a large number of couple ifferential equations. The present simplification lump all active packets together an all passive packets together to a two-compartment system. A mathematical analysis of the ifferences an approximation between the present moel an the istribute threshol moel is given in Appenix A, whereas the following list summarizes the main ifferences:. Steay state provision was moele with a sigmoi Emax function instea of the more complicate parametric function use in Incretin effects uring the OGTT were implemente for provision an packet activation. 3. The flow of insulin between passive an active packets is moele to be uniirectional, i.e. all reay releasable insulin is release from an active packet before it is eactivate.. Reistribution was moele to involve ranom activation rather than ranom change of threshol, see Appenix A.

3 Secretion Moel for IVGTT an OGTT All passive packets were approximate to contain the same amount of insulin. The central approximation is that all passive packets contain the same amount of insulin. This will not be influential uring simple glucose challenge tests, where glucose is single peake. But for multiple glucose peaks, e.g. for glucose oscillations, the response of the threshol istribution hypothesis will be ifferent from that of the present moel. Moel Equations As illustrate in Fig., the moel can be ivie into four components, () provision of new insulin, (2) a pool of reay releasable insulin (RRI), i.e. insulin in passive packets available for quick release as a consequence of abruptly increasing glucose concentrations, (3) a pool of insulin in active packets, which is quickly being release, an () a plasma compartment that represents insulin pharmacokinetics. The provision, P of new insulin is typically written as, P = α(p P(G, )) t P(G, ) is the glucose epenent steay state provision, escribe below. Glucose (G) is implemente as the linear interpolation between measure concentrations of plasma glucose, except uring the time interval between an min, where the baseline glucose (BG) value was use to ensure that the interpolate glucose oes not rise prior to glucose aministration at time zero. 2 When the system is not at steay state, e.g. ue to rapi changes in glucose, P(t) will be ifferent from P(G, ). Whereas changes in P(G, ) are immeiate, the changes in P will be elaye with time constant α, leaing to smooth changes. Note that this elay is not the only contributing factor to the elay in the secon phase insulin response, see Appenix B. Oral ingestion of nutrients is known to enhance insulin secretion, the incretin effect, leaing to higher insulin secretion uring an OGTT than from an experiment with matche glucose concentrations obtaine by IV infusion of glucose. 5 The incretin effect is meiate by insulinotropic intestinal hormones, as e.g. glucagon-like peptie-, which enhances both first an secon phase release, see e.g. the experiments by Fritsche et al., 8 an consequently also the provision. We have P(G, ) = E maxg h EC50 h + + E [G BG] + Gh OGTT AUC G BG where the last term is the incretin effect, moele with glucose above BG as a surrogate for incretin hormones, which are rarely measure in the OGTT. [G BG] + is the maximum of zero an (G BG), an AUC G BG is the area uner the curve (AUC) of [G BG] +. EC 50 is calculate via the initialization given below. E OGTT is the effect parameter for the incretin effect in an OGTT, which is zero for an IVGTT. The two-compartment moel for the passive an active pool, as erive in Appenix A, can be written: t I passive = ( f (G))P k r I passive Ph t I active = f (G)P + k r I passive + Ph mi active SR = mi active ( ) f (G) = G h 2 / G h 2 + FC h 2 50 FC 50 = BG ( F BG / FBG ) /h2 where I passive is the total amount of insulin in packets with threshols above G, an I active is the amount in the active packets contributing to the total secretion rate SR. f(g) is the threshol istribution function, i.e. the fraction of active packets at a certain glucose concentration. FC 50 is the glucose concentration that activates 50% of the packets, F BG is the fraction of packets activate at baseline, an h 2 is the hill coefficient. The phase component, Ph, is the rate of insulin removal from the passive to the active packets cause by packet activation ue to rising glucose concentration. The assumption that all passive packets contain the same amount of insulin allow us to calculate Ph as the amount of insulin per passive packet I passive /N( f(g)) times the rate of packet activation N f(g)/t, where N is the total number of packets. Ph is: Ph = I passive f (G) t G f (G) for ( t G > 0) 0 for ( t G 0) Ph is seen to be sensitive to glucose changes, giving us a first phase insulin release when glucose concentration increases abruptly. Incretin effects on the first phase release are explaine by increase packet activation with oral glucose aministration, which is implemente through two ifferent values of h 2 in the OGTT an the IVGTT, h OGTT an h IVGTT. Plasma insulin concentration is compute assuming first orer elimination, t I plasma = SR/V k I I plasma where I plasma is the plasma concentration of insulin. V is the apparent volume of istribution for insulin, an k I is the elimination rate constant. Since V an E max (an E OGTT ) can be shown not to be simultaneous ientifiable, V was fixe to the plasma volume (3 l), which is slightly higher than typical estimates of the central volume of istribution. 