## Abstract

Availability of quantitative indexes of insulin secretion is important for definition of the alterations in β-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 β-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: Φ_{s}, Φ_{d}, and Φ_{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 Φ_{s} = 18.8 ± 1.8 (10^{9}min^{−1}), Φ_{d} = 222 ± 30 (10^{9}), and Φ_{b} = 5.2 ± 0.4 (10^{9} min^{−1}). The model also allows one to quantify the β-cell times of response to increasing and decreasing glucose stimulus, equal to 5.7 ± 2.2 (min) and 17.8 ± 2.0 (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
- β-cell sensitivity
- mathematical model
- kinetics

several protocols are currently in use to define the alterations in β-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 β-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 β-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 β-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 β-cell function could be greatly enhanced if it were possible to obtain, in addition to ISR, quantitative indexes describing β-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 β-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 26.1 ± 1.7 kg/m^{2} . Glucose tolerance 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
Equation 1
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)
Equation 2SR_{s} is assumed to be equal to Y (pmol · l^{−1} · min^{−1}), the provision of new insulin to the β-cells
Equation 3which is controlled by glucose according to the following equation
Equation 4i.e., in response to an elevated glucose level, Y and thus SR_{s} tend with a time constant 1/α (min) toward a steady-state value linearly related via parameter β (min^{−1}) to glucose concentration G (mmol/l) above its basal level G_{b} (static glucose control). Parameter β describes the static control of glucose on β-cells.

SR_{d} is assumed to represent the secretion of insulin stored in the β-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
Equation 5The flux of insulin secretion, SR_{d}, is then proportional to the derivative of glucose
Equation 6Parameter* 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/d*t* 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 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
Equation 7According 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
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
Equation 8
Equation 9
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 Φ_{s} (min^{−1}) measures the stimulatory effect of a glucose stimulus on β-cell secretion at steady state. For both models
Equation 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
Equation 11For *model M1*, X_{0} is simply
Equation 12For *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
Equation 13If G_{t} < G_{max}, then the dynamic glucose control is active as long as G < G_{t}, and X_{0} becomes
Equation 14By normalizing X_{0} to the glucose increase, the dynamic sensitivity to glucose Φ_{d} (dimensionless) can be derived
Equation 15

##### BASAL.

The basal sensitivity index Φ_{b} (min^{−1}) measures basal insulin secretion rate over basal glucose concentration
Equation 16

#### Response times.

The models also allow one to quantify the β-cell response times (min) to a glucose stimulus. For both models, the β-cell response time to a decreasing glucose stimulus (T_{down}) is simply
Equation 17because in this case, secretion equals provision Y, which is described by *Eq. 4
*, with 1/α as time constant. When glucose increases, the additional amount X_{0} of insulin secreted due to the dynamic control of glucose accelerates the β-cell response. As detailed in the
, this is equivalent to reduction in the β-cell response time now indicated as T_{up}
Equation 18

### Model Identification

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 α, β,*k*
_{d} for *M1* or α, β,*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*
_{01} = 0.0600 ± 0.0006 min^{−1};*k*
_{21} = 0.0559 ± 0.0017 min^{−1}; *k*
_{12} = 0.0492 ± 0.0002 min^{−1}; and V_{1} = 4.06 ± 0.06 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.

## RESULTS

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 β-cell secretion (*Eqs. 8
* and *
9
*) is shown in Fig. 6; sensitivity indexes and response times are reported in Table3.

