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

Gianna Toffolo1, Elena Breda1, Melissa K. Cavaghan2, David A. Ehrmann2, Kenneth S. Polonsky3, and Claudio Cobelli1

1 Department of Electronics and Informatics, University of Padova, 35131 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 63110


    ABSTRACT
TOP
ABSTRACT
INTRODUCTION
MATERIALS AND METHODS
RESULTS
DISCUSSION
APPENDIX
REFERENCES

Availability of quantitative indexes of insulin secretion is important for definition of the alterations in beta -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 beta -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: Phi s, Phi d, and Phi 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 Phi s = 18.8 ± 1.8 (109 min-1), Phi d = 222 ± 30 (109), and Phi b = 5.2 ± 0.4 (109 min-1). The model also allows one to quantify the beta -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; beta -cell sensitivity; mathematical model; kinetics


    INTRODUCTION
TOP
ABSTRACT
INTRODUCTION
MATERIALS AND METHODS
RESULTS
DISCUSSION
APPENDIX
REFERENCES

SEVERAL PROTOCOLS are currently in use to define the alterations in beta -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 beta -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 beta -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 beta -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 beta -cell function could be greatly enhanced if it were possible to obtain, in addition to ISR, quantitative indexes describing beta -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 beta -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
TOP
ABSTRACT
INTRODUCTION
MATERIALS AND METHODS
RESULTS
DISCUSSION
APPENDIX
REFERENCES

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/m2 . 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
<A><AC>CP</AC><AC>˙</AC></A><SUB><IT>1</IT></SUB>(<IT>t</IT>)<IT>=</IT>−(<IT>k<SUB>01</SUB>+k<SUB>21</SUB></IT>)CP<SUB><IT>1</IT></SUB>(<IT>t</IT>) (1)

<IT>+k<SUB>12</SUB></IT>CP<SUB><IT>2</IT></SUB>(<IT>t</IT>)<IT>+</IT>SR(<IT>t</IT>) CP<SUB><IT>1</IT></SUB>(<IT>0</IT>)<IT>=0</IT>

<A><AC>CP</AC><AC>˙</AC></A><SUB><IT>2</IT></SUB>(<IT>t</IT>)<IT>=k<SUB>21</SUB></IT>CP<SUB><IT>1</IT></SUB>(<IT>t</IT>)<IT>−k<SUB>12</SUB></IT>CP<SUB><IT>2</IT></SUB>(<IT>t</IT>) CP<SUB><IT>2</IT></SUB>(<IT>0</IT>)<IT>=0</IT>
where the overdot indicates time derivative; CP1 (pmol/l) is C-peptide concentration (above basal) in compartment 1; CP2 (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; k12 and k21 (min-1) are transfer rate parameters between compartments; k01 (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(<IT>t</IT>)<IT>=</IT>SR<SUB>s</SUB>(<IT>t</IT>)<IT>+</IT>SR<SUB>d</SUB>(<IT>t</IT>) (2)
SRs is assumed to be equal to Y (pmol · l-1 · min-1), the provision of new insulin to the beta -cells
SR<SUB>s</SUB>(<IT>t</IT>)<IT>=</IT>Y(<IT>t</IT>) (3)
which is controlled by glucose according to the following equation
<A><AC>Y</AC><AC>˙</AC></A>(<IT>t</IT>)<IT>=</IT>−<IT>&agr;</IT>{Y(<IT>t</IT>)<IT>−&bgr;</IT>[G(<IT>t</IT>)<IT>−</IT>G<SUB>b</SUB>]} Y(<IT>0</IT>)<IT>=0</IT> (4)
i.e., in response to an elevated glucose level, Y and thus SRs tend with a time constant 1/alpha (min) toward a steady-state value linearly related via parameter beta  (min-1) to glucose concentration G (mmol/l) above its basal level Gb (static glucose control). Parameter beta  describes the static control of glucose on beta -cells.


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Fig. 1.   Model of C-peptide kinetics. CP1 and CP2 (pmol/l) are C-peptide concentrations in the accessible and peripheral compartments, respectively; kij (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.