5 Moel Implementation The moel was implemente as a non-linear mixeeffects moel in NONMEM V with FOCE. Parameters to be estimate: k I, α, E max, h, E OGTT, k r, m, F BG, h IVGTT, an h OGTT, are escribe in Table 2. E max,

4 36 OVERGAARD et al. FIGURE. Moel for observe insulin concentration in glucose tolerance tests. The moel inclues a one compartment moel for insulin pharmacokinetics, two compartments to escribe reay releasable insulin in the active an passive packets, an one ifferential equation to escribe the relationship between glucose an the provision of new insulin. The provision is ivie into the active an passive packets accoring to the fraction of active packets f(g), which is a function of the glucose concentration. k r, an E OGTT exhibit between subject variability (BSV) by the proportional moel, θ exp(η), an F BG exhibit BSV via the logistic function, F BG = exp(θ + η)/( + exp(θ + η)) to ensure values between zero an one, where θ is the fixe effect an η is a Gaussian ranom effect that varies between iniviuals. Data from the OGTT an the IVGTT were treate as if it was from ifferent iniviuals, such that no false correlation is introuce in the parameter estimates from the two trials. Observe (y) an preicte insulin plasma concentrations (I plasma ) were log-transforme for the resiuals ε to be Normal istribute, i.e. k I α E max h E OGTT k r m F BG TABLE 2. h IVGTT h OGTT log(y) = log(i plasma ) + ε Description of parameters to be estimate. The elimination rate of insulin The rate constant for provision to reach steay state Maximum value of the incretin inepenent part of the steay state provision rate Hill coefficient for the incretin inepenent part of the steay state provision rate Effect parameter for the incretin effect in an OGTT, which is zero for an IVGTT Reistribution rate constant from passive to active packets The rate of insulin release from active packets The fraction of packets activate at the initial steay state glucose concentration Hill coefficient for the threshol istribution function of the IVGTT Hill coefficient for the threshol istribution function of the OGTT Initialization The moel is initiate in steay state, uner the assumption that BG has prouce a steay state provision corresponing to the baseline insulin concentration I 0, where BG/I 0 is calculate as the average of measure glucose/insulin concentrations prior to glucose aministration. Initialization in steay state allow us to utilize the steay state equation to calculate EC 50, ( ) /h Emax EC 50 = BG I 0 Vk I Inexes The amount of RRI in the reay releasable pool (RRP) is known to change accoring to the history of glucose concentration, 9 whereas the size of the pool at fasting, i.e. the initial amount of insulin in the passive compartment A 0 is suggeste as a first phase inex. In the presente moel, A 0 is not estimate irectly, but erive using the following equation, A 0 = I 0 Vk I F BG k r It prove useful also to present a erive secon phase inex γ, which is the slope of the incretin inepenent part of the steay state provision with respect to glucose at BG, similar to inexes of other insulin secretion moels. 27,28 γ is compute by, γ = E maxhbg h EC h 50 ( BGh + EC h 50) 2

5 Secretion Moel for IVGTT an OGTT 37 TABLE 3. Parameter Unit Estimate Population Parameter Estimates. SE of estimate k I /min α /min E max pm/min % h E OGTT nmol % k r /min % m /min F BG % % h IVGTT h OGTT CV of parameter Moel Development Different moels were iscriminate base on, robustness, likelihoo function value, ability to capture iniviual insulin profiles, reproucibility, bias of parameter estimates compare to known physiological values, bias in the preictions mae by the typical set of parameters, an whether the implementation seeme physiologically reasonable. Base on the liste criteria for moel selection, a number of incretin effects on the first phase in the OGTT were attempte an iscare. These attempts inclue: () no first phase incretin effect, (2) some packets were activate uring the OGTT only, (3) the Ph input to active packets was elevate compare to removal from passive packets, corresponing to recruitment of insulin from packets not contributing to steay state secretion, () glucose above baseline or transit compartment functions