Critical evaluation of the combined model approach for
estimation of prehepatic insulin secretion
Richard M.
Watanabe1,2,
Garry
M.
Steil2, and
Richard N.
Bergman2
1 Department of Exercise
Science, University of Southern California, Los Angeles,
90082-0652; and 2 Department
of Physiology and Biophysics, University of Southern California
School of Medicine, Los Angeles, California 90033
 |
ABSTRACT |
The combined model approach uses kinetic
analysis of both plasma insulin and C-peptide dynamics to
estimate prehepatic insulin secretion rates and parameters of insulin
and C-peptide kinetics. The original model used
single-compartment kinetics to describe both insulin and
C-peptide despite knowledge that C-peptide follows two-compartment kinetics. The performance of the model under rapidly changing secretory conditions has come into question. Thus a more complex combined model is introduced, incorporating two-compartmental C-peptide disappearance. The addition of two-compartment
C-peptide kinetics required a novel numerical approach to
allow estimation of model parameters. This simulation study was
undertaken to 1) compare the
performance of the original combined model and
2) examine the numerical method used
to identify parameters for the extended combined model with
two-compartment C-peptide kinetics under simulated conditions
of rapidly changing insulin and C-peptide. Monte Carlo
simulation revealed that the original combined model does not provide
accurate estimates of prehepatic insulin secretion under rapid
kinetics. However, the extended combined model provides accurate
reconstruction of prehepatic insulin secretory profile without separate
quantification of C-peptide kinetics.
kinetic analysis; C-peptide; mathematical modeling; intravenous glucose tolerance test
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INTRODUCTION |
BOTH INSULIN RESISTANCE and
-cell failure are
hallmarks of non-insulin-dependent diabetes mellitus (NIDDM) (9, 17). Although it is clear that a relative insulin secretory defect is a
critical element in the pathogenesis of NIDDM, the temporal development
of metabolic defects in the natural history of the disease is not yet
clarified. Thus the ability to derive accurate and precise measures of
insulin secretion is paramount in detecting and understanding the
pathogenesis of NIDDM. The traditional measure of insulin secretory
function has been the plasma insulin concentration either at fasting or
after perturbation by glucose or other insulin secretagogues. However,
the peripheral insulin concentration does not accurately reflect
insulin secretory activity at the level of the
-cell due to
significant hepatic extraction of insulin by the liver during first
pass (7, 14, 21). Furthermore, there is evidence for variation in
hepatic extraction among physiological conditions (7, 14, 21) that
confounds comparison of plasma insulin concentrations.
The discovery of equimolar release of insulin and C-peptide
by the pancreatic
-cells (23, 25) led to development of improved methods for assessment of insulin secretion in vivo (8, 11-13, 22,
28). However, as previously noted (19, 27, 28, 30), insulin secretion
estimates based on kinetic analysis of peripheral C-peptide
concentration alone involve multiple experimental protocols or a priori
assumption of C-peptide kinetic parameters. As a simple alternative, we introduced the "combined model" approach, which uses both plasma insulin and C-peptide measurements from a
single experimental protocol to derive estimates of prehepatic insulin secretion (28, 30). Analysis of the kinetics of both peptides in plasma
provides two sources of information regarding the
insulin-C-peptide secretory profile. The combined model uses
the information that can be gleaned from the kinetics of both plasma
insulin and C-peptide to derive prehepatic insulin secretion
rates. This allows for estimation of prehepatic insulin secretion rates
without separate experimental assessment or a priori assumption of the
kinetic parameters of the model.
The accuracy of secretion estimates derived from the combined model has
been questioned due to the use of a single-compartment characterization for plasma C-peptide kinetics (18, 27). The two-compartment nature of C-peptide has been confirmed (11, 23, 24) following the original report by Faber et al. (13). Whereas the
second-order nature of C-peptide was not disputed (28), the
simplifying assumption of single-compartment C-peptide kinetics in the combined model was motivated by the desire to minimize
model complexity (3, 4). It was hoped that this simplification would be
an adequate representation of the system to allow characterization of
prehepatic insulin secretion. Although this assumption apparently holds
true under conditions of slow secretory dynamics (28), it is likely
that the performance of single-compartment C-peptide kinetics
will degrade as insulin secretion, and therefore changes in plasma
C-peptide kinetics, become increasingly rapid and the effect
of the "fast" C-peptide distribution compartment
becomes more important.