SRd is assumed to represent the secretion of insulin stored in the beta -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<IT>=k</IT><SUB>d</SUB>dG (5)
The flux of insulin secretion, SRd, is then proportional to the derivative of glucose
<AR><R><C>SR<SUB>d</SUB>(<IT>t</IT>)<IT>=</IT><FR><NU>dQ</NU><DE>d<IT>t</IT></DE></FR><IT>=k</IT><SUB>d</SUB> <FR><NU>dG</NU><DE>d<IT>t</IT></DE></FR></C><C>if <FR><NU>dG</NU><DE>d<IT>t</IT></DE></FR><IT>>0 </IT>and G(<IT>t</IT>)<IT>></IT>G<SUB>b</SUB></C></R><R><C><IT>      =0</IT></C><C>otherwise</C></R></AR> (6)
Parameter kd 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 dynamic control (Fig. 2): SRd is still proportional to the derivative of glucose, but the proportionality factor is allowed to vary with glucose concentration
<AR><R><C>SR<SUB>d</SUB>(<IT>t</IT>)</C><C><IT>=k</IT>(G) <FR><NU>dG</NU><DE>d<IT>t</IT></DE></FR><IT>=k</IT><SUB>d</SUB><FENCE><IT>1−</IT><FR><NU>G(<IT>t</IT>)<IT>−</IT>G<SUB>b</SUB></NU><DE>G<SUB>t</SUB><IT>−</IT>G<SUB>b</SUB></DE></FR></FENCE> <FR><NU>dG</NU><DE>d<IT>t</IT></DE></FR></C></R><R><C></C><C>if <FR><NU>dG</NU><DE>d<IT>t</IT></DE></FR><IT>>0 </IT>and G<SUB>b</SUB><IT><</IT>G(<IT>t</IT>)<IT><</IT>G<SUB>t</SUB></C></R><R><C></C><C><IT>=0   </IT>otherwise</C></R></AR> (7)
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 Gt able to promote the secretion of all stored insulin, i.e., an additional increase of glucose above Gt has no effect on insulin secretion. M2 is a generalization of M1: in fact, for elevated Gt, the term
<FENCE>1−<FR><NU>G(t)<IT>−</IT>G<SUB>b</SUB></NU><DE>G<SUB>t</SUB><IT>−</IT>G<SUB>b</SUB></DE></FR></FENCE>
approximates 1, and M2 reduces to M1.


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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).

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
M<IT>1</IT><IT>: </IT>ISR(<IT>t</IT>)<IT>=</IT>{SR<SUB>b</SUB><IT>+</IT>SR(<IT>t</IT>)}V<SUB><IT>1</IT></SUB>

=<FENCE>k<SUB>01</SUB>CP<SUB>1b</SUB><IT>+</IT>Y(<IT>t</IT>)<IT>+k</IT><SUB>d</SUB> <FR><NU>dG</NU><DE>d<IT>t</IT></DE></FR></FENCE>V<SUB><IT>1</IT></SUB>  if <FR><NU>dG</NU><DE>d<IT>t</IT></DE></FR><IT>>0 </IT>and G(<IT>t</IT>)<IT>></IT>G<SUB>b</SUB> (8)

<IT>=</IT>{<IT>k<SUB>01</SUB></IT>CP<SUB>1b</SUB><IT>+</IT>Y(<IT>t</IT>)}V<SUB><IT>1</IT></SUB>    otherwise

M<IT>2</IT><IT>: </IT>ISR(<IT>t</IT>)<IT>=</IT>{SR<SUB>b</SUB><IT>+</IT>SR(<IT>t</IT>)}V<SUB><IT>1</IT></SUB>

=<FENCE>k<SUB>01</SUB>CP<SUB>1b</SUB><IT>+</IT>Y(<IT>t</IT>)<IT>+k</IT><SUB>d</SUB><FENCE><IT>1−</IT><FR><NU>G(<IT>t</IT>)<IT>−</IT>G<SUB>b</SUB></NU><DE>G<SUB>t</SUB><IT>−</IT>G<SUB>b</SUB></DE></FR></FENCE> <FR><NU>dG</NU><DE>d<IT>t</IT></DE></FR></FENCE>V<SUB><IT>1</IT></SUB>  if <FR><NU>dG</NU><DE>d<IT>t</IT></DE></FR><IT>>0</IT> and G<SUB>b</SUB><IT><</IT>G(<IT>t</IT>)<IT><</IT>G<SUB>t</SUB> (9)

