Quantitative indexes of -cell function during graded up&down glucose infusion from C-peptide minimal models

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1 Am J Physiol Endocrinol Metab 280: E2 E10, Quantitative indexes of -cell function during graded up&down glucose infusion from C-peptide minimal models GIANNA TOFFOLO, 1 ELENA BREDA, 1 MELISSA K. CAVAGHAN, 2 DAVID A. EHRMANN, 2 KENNETH S. POLONSKY, 3 AND CLAUDIO COBELLI 1 1 Department of Electronics and Informatics, University of Padova, Padova, Italy; 2 Department of Medicine, The University of Chicago, Chicago, Illinois 60637; and 3 Department of Medicine, Washington University School of Medicine, St Louis, Missouri Received 24 February 2000; accepted in final form 24 August 2000 Toffolo, Gianna, Elena Breda, Melissa K. Cavaghan, David A. Ehrmann, Kenneth S. Polonsky, and Claudio Cobelli. Quantitative indexes of -cell function during graded up&down glucose infusion from C-peptide minimal models. Am J Physiol Endocrinol Metab 280: E2 E10, Availability of quantitative indexes of insulin secretion is important for definition of the alterations in -cell responsivity to glucose associated with different physiopathological states. This is presently possible by using the intravenous glucose tolerance test (IVGTT) in conjunction with the C-peptide minimal model. However, the secretory response to a more physiological slowly increasing/decreasing glucose stimulus may uncover novel features of -cell function. Therefore, plasma C-peptide and glucose data from a graded glucose infusion protocol (seven 40-min periods of 0, 4, 8, 16, 8, 4, and 0 mg kg 1 min 1 ) in eight normal subjects were analyzed by use of a new model of insulin secretion and kinetics. The model assumes a two-compartment description of C-peptide kinetics and describes the stimulatory effect on insulin secretion of both glucose concentration and the rate at which glucose increases. It provides in each individual the insulin secretion profile and three indexes of pancreatic sensitivity to glucose: s, d, and b, related, respectively, to the control of insulin secretion by the glucose level (static control), the rate at which glucose increases (dynamic control), and basal glucose. Indexes (means SE) were s (10 9 min 1 ), d (10 9 ), and b (10 9 min 1 ). The model also allows one to quantify the -cell times of response to increasing and decreasing glucose stimulus, equal to (min) and (min), respectively. In conclusion, the graded glucose infusion protocol, interpreted with a minimal model of C-peptide secretion and kinetics, provides a quantitative assessment of pancreatic function in an individual. Its application to various physiopathological states should provide novel insights into the role of insulin secretion in the development of glucose intolerance. insulin secretion; -cell sensitivity; mathematical model; kinetics Address for reprint requests and other correspondence: C. Cobelli, Dipartimento di Elettronica e Informatica, Via Gradenigo 6a, Padova, Italy ( cobelli@dei.unipd.it). SEVERAL PROTOCOLS are currently in use to define the alterations in -cell responsivity to glucose associated with different physiopathological states, including the intravenous glucose tolerance test (IVGTT), the hyperglycemic clamp, the graded glucose infusion, and the oscillatory glucose infusion. In view of the importance of -cell dysfunction in the physiopathology of type 2 diabetes, these tests play an important role in our understanding of this condition. All these tests are based on the assumption that the major defects in -cell function result in reduced or absent secretory response to glucose. On the other hand, the inability to sense a fall in glucose and to suppress insulin secretion appropriately should also be considered as a possible defect in -cell dysfunction. An advantage of the graded glucose infusion protocol is its ability to characterize the dose-response relationship between glucose and secretion rate during a physiological perturbation, first by reconstructing the insulin secretion rate (ISR) by deconvolution, and then by plotting the average ISR against the corresponding average glucose level during each glucose infusion period (4, 5, 7). The value of the graded glucose infusion as a measure of -cell function could be greatly enhanced if it were possible to obtain, in addition to ISR, quantitative indexes describing -cell sensitivity to glucose, similar to what is available for the IVGTT, interpreted with a C-peptide minimal model (14, 15). The aim of the present study was to investigate whether a detailed characterization of -cell function can also be obtained from a more physiological slowly increasing/decreasing glucose infusion protocol (up&down graded infusion) by using a model to interpret glucose and C-peptide data. MATERIALS AND METHODS Selection and Definition of Study Subjects Studies were performed in eight healthy nondiabetic subjects (7 females and 1 male). Mean age was 34 3 (SE) yr, and body mass index was kg/m 2. Glucose toler- The costs of publication of this article were defrayed in part by the payment of page charges. The article must therefore be hereby marked advertisement in accordance with 18 U.S.C. Section 1734 solely to indicate this fact. E /01 $5.00 Copyright 2001 the American Physiological Society