## DISCUSSION

The C-peptide minimal modeling approach, which has been successfully applied to IVGTT data (14, 15), has been used here to assess β-cell secretion during a more physiological glucose perturbation, in which a rising followed by a falling glucose concentration is 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 β-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 steady-state conditions is proportional, through parameter β, to the glucose stimulus, with a delay with respect to the glucose profile represented by 1/α. Parameter β thus represents the sensitivity Φ_{s} (static sensitivity index) of β-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, 18.8 ± 1.8, can be compared with the sensitivity in the basal state, Φ_{b} = 5.2 ± 0.4, because they are both steady-state secretory indexes. Our results (Φ_{s}significantly higher than Φ_{b}) indicate that a separate assessment of β-cell function in the basal state and during a glucose stimulus is important, because β-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 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 β-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 ΔG in glucose concentration, from G_{1} to G_{2} = G_{1}+ΔG, promotes the secretion of an amount of insulin proportional to ΔG but independent of the glucose levels G_{1} and G_{2}. Parameter*k*
_{d} represents the sensitivity Φ_{d}(dynamic sensitivity index) of β-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} (pmol/l) of insulin stored in the β-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 ΔG in glucose concentration promotes the secretion of an amount of insulin dependent not only on ΔG 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 β-cell dynamic sensitivity index Φ_{d} and 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 10–12 mmol/l in normal subjects and 18–20 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-steady-state relationships between glucose concentration and ISR during the graded infusion protocol. The model can also be used as a simulation tool to predict the steady-state 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
Equation 19From *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 Φ_{s} = β, when multiplied by V_{1}, is the slope of this relation, and (SR_{b} − βG_{b})V_{1} is the intercept.

The minimal model also allows one to estimate the β-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 β-cells respond to an increasing glucose stimulus. In normal subjects, the β-cell response time T_{up} during an increasing glucose step is 5.7 ± 2.2 (min), lower than the β-cell response time during a decreasing glucose step, T_{down} = 17.8 ± 2.0 (min), because of the dynamic control of glucose on the secretion of stored insulin.

### Up&Down Graded Infusion vs. IVGTT

Pancreatic indexes Φ_{s} and Φ_{d} estimated with the up&down graded glucose infusion (Table 3) can be compared with their IVGTT counterparts, the second-phase sensitivity Φ_{2}and the first-phase sensitivity Φ_{1}, obtained in normal subjects: Φ_{2} = 11.3 ± 1.1, 10.5 ± 0.6, 10.9 ± 1.4 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); Φ_{1} = 92 ± 15, 156 ± 18, 191 ± 29 in the same three groups. Both Φ_{s} and Φ_{d} are significantly higher than the IVGTT indexes Φ_{2} and Φ_{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 β-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 β-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.

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 β-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.

## Acknowledgments

This work was partially supported by National Institute of Diabetes and Digestive and Kidney Diseases Grants DK-31842, DK-20595, and DK-02742, and by the Blum Kovler Foundation.

## Appendix

The purpose here is to define the β-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 β-cell response time is simply 1/α, which represents the time at which Y approximates its steady-state level [Y_{ss} = β(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 β-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
Equation A1In *Eq. EA1
*, 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}/α is the reduction of this amount due to the β-cell response time.

A relation similar to *Eq. EA1
* 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
Equation A2where 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 β-cell response time 1/α 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
Equation A3By comparing *Eq. EA3
* with *Eq. EA2
*, 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 β-cell response time from 1/α to 1/α − X_{0}/Y_{max}.

In conclusion, the β-cell response time T_{down} during a decreasing glucose stimulus is simply
Equation A5because only the static control is active. During an increasing glucose stimulus, when both the static and the dynamic controls are active, the β-cell response time T_{up} becomes
Equation A6T_{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} ≈ β(G_{max} − G_{b}). When this approximation is used for Y_{max}, and *Eq. 15
* is used for X_{0}, *Eq. EA6
* becomes
Equation A7With our data, the use of *Eq. EA7
* instead of*A6* results in a modest overestimation of T_{up}, <10% as an average.

## Footnotes

Address for reprint requests and other correspondence: C. Cobelli, Dipartimento di Elettronica e Informatica, Via Gradenigo 6a, 35131 Padova, Italy (E-mail: cobelli{at}dei.unipd.it).

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.

- Copyright © 2001 the American Physiological Society