To deal with rapid C-peptide kinetics, a more complex
combined model is introduced, which incorporates the correct
two-compartmental structure for C-peptide disappearance. The
addition of two-compartment C-peptide kinetics requires a
novel numerical approach to allow estimation of the model parameters.
Most importantly, this extended combined model retains the ability to
derive estimates of prehepatic insulin secretion without separate
assessment or a priori assumption of C-peptide kinetics.
The current simulation study was undertaken with two goals. The first
goal was to examine the performance of the original combined model
under conditions of rapidly changing insulin and C-peptide.
The second goal was to examine the performance of the numerical
identification scheme used for the extended combined model with
two-compartment C-peptide kinetics under the identical computer-generated conditions. The a priori expectation is that the
extended combined model would perform better than the original combined
model simply due to the additional parameters introduced by the
inclusion of the second compartment. The goal for the extended combined
model was to examine whether the numerical approach used for the
parameter identification has a negative impact on the model's ability
to reproduce the insulin secretory profile.
The simulation study shows that, under conditions of rapid kinetics,
the original combined model is not able to provide accurate reconstruction of the insulin secretory profile. However, the extended
combined model, incorporating two-compartment C-peptide kinetics and utilizing an alternative parameter identification scheme,
does provide an accurate representation of prehepatic insulin secretion
from insulin and C-peptide measurements, even under
conditions of rapidly changing secretion. Thus, in principle, the
extended combined model can be used to calculate insulin secretion correctly without separate quantification of C-peptide
kinetics.
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METHODS |
The Models
The original combined model (model 1).
Details of the original combined model (Fig.
1A)
have been previously described (28, 30) and are briefly reviewed here. R(t) denotes the equimolar
insulin-C-peptide release by the
-cells. For the insulin
component of the model, a constant fraction of insulin is assumed to be
lost due to first-pass hepatic extraction. Thus F × R(t) represents insulin that
survives hepatic transit and distributes in the systemic circulation
with single-compartment kinetics. In contrast,
R(t) remains unchanged for
C-peptide, since C-peptide is known to survive
hepatic transit (22, 24). Model equations were derived as follows
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(1)
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(2)
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Where
VI and
VC are the single-compartment
distribution spaces for insulin and C-peptide, respectively.
Allowing r(t) to be the secretion
rate per unit C-peptide distribution volume
[R(t)/VC] and normalizing the fractional hepatic extraction rate (F) to the
ratio of the C-peptide and insulin distribution spaces
[f = F × (VC/VI)],
Eqs.
1 and 2 simplify to
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(3)
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(4)
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Because
the model is expressed in terms of insulin secretion per unit volume
C-peptide distribution, secretion rates in mass per time can
be obtained by correcting for the C-peptide distribution space, which has been previously assessed to be 12.5% body weight for
the single-compartment assumption (28).

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Fig. 1.
Schematic diagram of models. Original combined model
(model 1,
A) is shown. Insulin and
C-peptide are assumed to be secreted in equimolar fashion
[R(t)] by the -cells.
Insulin that survives hepatic transit appears in the systemic
circulation as F × R(t) and distributes within a single
compartment. C-peptide survives hepatic transit and is also
assumed to follow single-compartment kinetics. Extended combined model
(model 2,
B) is identical to
model 1 except that C-peptide is
assumed to follow 2-compartment distribution kinetics, where
compartment 1 represents plasma and
compartment 2 the extravascular space.
KI and
KC, fractional disappearance rates
for insulin and C-peptide, respectively.
K21 and
K12, exchange rates between 1st
and 2nd compartments. See METHODS for
details.
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The extended combined model (model 2).
The extended combined model (Fig.
1B) is identical to
model
1 in every aspect, except for
incorporation of two-compartment C-peptide kinetics as described by
Faber et al. (13). Model equations can be written as follows
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(5)
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(6)
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(7)
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where
C1 represents the plasma,
C2 represents the extravascular
compartments for C-peptide, and parameters
K21 and
K12 are the exchange rates between
the first and second compartments. If similar substitutions are made
for the secretion rate and fractional hepatic extraction as in
model
1, new equations can be derived as
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(8)
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(9)
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(10)
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where
insulin secretion is now estimated per unit C-peptide volume
in the first, presumably plasma, compartment
[r(t) = R(t)/VC1] and f = F × (VC1/VI).