2 E3 ance was determined by World Health Organization criteria during an oral glucose tolerance test (17). All subjects had a normal screening blood count and chemistries and took no medications known to affect glucose metabolism. All fasting plasma glucose levels were 98 mg/dl (5.4 mm), and glycosylated hemoglobin values were normal. The study protocol was approved by the Institutional Review Board at the University of Chicago, and all subjects gave written informed consent. Experimental Protocol All studies were performed in the Clinical Research Center at the University of Chicago, starting at 0800 in the morning after an overnight fast. Intravenous cannulas were placed in a forearm vein for blood withdrawal, and the forearm was warmed to arterialize the venous sample. A second catheter was placed in the contralateral forearm for administration of glucose. Subjects received graded glucose infusions at progressively increasing and then decreasing rates (0, 4, 8, 16, 8, 4, 0 mg kg 1 min 1 ). Each glucose infusion rate was administered for a total of 40 min. Glucose and C-peptide levels were measured at 10-min intervals during a 40-min baseline period before the glucose infusion and throughout the 240-min glucose infusion. Assay Plasma glucose was measured immediately by the glucose oxidase technique (Yellow Springs Instrument analyzer, Yellow Springs, OH). The coefficient of variation of this method is 2%. Plasma C-peptide was measured as previously described (10). The lower limit of sensitivity of the assay is 0.02 pmol/ml, and the average intra- and interassay coefficients of variation are 6 and 8%, respectively. Glycosylated hemoglobin was measured by boronate affinity chromatography, with an intra-assay coefficient of variation of 4% (Bio-Rad Laboratories, Hercules, CA). Models of C-peptide Secretion and Kinetics Because the secretion model is assessed from C-peptide measurements taken in plasma, it must be integrated into a model of whole body C-peptide kinetics. The well validated model, originally proposed in Ref. 9, has been assumed (Fig. 1): compartment 1, accessible to measurement, represents plasma and rapidly equilibrating tissues; compartment 2 represents tissues in slow exchange with plasma. Model equations are Fig. 1. Model of C-peptide kinetics. CP 1 and CP 2 (pmol/l) are C- peptide concentrations in the accessible and peripheral compartments, respectively; k ij (min 1 ) are kinetic parameters; SR (pmol l 1 min 1 ) is the pancreatic secretion normalized to the volume of distribution of compartment 1, and y is the C-peptide concentration measurement. CP 1 t k 01 k 21 CP 1 t k 12 CP 2 t SR t CP 1 0 (1) CP 2 t k 21 CP 1 t k 12 CP 2 t CP 2 0 where the overdot indicates time derivative; CP 1 (pmol/l) is C-peptide concentration (above basal) in compartment 1; CP 2 (pmol/l) is the equivalent concentration in compartment 2 (above basal), equal to the C-peptide mass in compartment 2 divided by the volume of the accessible compartment; k 12 and k 21 (min 1 ) are transfer rate parameters between compartments; k 01 (min 1 ) is the irreversible loss; and SR (pmol l 1 min 1 ) is the pancreatic secretion (above basal) entering the accessible compartment, normalized to the volume of distribution of compartment 1. As for the IVGTT model (14), the functional relationship between insulin secretion and plasma glucose concentration is derived from a previously proposed model (11, 12) based on the packet storage hypothesis of insulin secretion. SR is described as the sum of two components controlled, respectively, by glucose concentration (static glucose control) and by the rate of change of glucose concentration (dynamic glucose control) SR t SR s t SR d t (2) SR s is assumed to be equal to Y (pmol l 1 min 1 ), the provision of new insulin to the -cells SR s t Y t (3) which is controlled by glucose according to the following equation Ẏ t Y t G t G b Y 0 (4) i.e., in response to an elevated glucose level, Y and thus SR s tend with a time