={k<SUB>01</SUB>CP<SUB>1b</SUB><IT>+</IT>Y(<IT>t</IT>)}V<SUB><IT>1</IT></SUB>    otherwise
where SRb is insulin secretion in the basal state, and V1 (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 Phi s (min-1) measures the stimulatory effect of a glucose stimulus on beta -cell secretion at steady state. For both models
&PHgr;<SUB>s</SUB><IT>=&bgr;</IT> (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 X0 (pmol/l) as the amount of insulin (per unit of C-peptide distribution volume) released in response to the maximum glucose concentration Gmax achieved during the experiment
X<SUB><IT>0</IT></SUB><IT>=</IT><LIM><OP>∫</OP><LL>G<SUB>b</SUB></LL><UL>G<SUB>max</SUB></UL></LIM> dQ<IT>=</IT><LIM><OP>∫</OP><LL>G<SUB>b</SUB></LL><UL>G<SUB>max</SUB></UL></LIM><IT> k</IT>(G)dG (11)
For model M1, X0 is simply
X<SUB><IT>0</IT></SUB><IT>=k</IT><SUB>d</SUB>(G<SUB>max</SUB><IT>−</IT>G<SUB>b</SUB>) (12)
For model 2, two situations must be considered. If Gt > Gmax, i.e., the dynamic control of glucose on insulin secretion is active in the entire rising portion of the curve, then
X<SUB>o</SUB><IT>=</IT><LIM><OP>∫</OP><LL>G<SUB>b</SUB></LL><UL>G<SUB>max</SUB></UL></LIM><IT> k</IT>(G)dG<IT>=k</IT><SUB>d</SUB><FENCE><IT>1−</IT><FR><NU>G<SUB>max</SUB><IT>−</IT>G<SUB>b</SUB></NU><DE><IT>2</IT>(G<SUB>t</SUB><IT>−</IT>G<SUB>b</SUB>)</DE></FR></FENCE>(G<SUB>max</SUB><IT>−</IT>G<SUB>b</SUB>) (13)
If Gt < Gmax, then the dynamic glucose control is active as long as G < Gt, and X0 becomes
X<SUB><IT>0</IT></SUB><IT>=</IT><LIM><OP>∫</OP><LL>G<SUB>b</SUB></LL><UL>G<SUB><IT>t</IT></SUB></UL></LIM><IT> k</IT>(G)dG<IT>=k</IT><SUB>d</SUB>(G<SUB>t</SUB><IT>−</IT>G<SUB>b</SUB>)<IT>/2</IT> (14)
By normalizing X0 to the glucose increase, the dynamic sensitivity to glucose Phi d (dimensionless) can be derived
&PHgr;<SUB>d</SUB><IT>=</IT><FR><NU>X<SUB><IT>0</IT></SUB></NU><DE>G<SUB>max</SUB><IT>−</IT>G<SUB>b</SUB></DE></FR> (15)

BASAL. The basal sensitivity index Phi b (min-1) measures basal insulin secretion rate over basal glucose concentration
&PHgr;<SUB>b</SUB><IT>=</IT><FR><NU>SR<SUB>b</SUB></NU><DE>G<SUB>b</SUB></DE></FR><IT>=</IT><FR><NU><IT>k<SUB>01</SUB></IT>CP<SUB>1b</SUB></NU><DE>G<SUB>b</SUB></DE></FR> (16)

Response times. The models also allow one to quantify the beta -cell response times (min) to a glucose stimulus. For both models, the beta -cell response time to a decreasing glucose stimulus (Tdown) is simply
T<SUB>down</SUB><IT>=</IT><FR><NU><IT>1</IT></NU><DE><IT>&agr;</IT></DE></FR> (17)
because in this case, secretion equals provision Y, which is described by Eq. 4, with 1/alpha as time constant. When glucose increases, the additional amount X0 of insulin secreted due to the dynamic control of glucose accelerates the beta -cell response. As detailed in the APPENDIX, this is equivalent to reduction in the beta -cell response time now indicated as Tup
T<SUB>up</SUB><IT>=</IT><FR><NU><IT>1</IT></NU><DE><IT>&agr;</IT></DE></FR><IT>−</IT><FR><NU><IT>&PHgr;</IT><SUB>d</SUB></NU><DE><IT>&PHgr;</IT><SUB>s</SUB></DE></FR> (18)

Model Identification

For both models M1 and M2, all parameters are a priori uniquely identifiable (6, 8), i.e., kinetic parameters k01, k21, k12 , and secretory parameters alpha , beta , kd for M1 or alpha , beta , kd, Gt 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 k01 = 0.0600 ± 0.0006 min-1; k21 = 0.0559 ± 0.0017 min-1; k12 = 0.0492 ± 0.0002 min-1; and V1 = 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
TOP
ABSTRACT
INTRODUCTION
MATERIALS AND METHODS
RESULTS
DISCUSSION
APPENDIX
REFERENCES

Mean plasma glucose and C-peptide concentration values during the up&down graded glucose infusion protocol are shown in Fig. 3.


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Fig. 3.   Mean plasma glucose and C-peptide concentration during the up&down graded glucose infusion (n = 8).