was use to elevate packets activation. The escribe implementation of incretin effects for the secon phase was chosen above the following list of iscare attempts. () transit compartments were use to moel an incretin profile, (2) proportional rather than aitive incretin effects, (3) incretin effects irectly on the provision rather than on the steay state provision, () separate levels of E max an/or hill coefficient h were use for the IVGTT an OGTT. Besies investigations of ifferent structural moels, several combinations of between-subject variation were teste. BSV on the hill coefficients were juge to run unstable, whereas BSV on α an k I worsene the correlation between IVGTT estimates an OGTT estimates, possibly because iniviual values of α an k I cannot be estimate robustly from the OGTT. RESULTS The estimate typical parameter values for the stuy population are presente in Table 3, along with the stanar error (SE) of the estimate an the coefficient of variation (CV) between subjects in the stuy. Moel Preictions Two kins of moel preictions are calculate for this moel, () iniviual preictions are base on the iniviually estimate parameter values, an (2) population preictions utilize the typical parameter values to compute a typical insulin concentration profile for a given glucose profile. The geometric mean of the iniviual- an the population-preictions are compare to ata in Fig. 2, emonstrating ability of the moel to capture ifferences between insulin response in the OGTT an the IVGTT. Reproucibility an Similarity of Parameters Parameter estimates obtaine from the IVGTT are plotte against those obtaine from the OGTT in Fig. 3, an the correlation coefficients between estimates are presente (for parameters plotte on a log-scale the correlation coefficient for the log-transforme parameters are presente). γ, A 0, k r, exhibit a clear correlation between experiments (correlation coefficient aroun 0.7), whereas FC 50, F BG, an E max emonstrate an intermeiate correlation (correlation coefficient aroun 0.5). The high egree of correlation emonstrates parameter reproucibility, in the sense that subjects characterize with high values in one experiment are most likely associate with high values in the other experiment. All parameters except FC 50 are close to the line of unity, emonstrating similarity of the parameter values, giving a hint to a similar physiological origin of parameters from the ifferent experiments. The bias in FC 50 is ue to the implemente incretin effect, where a higher hill coefficient results in a higher fraction of packet activation when glucose starts to rise, resulting in a ecrease in the level of glucose necessary to activate 50% of the packets. Prognostic Inexes The four subjects that subsequent to the stuy have evelope type 2 iabetes are associate with a low A 0, alowf BG, a high FC 50, an a high k r, especially k r for which all exhibite a high value. For the secon phase, a low E max seems to inicate iabetes for the OGTT, but not for the IVGTT, whereas γ an EC 50 oes not appear to be particularly preictive. As expecte, also more escriptive inices, such as high BG an high AUC for glucose above BG, appear to implicate higher risk for evelopment of type 2 iabetes. Of the four subjects in a pre-iabetic state, the one with lowest BG was foun to have the lowest A 0 among the total stuy population, see Fig., inicating that the first phase inexes (A 0, k r, an F BG ) carry separate prognostic information to that of BG. DISCUSSION We have suggeste a new approximation of the istribute threshol hypothesis for parameter estimation that

6 38 OVERGAARD et al. IVGTT OGTT Glucose Concentration [mm] Glucose Concentration [mm] Time [min] Time [min] IVGTT OGTT Insulin Concentration [pm] Insulin Concentration [pm] Time [min] Time [min] FIGURE 2. All observe glucose (top) an insulin (bottom) concentrations (small light colore circles), geometric mean of insulin an mean of glucose (large black squares), geometric mean of iniviual preictions (light colore line), an geometric mean of population preictions (black line), are presente for the IVGTT (left), an the OGTT (right). Emax, Corr= 0. FC50, Corr= 0.52 KRD, Corr= 0.67 IVGTT Estimate IVGTT Estimate IVGTT Estimate OGTT Estimate OGTT Estimate OGTT Estimate GAMMA, Corr= 0.75 FBG, Corr= 0. A0, Corr= 0.67 IVGTT Estimate IVGTT Estimate IVGTT Estimate OGTT Estimate OGTT Estimate OGTT Estimate FIGURE 3. Iniviual parameter values obtaine from the IVGTT are compare to those obtaine from the OGTT. Triangles signify control subjects, circles signify first egree relatives of iabetics, an numbers signify the four first egree relatives of iabetics that have evelope type 2 iabetes within the 0 years since stuy completion. The correlation coefficient presente above each plot is calculate for the log transforme parameter for k r, FC 50,anA 0.