Secretion rates estimated by this model can be corrected to mass/time
by assuming a C-peptide distribution space of 6.02% body
weight (27), the estimate for the plasma compartment for a
two-compartment representation.
The Simulation Study Protocol
The study was designed to 1)
determine how well the original combined model
(model
1) could account for the rapid
plasma insulin and C-peptide kinetics observed during a
typical intravenous glucose tolerance test (IVGTT) and
2) assess the practical
applicability and accuracy of the new numerical approach applied to the
extended combined model (model
2). To test this, it was assumed
that single-compartment insulin and two-compartment C-peptide
kinetics were a correct representation of the in vivo system.
The simulation study protocol is outlined schematically in Fig.
2. Known kinetic parameters and IVGTT-like
secretory profiles were submitted to
model
2 to simulate plasma insulin and
C-peptide data. Data representing typical
"experimental" IVGTT data were generated by adding random error
to each of the "perfect" simulated sets of data. Random Gaussian
error with coefficients of variation of 3, 5, and 10% were added to
the insulin and C-peptide time courses. Thirty sets of
"experimental" data were generated at each error level (2 profiles × 3 error levels × 30 sets/error level,
n = 180 overall). Each of the
"experimental" data sets was analyzed using both models to assess
the ability of each to reproduce the known kinetic parameters and
insulin secretion rates.

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Fig. 2.
Simulation study scheme. A schematic diagram of the simulation study is
shown. A known secretory profile and model parameters are assigned to
model 2, and plasma insulin and
C-peptide data are simulated. Random error is added to
simulated data to create "noisy" data. Noisy data are then
analyzed using both models 1 and
2. Results from both models are then
compared with known parameters and secretory profile.
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From a theoretical perspective, the a priori expectation would be for
model
2 to perform better than
model
1 and provide accurate estimates of
both the model parameters and prehepatic insulin secretion rates. This
is simply due to the fact that model
2 is used to generate the simulated
data to be analyzed. It must be emphasized that the goal of the
simulation study was not to compare the performance of
model
1 with
model
2. In terms of
model
2, it was equivocal whether the
numerical approach used to estimate the kinetic parameters would
perform well in the face of experimental data, i.e., data with error.
Thus the evaluation of model
2 was for the assessment of the
numerical method used for parameter identification rather than the
actual performance of the model per se.
The secretory profiles used for the simulation study are shown in Fig.
3. Profile
A was taken from work previously
published by this group (30) and
profile
B was adapted from the studies by
Shapiro et al. (26). Parameters for the simulations were taken from the
literature (26, 27, 30) and are shown in Table
1. The plasma insulin and
C-peptide time courses generated by simulating with these
known parameters and secretory profiles are shown in Fig. 3.

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Fig. 3.
Known secretion profiles and simulated data.
Top panels show known insulin
secretory profiles (A and
B) used for simulation study.
Resultant simulated plasma insulin ( ) and C-peptide ( ) data
generated by simulating model 2 with known profiles and model
parameters are shown below. See
METHODS for details.
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Numerical Methods
Identification of model 1.
Nonlinear weighted least-squares methods were used for the
identification of parameters and insulin secretion rates for both models. Details of numerical analyses for
model
1 have previously been described (30).
The only difference in the current analysis was a modification in the
weighting matrix. For identification of
model
2 parameters an algorithmic weighting
function was used (see Data
analysis). Thus, to maintain relative compatibility with the identification of model
2,
model
1 identifications used the weighting
matrix generated from model
2 analyses.
Identification of model 2.
The traditional approach of simultaneously identifying model parameters
and secretion rates, such as that used for
model
1, was not applicable to
model
2. This is because the additional two
parameters required to identify parameters for two-compartment C-peptide kinetics, coupled with the large number of
parameters associated with characterizing the prehepatic insulin
secretion rate (n-1 samples), result in a system that is
unidentifiable; i.e., unique solutions for the individual model
parameters and the insulin secretion rates cannot be determined.
Therefore, a unique numerical approach was used to circumvent these
problems, which allowed for identification of the kinetic parameters
independent of the secretion rate.