constant 1/ (min) toward a steady-state value linearly related via parameter (min 1 ) to glucose concentration G (mmol/l) above its basal level G b (static glucose control). Parameter describes the static control of glucose on -cells. SR d is assumed to represent the secretion of insulin stored in the -cells in a promptly releasable form (labile insulin). Labile insulin is not homogeneous with respect to the glucose stimulus: for a given glucose step, only a fraction of labile insulin is mobilized, so that more insulin can be rapidly released in response to a subsequent more elevated glucose step. It is first assumed that the amount of released insulin (dq) in response to a glucose increase from G to G dg is proportional to the glucose increase dg dq k d dg (5) The flux of insulin secretion, SR d, is then proportional to the derivative of glucose SR d t dq dt k dg dg d if dt dt 0 and G t G b (6) 0 otherwise Parameter k d describes the dynamic control of glucose on insulin secretion, i.e., the effect of the rate of change of glucose on insulin secretion when glucose concentration is increasing (dg/dt positive). As will be detailed in RESULTS, the model described so far, hereafter indicated as model M1, is able to describe the C-peptide data of most, but not all subjects. We therefore tested a second model, called model M2, which differs from M1 in that it incorporates a more flexible description of the

3 E4 Fig. 2. Parameter k(g) of the glucose dynamic control, equal to the ratio between secretion rate of stored insulin and the rate of change of glucose, for model M1 (A) and model M2 (B). dynamic control (Fig. 2): SR d is still proportional to the derivative of glucose, but the proportionality factor is allowed to vary with glucose concentration SR d t k G dg dt k d 1 G t G b G t G b dg dt 0 if dg dt 0 and G b G t G t otherwise According to Eq. 7, the dynamic control is maximum when glucose increases just above its basal value; then it decreases linearly with glucose concentration and vanishes when glucose concentration exceeds the threshold level G t able to promote the secretion of all stored insulin, i.e., an additional increase of glucose above G t has no effect on insulin secretion. M2 is a generalization of M1: in fact, for elevated G t, the term 1 G t G b G t G b approximates 1, and M2 reduces to M1. Model Assessment of Insulin Secretion Insulin secretion profile. Models M1 and M2 allow one to reconstruct the profile of insulin secretion ISR (pmol/min) during the up&down graded infusion as (7) s (10) DYNAMIC. The dynamic sensitivity to glucose measures the stimulatory effect of the rate of change of glucose on secretion of stored insulin. To calculate this index, it is useful to define first the parameter X 0 (pmol/l) as the amount of insulin (per unit of C-peptide distribution volume) released in response to the maximum glucose concentration G max achieved during the experiment X 0 Gb G max dq Gb G max k G dg (11) For model M1, X 0 is simply X 0 k d G max G b (12) For model 2, two situations must be considered. If G t G max, i.e., the dynamic control of glucose on insulin secretion is active in the entire rising portion of the curve, then G max X o k G dg k d 1 G max G b max G b (13) 2 G t G b G Gb If G t G max, then the dynamic glucose control is active as long as G G t, and X 0 becomes X 0 Gb G t k G dg k d G t G b /2 (14) By normalizing X 0 to the glucose increase, the dynamic sensitivity to glucose d (dimensionless) can be derived X 0 d (15) G max G b BASAL. The basal sensitivity index b (min 1 ) measures basal insulin secretion rate over basal glucose concentration b SR b k 01CP 1b (16) G b G b M1: ISR t SR b SR t V 1 k 01 CP 1b Y t k d dg dt V 1 if dg dt 0 and G t G b (8) k 01 CP 1b Y t V 1 otherwise M2: ISR t SR b SR t V 1 k 01 CP 1b Y t k d 1 G t G b G t G b dg 1 dt V k 01 CP 1b Y t V 1 if dg dt 0 and G b G t G t (9) otherwise where SR b is insulin secretion in the basal state, and V 1 (in liters) is the C-peptide volume of distribution in the accessible compartment. Sensitivity indexes. Three sensitivity indexes can be defined. STATIC. The static sensitivity to glucose s (min 1 ) measures the stimulatory effect of a glucose stimulus on -cell secretion at steady state. For both models Response times. The models also allow one to quantify the -cell response times (min) to a glucose stimulus. For both models, the -cell response time to a decreasing glucose stimulus (T down ) is simply T down 1 (17)