7 Secretion Moel for IVGTT an OGTT 39 OGTT Estimate KRD OGTT Estimate A IVGTT Estimate A OGTT Estimate Eogtt OGTT Estimate BG IVGTT EstimateBG FIGURE. Iniviual parameter values are compare to illustrate separate prognostic relevance. Triangles signify control subjects, circles signify first egree relatives of iabetics, an the numbers signify the four subjects that have evelope type 2 iabetes within the 0 years since stuy completion. k r is compare to E OGTT for the OGTT(left), A 0 is compare to BG for the OGTT(mile), an A 0 is compare to BG for the IVGTT(right). conserves funamental mechanisms of threshol istribution, active an passive packets, an reistribution among packets. Effects of incretin hormones were inclue on packet activation an provision of insulin, to allow quantification of beta cell function, both for the IVGTT an the OGTT. The parameter estimates exhibite similarity an reproucibility, an several parameters are promising caniates for an early iagnostic for evelopment of type 2 iabetes. However, in spite of the usefulness, the reproucibility was less than perfect, which eserves some iscussion. Reproucibility Three factors can be name to cause reuce correlation, as that observe between some of the parameter estimates in Fig. 3.. Between-occasion variability (BOV). For example, the less then perfect correlation of BG (r = 0.70) an I 0 (r = 0.76), ata not shown, must be cause by BOV. 2. Suboptimal esign, e.g. timing of measurements an chosen glucose aministration. Compare to the IVGTT, the OGTT exhibit some technical esign problems: (a) inevitably, incretin an plasma glucose effects occur over the same time perio, making them ifficult to separate. (b) Glucose rises slowly, making separation of first an secon phase ifficult. (c) Glucose elevation is lower, making estimation of E max ifficult. 3. Lacking physiological precision, in the sense that parameters in the OGTT an the IVGTT have separate physiological meanings. However, the similarity in parameter values obtaine from the OGTT an IVGTT inicates that this last point is not a ominant issue. Sampling Design In analogy with other moels, 2 the robustness of the iniviual parameter estimates in the OGTT were investigate for ifferent sampling esigns, an similar conclusions were obtaine, i.e. the accuracy of the first phase parameter estimates rops significantly when fewer measurements are inclue aroun times 0 50 min. The original esign inclues sampling at (0, 5, 0, 5, 20, 30, 0, an 50 min). When sustaining the (0, 0, 20, an 30 min) samples, the first phase inexes (A 0 an k r ) are robust, with high correlation to the original estimates (r 0.95). This correlation is reuce to (r 0.86), when using only the (0, 20, an 0 min) samples, an to (r 0.77) when using only the (0 an 30 min) samples. Also the estimates of E OGTT an E max were sensitive to reuce sampling esign. Secon Phase Inexes γ an E max were highly correlate for the IVGTT (r = 0.85), but less correlate in the OGTT (r = 0.52). Whereas E max was estimate, the introuction of γ was motivate by () γ is theoretically similar to previous successful secon phase inexes, 27,28 (2) a simulation stuy verifie that γ is more robustly estimate than E max in the OGTT, which is also seen as a higher correlation in Fig. 3. Both parameters were well estimate from the IVGTT, inicating esign problems with the OGTT. (3) A clear relationship exists between parameter estimates of E max an E OGTT, but not between γ an E OGTT. Incretin Effect For the OGTT, glucose above baseline exhibit similar ynamics as incretin hormones, see e.g. experiments by Rask et al., 2 an was foun the best available surrogate for their effect. However, the implementation leas to suspicion concerning the physiological origin (glucose or incretin hormones) of E OGTT. Three observations inicate successful estimation of the incretin effects: () the similarity of E max an γ estimates obtaine in the IVGTT an the OGTT. (2) No correlation was foun between E OGTT an AUC G GB, which woul be expecte if glucose alone was causing what was estimate as incretin effects. (3) A simulation stuy confirme that γ an E OGTT can be estimate simultaneously from the OGTT, with a correlation