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 Gt 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 beta -cell secretion (Eqs. 8 and 9) is shown in Fig. 6; sensitivity indexes and response times are reported in Table 3.

                              
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Table 1.   Estimated secretory parameters



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Fig. 4.   Fit of model M1 in the 8 subjects.


                              
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Table 2.   Akaike information criterion



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Fig. 5.   Weighted residuals of model M2 (continuous line) against those of model M1 (dashed line) in subjects 5 and 8.



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Fig. 6.   Mean beta -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.


                              
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Table 3.   Quantitative indexes of beta -cell function


    DISCUSSION
TOP
ABSTRACT
INTRODUCTION
MATERIALS AND METHODS
RESULTS
DISCUSSION
APPENDIX
REFERENCES

The C-peptide minimal modeling approach, which has been successfully applied to IVGTT data (14, 15), has been used here to assess beta -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 beta -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 beta , to the glucose stimulus, with a delay with respect to the glucose profile represented by 1/alpha . Parameter beta  thus represents the sensitivity Phi s (static sensitivity index) of beta -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, Phi b = 5.2 ± 0.4, because they are both steady-state secretory indexes. Our results (Phi s significantly higher than Phi b) indicate that a separate assessment of beta -cell function in the basal state and during a glucose stimulus is important, because beta -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) = SRs (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 beta -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) = kd, i.e., it does not depend on G. This means that an increase Delta G in glucose concentration, from G1 to G2 = G1+Delta G, promotes the secretion of an amount of insulin proportional to Delta G but independent of the glucose levels G1 and G2. Parameter kd represents the sensitivity Phi d (dynamic sensitivity index) of beta -cells to the glucose rate of change. The product of kd and the total increase in glucose concentration in the rising portion of the data measures the total amount X0 (pmol/l) of insulin stored in the beta -cells before the experiment and thus released during the experiment.


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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.

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 Delta G in glucose concentration promotes the secretion of an amount of insulin dependent not only on Delta G but also on the glucose levels G1 and G2. 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 Gt 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, kd, and the threshold glucose concentration Gt. M2 is a generalization of M1, because M2 reduces to M1 when the threshold value Gt 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 beta -cell dynamic sensitivity index Phi d and the total amount X0 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.


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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 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
ISR<SUB>ss</SUB><IT>=</IT>{SR<SUB>b</SUB><IT>+&bgr;</IT>[G<SUB>ss</SUB><IT>−</IT>G<SUB>b</SUB>]}V<SUB><IT>1</IT></SUB> (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 Phi s = beta , when multiplied by V1, is the slope of this relation, and (SRb - beta Gb)V1 is the intercept.

The minimal model also allows one to estimate the beta -cell response times Tdown and Tup 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 beta -cells respond to an increasing glucose stimulus. In normal subjects, the beta -cell response time Tup during an increasing glucose step is 5.7 ± 2.2 (min), lower than the beta -cell response time during a decreasing glucose step, Tdown = 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 Phi s and Phi d estimated with the up&down graded glucose infusion (Table 3) can be compared with their IVGTT counterparts, the second-phase sensitivity Phi 2 and the first-phase sensitivity Phi 1, obtained in normal subjects: Phi 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); Phi 1 = 92 ± 15, 156 ± 18, 191 ± 29 in the same three groups. Both Phi s and Phi d are significantly higher than the IVGTT indexes Phi 2 and Phi 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 beta -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 beta -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 beta -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
TOP
ABSTRACT
INTRODUCTION
MATERIALS AND METHODS
RESULTS
DISCUSSION
APPENDIX
REFERENCES