4 E5 because in this case, secretion equals provision Y, which is described by Eq. 4, with 1/ as time constant. When glucose increases, the additional amount X 0 of insulin secreted due to the dynamic control of glucose accelerates the -cell response. As detailed in the APPENDIX, this is equivalent to reduction in the -cell response time now indicated as T up Model Identification T up 1 d s (18) For both models M1 and M2, all parameters are a priori uniquely identifiable (6, 8), i.e., kinetic parameters k 01, k 21, k 12, and secretory parameters,, k d for M1 or,, k d,g t for M2. However, numerical identification of the models requires knowledge of C-peptide kinetics. Kinetic parameters were fixed to standard values by following the method proposed in Ref. 16. Their average values (means SE) were k min 1 ; k min 1 ; k min 1 ; and V liters. The secretory parameters of both models were then estimated for each subject, together with a measure of their precision, by applying weighted nonlinear least square methods (6, 8) to C-peptide data by using the SAAMII software (3). Weights were chosen optimally, i.e., equal to the inverse of the variance of the measurement errors, which were assumed to be independent, gaussian, and zero mean with a constant standard deviation, which has been estimated a posteriori. Glucose concentration, linearly interpolated between data, and its time derivative, calculated by means of a spline function interpolation of glucose data, have been assumed as error-free model inputs. The comparison between models was made on the basis of criteria such as independence of residuals, precision of the estimates, and the principle of parsimony as implemented by the Akaike Information Criterion (AIC) (6, 8). Statistical Analysis Values are reported as means SE. The statistical significance of differences has been calculated by the two-tailed Student s t-test. The independence of residuals has been assessed by use of the runs test (2). P 0.05 was considered statistically significant. Table 1. Estimated secretory parameters Subject No. RESULTS, 10 9 min 1, min 1 k d,10 9 G t, mmol/l Model M (2%) (7%) 136 (22%) (3%) (14%) 399 (17%) (6%) (37%) 145 (50%) (4%) (18%) 298 (34%) (4%) (14%) 208 (27%) (3%) (13%) 181 (27%) (3%) (13%) 224 (16%) (5%) (24%) 364 (37%) Model M (2%) (7%) 166 (40%) 31.5 (135%) (3%) (14%) 418 (18%) 99.1 (401%) (6%) (45%) 160 (69%) 43.0 (475%) (4%) (19%) 308 (35%) 100 ( 1,000%) (2%) (9%) 578 (16%) 11.2 (10%) (3%) (14%) 199 (46%) 68.6 (1335%) (3%) (14%) 292 (16%) 28.4 (42%) (3%) (21%) 905 (42%) 12.4 (26%) Precision of the parameter estimate is expressed as percent coefficient of variation and shown in parentheses. Parameter, static sensitivity index; parameter, time constant; k d, dynamic control of glucose on insulin secretion; G t, glucose threshold level. Mean plasma glucose and C-peptide concentration values during the up&down graded glucose infusion protocol are shown in Fig. 3. Individual secretion parameters of models M1 and M2 are summarized in Table 1, together with their precision. The ability of model M1 to fit the individual data is shown in Fig. 4. From Table 1, precise estimates are obtained with M1 in all of the eight subjects. With M2, precise estimates of all parameters are obtained only in subjects 5, 7, and 8. In these subjects, model M2 performs better than M1, as indicated by a lower AIC value (Table 2). In particular, it performs notably better than M1 in subjects 5 and 8, for whom M1 produces a systematic underestimation of the initial portion of the data (Fig. 4). In these subjects, residuals are independent with M2 but not with M1 (Fig. 5). In subject 7, M2 performance slightly improves, because residuals are independent for both models, but AIC is lower with M2. However, M2 cannot be resolved in subjects 1, 2, 3, 4, and 6, because G t estimates are very high and affected by poor precision (Table 1) with no improvement in model fit, i.e., M2 tends to reduce to M1. Therefore, insulin secretion has been assessed by using M1 for subjects 1, 2, 3, 4, and 6 and M2 for subjects 5, 7, and 8; the mean profile of -cell secretion (Eqs. 8 and 9) is shown in Fig. 6; sensitivity indexes and response times are reported in Table 3. DISCUSSION Fig. 3. Mean plasma glucose and C-peptide concentration during the up&down graded glucose infusion (n 8). The C-peptide minimal modeling approach, which has been successfully applied to IVGTT data (14, 15), has been used here to assess -cell secretion during a more physiological glucose perturbation, in which a rising followed by a falling glucose concentration is