8 350 OVERGAARD et al. between simulate an estimate values of (r = 0.9) for γ an (r = 0.95) for E OGTT. First Phase Inexes The AUC of insulin above I 0 uring the first 0 min after glucose aministration has been use as an inex for first phase secretion. 26 This inex correlate with A 0 (r = 0.96), k r (r = 0.63), an F BG (r = 0.30), emonstrating a clear ifference between the three inexes, where A 0 clearly represents the more traitional first phase inex. Insulin response to glucose in the OGTT has previously been successfully escribe by a static an a ynamic inex φ D, 2 where φ D epen on the erivative of glucose, which is similar to Ph in the present moel. An insulin base calculation of φ D emonstrate that φ D an A 0 obtaine from the OGTT correlates equally well to the 0 min AUC of insulin above baseline for the IVGTT (r = 0.65) an (r = 0.67). However, compare to previous moels, the present moel brings new an potentially important inexes to escribe beta cell function. Does Reistribution Represent a Funamental Factor for Beta Cell Dysfunction? Uner the assumption that packets represent beta cells, as elue to in other work, 23 reistribution woul be the activity of passive cells, e.g. by ranom activation, so that fast reistribution will represent a high frequency of cells releasing their content. Note that the precise formulation of reistribution has change slightly compare to the istribute threshol moel, see Appenix A, an that reistribution has not yet been unerstoo in a cell biological framework. In the whole boy system, glucose is known to have potentiating effects on the first phase release, 0 an reistribution is a likely mechanism for the normalization of an elevate RRP, making k r a etermining factor for the steay state RRP. Since provision of insulin to the RRP is necessary, steay state provision is another likely etermining factor for the steay state RRP. For subjects in Fig. 3, k r is large an A 0 is small, but k r is more pronounce than A 0, which coul inicate that reistribution is the more funamental factor for beta cell ysfunction. This makes sense, since A 0 epen upon baseline provision, which is known to increase uring evelopment of insulin resistance. Hence, A 0 may be a composite inex, influence in opposite irections by beta cell ysfunction an insulin resistance, whereas k r may be a more funamental factor for beta cell ysfunction. Provision The provision of insulin to the RRP is a simple function of glucose that lump a number of intracellular processes together, such as the glucokinase, up regulation of proinsulin, an increase formation of new insulin granules. This clear approximation of reality means that the rate constant α an E max, may not be vali for other experiments on longer time horizons. Threshol Distribution In the light of the fact that some 22,23,29 believe heterogeneity in beta cells activation threshol to cause the biphasic insulin release, this heterogeneity shoul relate to the estimate threshol istribution. In fact, uner the assumption that the potential size of the RRP in each beta cell is ientical, the estimate istribution of threshols an the measure istribution of activate beta cells shoul be ientical. Some experiments have foun that 53% of beta cells secrete etectable amounts of insulin at 5.6 mm glucose, which is in the high en of the range of iniviual estimates of F BG between 0. an 0.6. Two concerns for this comparison are that () reistribution from passive cells coul also lea to insulin secretion, allowing some passive cells to be estimate as active, (2) oscillations of insulin secretion coul interfere with the estimate activity frequency, which is not accounte for in the moel. It is worth noting that reistribution was estimate more robustly than F BG, an also that the inclusion of BSV on reistribution was more important than for F BG, to explain variations in the first phase response, possibly inicating a higher egree of uncertainty in the estimate values for F BG. Iniviual Secretion Profiles In Fig. 5, iniviual an population preictions are compare to ata for a few selecte iniviuals. For subject to 3, a small RRP was estimate for both experiments, corresponing to a lower than typical first phase for the IVGTT, an a lower than typical quick increase in plasma insulin concentration for the OGTT. Both subject to 3 have evelope type 2 iabetes post stuy, which is in agreement with the common unerstaning that a low first phase secretion is an early iagnostic marker for type 2 iabetes. Also subject has evelope type 2 iabetes, but for this iniviual the first phase response appears normal, an A 0 is among the highest 50%. Interestingly, this iniviual was associate with a relatively low F BG (among the lowest 30%) an a relatively high k r (among the highest 30%), which shoul inicate a low first phase. The apparent slightly above normal first phase originates from the provision of new insulin at baseline glucose, for which the subject ha the largest value among the 0 subjects. Subject 5 an 6 exhibit an above average amount of RRI, an as expecte none of these have evelope iabetes. Subject 6 has an E max aroun average in both experiments an E OGTT was the secon largest in the stuy, explaining why insulin concentrations uring the later stage of the IVGTT are close to typical, while they are far above the population preictions in the OGTT.