The purpose here is to define the beta -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 beta -cell response time is simply 1/alpha , which represents the time at which Y approximates its steady-state level [Yss = beta (Gmax - Gb)] by 1/e = 63%, in response to a glucose step increase from basal (G = Gb) to an elevated level (G = Gmax). Under these experimental conditions, the beta -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 t1, at which Y well approximates its steady-state level
<LIM><OP>∫</OP><LL>0</LL><UL>t<SUB>1</SUB></UL></LIM> Y(<IT>t</IT>)d<IT>t=</IT>Y<SUB>ss</SUB><IT>t<SUB>1</SUB>−</IT>Y<SUB>ss</SUB> <FR><NU><IT>1</IT></NU><DE><IT>&agr;</IT></DE></FR> (A1)
In Eq. A1, Ysst1 represents the amount of insulin that would be secreted (above basal) in the 0-t1 interval if the response were immediate, and Yss/alpha is the reduction of this amount due to the beta -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) = Gb, Y(0) = 0] to elevated levels [G(t1) = Gmax, Y(t1) = Ymax] with time-varying patterns, because by integrating Eq. 4 one has
<LIM><OP>∫</OP><LL>0</LL><UL>t<SUB>1</SUB></UL></LIM> Y(<IT>t</IT>)d<IT>t=&bgr; </IT><LIM><OP>∫</OP><LL><IT>0</IT></LL><UL><IT>t<SUB>1</SUB></IT></UL></LIM> (G(<IT>t</IT>)<IT>−</IT>G<SUB>b</SUB>)d<IT>t−</IT>Y<SUB>max</SUB> <FR><NU><IT>1</IT></NU><DE><IT>&agr;</IT></DE></FR> (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-t1 interval from provision Y if the response were immediate. As before, the beta -cell response time 1/alpha 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 X0 of stored insulin. Therefore, the total amount of secreted insulin is
<LIM><OP>∫</OP><LL>0</LL><UL>t<SUB>1</SUB></UL></LIM> Y(<IT>t</IT>)d<IT>t+</IT>X<SUB><IT>0</IT></SUB><IT>=&bgr; </IT><LIM><OP>∫</OP><LL><IT>0</IT></LL><UL><IT>t<SUB>1</SUB></IT></UL></LIM> (G(<IT>t</IT>)<IT>−</IT>G<SUB>b</SUB>)d<IT>t−</IT>Y<SUB>max</SUB> <FR><NU><IT>1</IT></NU><DE><IT>&agr;</IT></DE></FR><IT>+</IT>X<SUB><IT>0</IT></SUB>

=&bgr; <LIM><OP>∫</OP><LL>0</LL><UL>t<SUB>1</SUB></UL></LIM> (G(<IT>t</IT>)<IT>−</IT>G<SUB>b</SUB>)d<IT>t−</IT>Y<SUB>max</SUB><FENCE><FR><NU><IT>1</IT></NU><DE><IT>&agr;</IT></DE></FR><IT>−</IT><FR><NU>X<SUB><IT>0</IT></SUB></NU><DE>Y<SUB>max</SUB></DE></FR></FENCE> (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 beta -cell response time from 1/alpha to 1/alpha  - X0/Ymax.

In conclusion, the beta -cell response time Tdown during a decreasing glucose stimulus is simply
T<SUB>down</SUB><IT>=</IT><FR><NU><IT>1</IT></NU><DE><IT>&agr;</IT></DE></FR> (A5)
because only the static control is active. During an increasing glucose stimulus, when both the static and the dynamic controls are active, the beta -cell response time Tup becomes
T<SUB>up</SUB><IT>=</IT><FR><NU><IT>1</IT></NU><DE><IT>&agr;</IT></DE></FR><IT>−</IT><FR><NU>X<SUB>o</SUB></NU><DE>Y<SUB>max</SUB></DE></FR> (A6)
Tup can be expressed as a function of sensitivity indexes if the system approximates steady-state conditions at time t1, so that Y(t1) = Ymax approx  beta (Gmax - Gb). When this approximation is used for Ymax, and Eq. 15 is used for X0, Eq. A6 becomes
T<SUB>up</SUB><IT>=</IT><FR><NU><IT>1</IT></NU><DE><IT>&agr;</IT></DE></FR><IT>−</IT><FR><NU>X<SUB><IT>0</IT></SUB></NU><DE>Y<SUB>max</SUB></DE></FR><IT>≈</IT><FR><NU><IT>1</IT></NU><DE><IT>&agr;</IT></DE></FR><IT>−</IT><FR><NU><IT>&PHgr;</IT><SUB>d</SUB>(G<SUB>max</SUB><IT>−</IT>G<SUB>b</SUB>)</NU><DE><IT>&bgr;</IT>(G<SUB>max</SUB><IT>−</IT>G<SUB>b</SUB>)</DE></FR><IT>=</IT><FR><NU><IT>1</IT></NU><DE><IT>&agr;</IT></DE></FR><IT>−</IT><FR><NU><IT>&PHgr;</IT><SUB>d</SUB></NU><DE><IT>&PHgr;</IT><SUB>s</SUB></DE></FR> (A7)
With our data, the use of Eq. A7 instead of A6 results in a modest overestimation of Tup, <10% as an average.


    ACKNOWLEDGEMENTS

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.


    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.

Received 24 February 2000; accepted in final form 24 August 2000.


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ABSTRACT
INTRODUCTION
MATERIALS AND METHODS
RESULTS
DISCUSSION
APPENDIX
REFERENCES

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Am J Physiol Endocrinol Metab 280(1):E2-E10
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