5 E6 Fig. 4. Fit of model M1 in the 8 subjects. produced by an exogenous intravenous glucose infusion. A novel version of the model is proposed, which incorporates the assumption that glucose stimulates pancreatic insulin secretion by exerting both a static control, i.e., proportional to its concentration, and a dynamic control, i.e., proportional to its rate of change. Similar assumptions are not new in modeling hormone secretory processes. In the present study, they have been used to interpret the data mechanistically, because they have been derived by building on specific assumptions about the physiology of insulin secretion, first formulated in the classical packet storage insulin secretion model (11, 12) and then incorporated in the minimal model of insulin secretion and kinetics during IVGTT (14, 15). More specifically, the model assumes the presence in the -cells of a pool of promptly releasable insulin, which can be rapidly secreted when glucose increases above its basal value, and an insulin provision process, which accounts for a slower component of secretion by allowing the formation of new insulin from insulin precursors and/or conversion of insulin from a storage to a labile form. The Static Control of Glucose on Insulin Secretion It is assumed that insulin provision under steadystate conditions is proportional, through parameter, to the glucose stimulus, with a delay with respect to the glucose profile represented by 1/. Parameter thus represents the sensitivity s (static sensitivity index) of -cells to the glucose stimulus, because it measures the relation between secretion rate (above basal) at steady state and the glucose stimulus (above basal). Its value, , can be compared with the sensitivity in the basal state, b , because they are both steady-state secretory indexes. Our results ( s significantly higher than b ) indicate that a separate assessment of -cell function in the basal state and during a glucose stimulus is important, because -cells are more sensitive to a suprabasal glucose stimulus than to the basal glucose level. The Dynamic Control of Glucose on Insulin Secretion The assumption of a static glucose control is not sufficient to provide a reliable description of the C- peptide data when the glucose infusion rate is first increased and then decreased; the model fit obtained by coupling the model of C-peptide kinetics (Eq. 1) with a secretion rate coming from provision only, i.e., SR(t) SR s (Eqs. 3 and 4) produces a systematic underestimation, especially in the rising portion of C-peptide data, as shown in Fig. 7. These findings suggest the existence of an additional secretion term that is active when glucose increases and represents the counterpart of the IVGTT first-phase secretion observed immediately after the glucose bolus injection. However, the increase in glucose concentrations from basal to maximum levels during the up&down graded infusion protocol (120 min) is much slower than during Table 2. Akaike information criterion Subject No. M1 M Fig. 5. Weighted residuals of model M2 (continuous line) against those of model M1 (dashed line) in subjects 5 and 8.

6 E7 Fig. 6. Mean -cell secretion during the up&down graded glucose infusion predicted by use of models M1 (for subjects 1, 2, 3, 4, and 6) and M2 (subjects 5, 7, and 8). ISR, insulin secretion rate. the IVGTT (2 3 min). The description adopted for the up&down graded infusion was therefore different from that used for the IVGTT, albeit based on similar assumptions, namely the packet storage hypothesis of insulin secretion (11, 12). According to this hypothesis, a bulk of insulin is stored in the -cells in a promptly releasable form and is secreted, when glucose exceeds its basal level, with a nonhomogeneous response: for a given increase in glucose concentration, only a portion of labile insulin is secreted, so that subsequent more elevated glucose concentration steps are able to stimulate the secretion of additional insulin. By assuming that the amount of insulin secreted in a given period of time depends on the glucose increase in that period, one finds that insulin secretion is controlled by the glucose rate of change through a proportionality constant k(g), which in principle depends upon G. Two different descriptions have been tested for k(g), thus leading to two different versions of the minimal model of C-peptide secretion during the up&down glucose infusion, denoted as models M1 and M2, respectively. In the former, it has been assumed simply that k(g) is constant, k(g) k d, i.e., it does not depend on G. This means that an increase G in glucose concentration, from G 1 to G 2 G 1 G, promotes the secretion of an amount of insulin proportional to G but independent of the glucose levels G 1 and G 2. Parameter k d represents the sensitivity d (dynamic sensitivity index) of -cells to the glucose rate of change. The product of k d