9 Secretion Moel for IVGTT an OGTT 35 FIGURE 5. Observe insulin concentrations (circles), are presente together with iniviual preictions (light colore line) an population preictions (black line) for selecte iniviuals. All 6 iniviuals were healthy at the time of the stuy, but iniviuals with i to have subsequently evelope type 2 iabetes, whereas iniviuals with i 5 an 6 have remaine healthy. Using Insulin vs. C-peptie for Assessment of Beta Cell Function Beta cells release an equimolar mixture of insulin an C-peptie, but C-peptie is cleare slower from plasma an oes not suffer the same first pass effect of the liver, hence C-peptie has been use in many moels an moel base methos to asses beta cell function. 2 In the absence of C-peptie ata, the present analysis was performe using insulin ata, so the obtaine secretion rate reflects a post hepatic secretion. One avantage of the higher elimination rate of insulin is that changes in secretion lea to more pronounce changes in plasma concentration, which we believe to prouce high estimation accuracy. Limitations an Potential Future Implementations The presente metho uses ata from the IVGTT an the OGTT simultaneously to fin all parameters. In orer to use the moel for estimation in a single experiment, it may be necessary to use Bayesian techniques or to fix some parameters. In our analysis, we moele the OGTT alone by fixing some parameters to the values foun by the simultaneous analysis, while estimating only parameters with BSV. The present moel is evelope from oral an intravenous aministration of glucose, using ata within 3 h of glucose aministration. If one wish to moel longer experiments or use other combinations of nutrients, then the limitations of the moel in its present formulation has been exceee. Compare to more empirical beta cell moels, one avantage of the present more mechanistic moel is that it can more naturally be extene an ajuste to account for new situations. Fruitful moel extensions an ajustments coul possibly inclue: () a escription of other experiments such as the HGC, to inclue new features at sustaine high glucose challenges (2) moel extensions to inclue simultaneous moels of the MTT an the OGTT coul possibly escribe the incretin effects from various compositions an amounts of nutrients, an possibly link the incretin effect to measure incretin hormones, (3) moel extensions that inclue a triple meal test coul investigate whether the potentiating effects of insulin release 8 is cause by the RRP, or possibly another pool, () moeling of the moifie IVGTT, to test whether the large tolbutamie riven insulin release can be relate to the RRP inclue in the moel, (5) extensions to inclue covariates for characterization of ifferences between patient populations, an (6) uring evelopment of rugs that target beta cells, escription, unerstaning, an preictions of results in future trial esigns coul be aie by the presente moel. CONCLUSION A new approximation of the istribute threshol hypothesis has been formulate, an it was verifie that it can be use for parameter estimation of the IVGTT an the OGTT. With this initiative, we approach a population moel for quantification of beta cell function that can be use for several of the ifferent tolerance tests available. The present work focuse on similarity, reproucibility, an prognostic relevance of iniviual parameters estimate from an IVGTT an an OGTT, for which valiation further inclue comparison to other inexes, simulation stuies, an preiction of iniviual profiles. First phase inexes comprise the steay state amount of reay releasable insulin, similar to traitional first phase inexes, an the rate of reistribution, which represents a new an possibly more funamental