and the total increase in glucose concentration in the rising portion of the data measures the total amount X 0 Fig. 7. Inadequacy of a model that simply assumes a static glucose control on insulin secretion, shown as its mean fit against mean C-peptide concentration. (pmol/l) of insulin stored in the -cells before the experiment and thus released during the experiment. Model M1 was able to accurately describe the C- peptide data of all except two subjects, where it produced a systematic underestimation of the initial portion of the data. A preliminary analysis of data obtained from the up&down graded glucose infusion protocol in physiopathological states, i.e., severe obesity and impaired glucose tolerance (unpublished observations), confirmed the inadequacy of M1 to reproduce C-peptide data of a portion of subjects and suggested the use of a more flexible description of k(g). Therefore, model 2 was introduced, with k(g) linearly dependent on G, i.e., an increase G in glucose concentration promotes the secretion of an amount of insulin dependent not only on G but also on the glucose levels G 1 and G 2. Model M2 assumes that the sensitivity of the dynamic glucose control is maximal when G varies (increases) around basal, and then decreases with higher G so as to vanish at the threshold glucose level G t able to promote the secretion of the totality of stored insulin. k(g) is then described by two parameters, the maximal sensitivity at basal glucose, k d, and the threshold glucose concentration G t. M2 is a generalization of M1, because M2 reduces to M1 when the threshold value G t becomes very large. This is confirmed by our results: M2 significantly improves upon M1 in those subjects for whom M1 was not adequate and reduces to M1 in the other subjects (Fig. 2). As with M1, the -cell dynamic sensitivity index d and Table 3. Quantitative indexes of -cell function Subject No. s,10 9 min 1 d,10 9 X 0, pmol/l b,10 9 min 1 T up, min T down, min , , , , , , , , Mean SE , s, d, and b, static, dynamic, and basal indexes of pancreatic sensitivity to glucose, respectively; X 0, amount of insulin released in response to G max per unit C-peptide volume; T up and T down, -cell response times.

7 E8 Fig. 8. Relationship between average deconvolution-derived insulin secretion rate (ISR) and average glucose concentration during the up&down graded infusion experiment. The model-predicted relationship is shown by the dashed line. the total amount X 0 of stored insulin can be measured from M2 parameters. Minimal Model Indexes vs. Quasi-Steady-State Analysis In the literature, the low-dose (glucose doses 2, 3, 4, 6, and 8 mg kg 1 min 1 ) graded glucose infusion experiments were used to explore the relationship between glucose stimulus and insulin secretion response in various physiopathological states (4, 5, 7). In those studies, the pancreatic secretion profile (ISR) was reconstructed by deconvolution from plasma C-peptide data by assuming the two-compartment model of C- peptide kinetics (Fig. 1), with parameters either derived (4) from a bolus intravenous C-peptide injection performed in the same subjects or fixed (5, 7) to standard values that follow the method proposed in Ref. 16. During each glucose infusion period, average ISR was calculated and plotted against the corresponding average glucose level to describe the dose-response relation between the two variables. These studies demonstrated a linear relationship across glucose concentrations spanning the glucose physiological range, i.e., up to mmol/l in normal subjects and mmol/l in non-insulin-dependent diabetes mellitus patients. This is confirmed by our data, because the relationship between average ISR derived by deconvolution and the corresponding average glucose concentration (Fig. 8) is approximately linear during increasing glucose steps. During decreasing glucose steps, the relationship shows an hysteresis, i.e., ISR appears to be higher than with increasing glucose steps. However, it is worth noting that the use of a quasi-steady-state method of data analysis to interpret a non-steady-state situation, like the one between plasma glucose and C-peptide concentration during the graded glucose infusion, is not entirely accurate, and particularly so with the protocol adopted in this study, because average glucose concentration and average ISR calculated during each step underestimate the steady-state values during the increasing steps and overestimate them during the decreasing steps. The minimal model approach overcomes these problems because model equations describe the non-steadystate relationships between glucose concentration and ISR during the graded infusion protocol. The model can also be used as a simulation tool to predict the steadystate relationship between glucose concentration and ISR, as if an ideal up&down graded infusion experiment were performed in which each glucose infusion step lasts until glucose and then ISR reach their steady-state levels. By denoting steady state with the subscript ss, the model-derived relationship, also shown in Fig. 8, is ISR ss SR b G ss G b V 1 (19) From Eq. 19 it is evident that the minimal model assumes a linear steady-state relationship between glucose stimulus and ISR but provides reliable estimates of its parameters from non-steady-state data, such as those measured during an up&down graded glucose infusion experiment: index s, when multiplied by V 1, is the slope of this relation, and (SR b G b )V 1 is the intercept. The minimal model also allows one to estimate the -cell response times T down and T up during a decreasing and an increasing glucose step. The former coincides with the time constant of insulin provision, whereas the second is an equivalent parameter that also takes into account the ability of the dynamic glucose control to accelerate the rate with which -cells respond to an increasing glucose stimulus. In normal subjects, the -cell response time T up during an increasing glucose step is (min), lower than the -cell response time during a decreasing glucose step, T down (min), because of the dynamic control of glucose on the secretion of stored insulin. Up&Down Graded Infusion vs. IVGTT Pancreatic indexes s and d estimated with the up&down graded glucose infusion (Table 3) can be compared with their IVGTT counterparts, the secondphase sensitivity 2 and the first-phase sensitivity 1, obtained in normal subjects: , , from, respectively, standard IVGTT at 500 mg/kg dose (14), standard IVGTT at 300 mg/kg dose (1, 13, 18), and insulin-modified IVGTT at 300 mg/kg dose (15); , , in the same three groups. Both s and d are significantly higher than the IVGTT indexes 2 and 1. However, both the profile and the range of glucose, and thus of C-peptide concentrations, are markedly different and higher on average in the up&down graded infusion experiment compared with IVGTT, thus indicating an effect of the glucose perturbation pattern and/or glucose range on static and dynamic glucose control. In particular, these results suggest that -cells are more sensitive to a slow glucose increase, as observed during the graded glucose infusion protocol, than to the brisk rise in glucose concentration observed after an IVGTT. Conversely, the -cell response time to a decreasing glucose stimulus, estimated from the up&down graded glucose infusion, varies in a range (11 28 min) similar to the one observed with the standard IVGTT.

8 E9 In conclusion, the dynamic insulin secretory responses to increasing and decreasing glucose concentrations can be modeled using modifications of the minimal model approach. The new models allow the characterization of both basal and dynamic insulin secretory responses as well as parameters of -cell sensitivity. The application of this model to various physiopathological states associated with alterations in insulin secretion and/or action should provide novel insights into the role of these processes in the development of glucose intolerance. APPENDIX The purpose here is to define the -cell response time by considering both secretion components: secretion from provision, controlled by glucose (static control), and secretion of stored insulin, controlled by the glucose rate of change (dynamic control). For insulin provision Y (Eq. 4), the -cell response time is simply 1/, which represents the time at which Y approximates its steady-state level [Y ss (G max G b )] by 1/e 63%, in response to a glucose step increase from basal (G G b ) to an elevated level (G G max ). Under these experimental conditions, the -cell response time causes a reduction in the amount of secreted insulin, which can be evaluated by integrating Eq. 4 from time 0 to a time t 1, at which Y well approximates its steady-state level t 1 1 Y t dt Y sst 1 Y ss (A1) In Eq. A1, Y ss t 1 represents the amount of insulin that would be secreted (above basal) in the 0-t 1 interval if the response were immediate, and Y ss / is the reduction of this amount due to the -cell response time. A relation similar to Eq. A1 also holds for the up&down protocol, where glucose and Y increase from basal [G(0) G b, Y(0) 0] to elevated levels [G(t 1 ) G max,y(t 1 ) Y max ] with time-varying patterns, because by integrating Eq. 4 one has t 1 t 1 1 Y t dt G t G b dt Y max (A2) where the first term of the right hand side still