10 352 OVERGAARD et al. first phase inex. The most useful secon phase inex γ is theoretically similar to the secon phase inex of other moels, an it is precisely estimate for the IVGTT, an to some exten believe separable from the incretin effects of the OGTT. Lack of perfect correlation between parameters estimates from the two experiments is likely cause by between-occasion variability, an by the esign of the OGTT, which yiel more imprecise parameter estimates, but enables estimation also of the incretin effect. APPENDIX A In the present appenix we use the original threshol istribution hypothesis 0 to erive the equations for the active an passive amounts of insulin presente in the main text. This erivation involves three approximations/alterations to the original moel that can be applie in arbitrary orer, for instance as escribe below. The funamental hypothesis in the original istribute threshol moel is that the reaily releasable insulin is store in small packets, where the ifferent packets have ifferent threshols, secreting insulin into the plasma only when the glucose concentration has exceee this threshol. The amount of reaily releasable insulin in packets with threshol between θ an θ + θ is given by ξ(θ,t)θ, so the threshol ensity istribution function can be use to moel the total secretion into plasma. The original istribute threshol moel, 0 can be written as, ξ(θ,t) = mξ(θ,t)h(g θ) + f (θ)p(g, t) t SR = m k r ξ(θ,t) + k r f (θ) G 0 ξ(θ, t)θ 0 ξ(θ, t)θ The secretion of insulin into the plasma is realize through the first term, where H( ) is the Heavisie function. The secon term is the provision of new insulin, an the last two terms are name reistribution terms corresponing to a ranom change of threshols of the ifferent packets. Note that f (θ) = /θ f (θ), is the istribution ensity function. In the original analysis glucose starte at zero, so that f (θ) gives the initial insulin istribution ensity function ξ(θ,0). I passive an I active can be written as, I passive = G ξ(θ, t)θ ; I active = G 0 ξ(θ, t)θ The ifferential equations for I passive an I active can be erive analytically, t I passive = ( f (G))P k r ( f (G))I passive + k r f (G)I active ξ(g, t) G t t I active = mi active + f (G)P + k r ( f (G))I passive k r f (G)I active + ξ(g, t) G t The first suggeste alteration to the istribute threshol moel involves uniirectional flow of insulin from passive packets to active packets. Note that the beta cell action potential will either spike or not, where a spike will lea to exocytosis. It is not possible to stop the spike by a ecrease in glucose an thereby stopping exocytosis of insulin that is currently being release, motivating a uniirectional flow. Whereas this change oes simplify the moel structure, the rate constant m is so large that it oes not alter the results. We get, t I passive = ( f (G))P k r ( f (G))I passive ξ(g, t) G ( ) G t H t t I active = mi active + f (G)P + k r ( f (G))I passive + ξ(g, t) G ( ) G t H t In the istribute threshol moel, reistribution can be unerstoo as a ranom change in the sensitivity to glucose, where passive packets may change threshol but still be passive. This formulation is slightly change. In the present moel, reistribution involves ranom activation of packets, in the sense that a passive packet may by a ranom mechanism release all insulin in its reay releasable pool, an then return to the passive state. By this mechanism, reistribution will not ecrease to zero when all packets are passive. The equations become, t I passive = ( f (G))P k r I passive ξ(g, t) G t H ( ) G t t I active = mi active + f (G)P + k r I passive + ξ(g, t) G t H ( ) G t The two alternative moels for reistribution were compare uring moel evelopment, an the chosen formulation prouce superior correlations between OGTT an IVGTT parameter estimates, superior objective function value, an was numerically more robust when changing initial estimates in the estimation. The central approximation in the present approach is that all passive packets contain the same amount of insulin, so that ξ(g, t) = I passive f (G)/( f (G)). This approximation will not be influential uring simple glucose challenge tests, where glucose is single peake. But if a secon an ientical peak is seen immeiately after the first peak, the