represents the amount of insulin that would be secreted (above basal) in the 0-t 1 interval from provision Y if the response were immediate. As before, the -cell response time 1/ determines a reduction in the total amount of secreted insulin that is proportional to this time and to the maximum value of provision Y. The dynamic control of insulin secretion by glucose causes the additional secretion of an amount X 0 of stored insulin. Therefore, the total amount of secreted insulin is t 1 t 1 1 Y t dt X 0 G t G b dt Y max X 0 t 1 G t G b dt Y max 1 X 0 Y max (A3) By comparing Eq. A3 with Eq. A2, the additional insulin secreted due to the dynamic control of glucose causes a reduction in the delay between the glucose stimulus and the insulin response equivalent to a reduction of -cell response time from 1/ to 1/ X 0 /Y max. In conclusion, the -cell response time T down during a decreasing glucose stimulus is simply T down 1 (A5) because only the static control is active. During an increasing glucose stimulus, when both the static and the dynamic controls are active, the -cell response time T up becomes T up 1 X o (A6) Y max T up can be expressed as a function of sensitivity indexes if the system approximates steady-state conditions at time t 1,so that Y(t 1 ) Y max (G max G b ). When this approximation is used for Y max, and Eq. 15 is used for X 0, Eq. A6 becomes T up 1 X 0 1 Y max d G max G b G max G b 1 d (A7) s With our data, the use of Eq. A7 instead of A6 results in a modest overestimation of T up, 10% as an average. This work was partially supported by National Institute of Diabetes and Digestive and Kidney Diseases Grants DK-31842, DK , and DK-02742, and by the Blum Kovler Foundation. REFERENCES 1. Avogaro A, Toffolo G, Valerio A, and Cobelli C. Epinephrine exerts opposite effects on peripheral glucose disposal and glucose stimulated insulin secretion. Diabetes 45: , Bard Y. Nonlinear Parameter Estimation. New York: Academic, Barret PHR, Bell BM, Cobelli C, Golde H, Schumitzky A, Vicini P, and Foster D. SAAMII: simulation, analysis and modeling software for tracer and pharmacokinetic studies. Metabolism 47: , Byrne MM, Sturis J, and Polonsky KS. Insulin secretion and clearance during low-dose graded glucose infusion. Am J Physiol Endocrinol Metab 268: E21 E27, Byrne MM, Sturis J, Sobel RJ, and Polonsky KS. Elevated plasma glucose 2 h postchallenge predicts defects in -cell function. Am J Physiol Endocrinol Metab 270: E572 E579, Carson ER, Cobelli C, and Finkelstein L. The Mathematical Modeling of Metabolic and Endocrine Systems. New York: Wiley, Cavaghan MK, Ehrmann DA, Byrne MM, and Polonsky KS. Treatment with the oral antidiabetic agent troglitazone improves beta cell responses to glucose in subjects with impaired glucose tolerance. J Clin Invest 100: , Cobelli C, Foster D, and Toffolo G. Tracer Kinetic in Biomedical Research: From Data to Model. New York: Kluwer Academic/ Plenum, Eaton RP, Allen RC, Schade DS, Erickson KM, and Standefer J. Prehepatic insulin production in man: kinetic analysis using peripheral connecting peptide behaviour. J Clin Endocrinol Metab 51: , Faber OK, Binder C, Markussen J, Heding LG, Naithani VK, Kuzuya H, Blix P, Horwitz DL, and Rubenstein AH. Characterization of seven C-peptide antisera. Diabetes 27, Suppl 1: , Grodsky G. A threshold distribution hypothesis for packet storage of insulin and its mathematical modeling. J Clin Invest 51: , Licko V. Threshold secretory mechanism: a model of derivative element in biological control. Bull Math Biol 35: 51 58, 1973.

9 E Macor A, Ruggeri A, Mazzonetto P, Federspil G, Cobelli C, and Vettor R. Visceral adipose tissue impairs insulin secretion and insulin sensitivity but not energy expenditure in obesity. Metabolism 46: , Toffolo G, De Grandi F, and Cobelli C. Estimation of -cell sensitivity from IVGTT C-peptide data. Knowledge of the kinetics avoids errors in modeling the secretion. Diabetes 44: , Toffolo G, Cefalu W, and Cobelli C. -Cell function during insulin modified IVGTT successfully assessed by the C-peptide minimal model. Metabolism 48: , Van Cauter E, Mestrez F, Sturis J, and Polonsky KS. Estimation of insulin secretion rates from C-peptide levels. Comparison of individual and standard kinetic parameters for C- peptide clearance. Diabetes 41: , WHO Expert Committee. Second Report on Diabetes Mellitus. World Health Organization, (Tech Rep Ser 646) 18. Zachwieja JZ, Toffolo G, Cobelli C, Bier DM, and Yarasheski KE. Resistance exercise and growth hormone administration in older men: effects on insulin sensitivity and secretion during a stable label intravenous glucose tolerance test. Metabolism 45: , 1996.