11 Secretion Moel for IVGTT an OGTT 353 threshol istribution hypothesis will preict the secon peak to give no first phase, because the passive packets involve are empty, whereas the present approximation will preict a nonzero first phase, since all passive packets contain the same amount of insulin. This ifference may be of particular relevance for preictions of insulin response to rapi oscillations in glucose. Following this approximation we get the equations for the amount of insulin in the passive an active packets that were presente in the main text, t I passive = ( f (G))P k r I passive I passive f (G) G f (G) t H ( ) G t t I active = mi active + f (G)P + k r I passive + I passive f (G) G f (G) t H APPENDIX B ( ) G t The present appenix provies the exact solution to the amount of insulin in the passive an active packets following a step increase in glucose from G to G 2,attimet = 0. Starting at the steay state solution: P(0 ) = P(G, ) I passive (0 ) = P(G, )( f (G )) k r I active (0 ) = P(G, ) m For a step increase in glucose, the Ph contribution can be compute via a Dirac elta-function, Ph = P(G, )( f (G 2 ) f (G )) δ(t) k r The amount of insulin in the passive packets can be compute as I passive (t) = ( f (G 2))P(G, ) + D(t); t > 0 k r D(t) = ( f (G 2 ))(P(G 2, ) P(G, )) α ( e ) k rt k r ( e αt ) k r (α k r ) where the first term reflects the immeiate removal of insulin ue to packet activation, corresponing to the first phase release. The secon term D(t) represents the elevation in the amount of insulin in the passive packets coming from an elevate provision, i.e. the secon phase contribution to the passive packets. α an k r constitute the two timescales for the secon phase contribution. Since α is more than a factor 0 larger than k r, the main contribution of the elaye increase of the secon phase comes from k r. The amount of insulin in the active packets can be compute as, I active = P(G, ) m + D 2 (t); t > 0 + P(G, )( f (G 2 ) f (G )) e mt k r D 2 (t) = e mt t 0 e mx ( f (G 2 ) (P(G 2, ) P(G, )) ( e αx ) + k r D(x) ) x where the first term in I active correspons to the initial amount, the secon term is the first phase contribution, an the thir term D 2 gives the secon phase contribution. Since m leas to a rather fast ecay compare to the remaining time scales, it is reasonable to approximate D 2 (t)as, ( D 2 (t) (P(G 2, ) P(G, )) ( f (G 2 )( e αt ) m + ( fg 2 )) α ( e ) ) k rt k r ( e αt ) k r (α k r ) One part originates from the immeiate increase in provision an another part originates from the steay increase in passive packets. REFERENCE Beal, S. L., L. B. Sheiner. NONMEM user s guies. NONMEM Project Group. San Francisco: University of California, Brea, E., M. K., Cavaghan, G. Toffolo, K. S., Polonsky, an C. Cobelli. Oral glucose tolerance test minimal moel inexes of beta-cell function an insulin sensitivity. Diabetes 50():50 58, Bruce, D. G., D. J. Chisholm, L. H. Storlien, an E. W. Kraegen. Physiological importance of eficiency in early pranial insulin secretion in non-insulin-epenent iabetes. Diabetes 37(6):736 7, 988. Cerasi, E. An analogue computer moel for the insulin response to glucose infusion. Acta Enocrinol. (Copenh) 55():63 83, Dea, M. K., M. Hamilton-Wessler, M. Aer, D. Moore, L. Schaffer, M. Loftager, A. Volun, R. N. Bergman. Albumin bining of acylate insulin (NN30) oes not eter action to stimulate glucose uptake. Diabetes 5(3): , Del Prato, S., P. Marchetti, an R. C. Bonaonna. Phasic insulin release an metabolic regulation in type 2 iabetes. Diabetes 5(Suppl ):S09 S6, Expert Committee Report of the expert committee on the iagnosis an classification of iabetes mellitus. Diabetes Care 26(Suppl ):S5 S20, Fritsche, A., N. Stefan, E. Hart, H. Haring, M. Stumvoll. Characterisation of beta-cell ysfunction of impaire glucose tolerance: evience for impairment of incretin-inuce insulin secretion. Diabetologia 3(7): , Gerich, J. E. Is reuce first-phase insulin release the earliest etectable abnormality in iniviuals estine to evelop type 2 iabetes? Diabetes 5(Suppl ):S7 S2, Grosky, G. M.. A threshol istribution hypothesis for packet storage of insulin an its mathematical moeling. J. Clin. Invest 5(8): , 972. Henriksen, J. E., F. Alfor, A. Hanberg, A. Vaag, G. M. War, A. Kalfas, an H. Beck-Nielsen. Increase glucose effectiveness in normoglycemic but insulin-resistant relatives of patients with

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