The method used for estimation of both prehepatic insulin secretion and
kinetic parameters for insulin and C-peptide exploits the
fact that insulin and C-peptide secretion rates are equal. Algebraic manipulation is used to factor out the insulin secretion rate
R(t) from the model equations. This
yields a set of equations relating the concentrations of insulin and
C-peptide in plasma in which the actual secretion rate,
R(t), is not included. The parameters of these equations, which are identical to the parameters of
model
2, can then be estimated by fitting
the C-peptide concentration, given insulin, or the opposite.
The numerical approach applied to
model
2 is described in detail in the
APPENDIX. The final equations used to
estimate the kinetic parameters for
model
2 are
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(11)
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(12)
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(13)
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(14)
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(15)
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(16)
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Equations
11-13
describe the kinetics of C-peptide in plasma
[C1(t)],
in which the plasma insulin profile
[I(t)] is used as a
surrogate representation of the prehepatic insulin secretion rate.
Parameters
1 and
2 characterize the kinetics of
C-peptide in plasma, and the inverse of these two parameters
represents the fast and slow time constants for C-peptide
disappearance. Equations
14-16
describe the alternative formulation of the model in which the kinetics
of plasma insulin are described using plasma C-peptide as the
surrogate representation of the prehepatic insulin secretion rate.
This alteration of the model equations allows identification of model
parameters independent of the insulin secretion rate. Once estimates
for the model parameters are obtained, deconvolution is used to
calculate the insulin secretion rates. The extended combined model
differs from the Eaton method in that the estimations of model
parameters are derived from plasma insulin and C-peptide measurements of the experiment of interest alone; i.e., the advantages of the original combined model method are retained.
For model
2 parameter identifications, the
simulated plasma insulin and C-peptide data were
simultaneously fit to Eqs.
11-16 to estimate parameters f, KI,
K12,
1, and
2 using weighted nonlinear least squares. Once individual parameter estimates were obtained, the
prehepatic secretory rate was reconstructed using deconvolution. Deconvolution was performed independently on both insulin and C-peptide data; then the results were averaged to provide a
final estimate of the prehepatic insulin secretion rate. The
model-predicted insulin and C-peptide concentrations and
their respective derivatives were used for the deconvolution
calculation. Model
2 parameters for C-peptide
kinetics can be resolved into the parameters from the standard
two-compartment configuration by the following relationships
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(17)
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(18)
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(19)
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Data analysis.
Simulations for data generation and all identifications for
model
2 were performed using MLAB (Civilized
Software, Bethesda, MD) on a personal computer. MLAB uses a
Marquardt-Levenburg weighted least-squares algorithm for parameter
identifications. Weights for model
2 identifications were estimated using
MLAB's internal EWT weighting function. This function uses a
five-point moving average to derive an estimate of the standard
deviation in the data and then computes the weight as the inverse
variance. All identifications for
model
1 were performed on an Alliant FX/1 super minicomputer or a SUN workstation using in-house software written
in FORTRAN-77 using IMSL subroutines, as previously described (30).
Weighting matrices from the model
2 identifications performed using MLAB
were used for all model
1 identifications.
All statistics were performed using SAS implemented on a personal
computer (SAS, Cary, NC). Data are reported as means ± SE unless
otherwise stated.
RESIDUALS.
The ability of a given model to account for the simulated insulin and
C-peptide data was assessed by residual analysis. Studentized residuals were calculated according to Draper and Smith (10), and
residual time courses were tested for systematic patterns using the
Runs Test. Use of the Studentized residual partially accounts for the
differences in degrees of freedom between the two sets of model fits
due to the differences in the number of estimated parameters.
Comparison of model fits between
models 1 and
2 was accomplished by qualitative
comparison of the residual time courses and comparison of the
sums-of-squares using the Akaike Information Criterion (AIC) (1).
PARAMETER ESTIMATES.
Model
1 estimates for both the fractional
disappearance rate for insulin
(KI) and index of fractional
hepatic extraction (f) can be directly compared with the known values
used for the simulation. For the simulation, parameter f represented
the true fraction normalized to the ratio of the insulin and the
two-compartmental C-peptide distribution spaces
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(20)
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The
parameter estimated by model
1 represents the fractional hepatic
extraction normalized to the ratio of the insulin and one-compartmental
C-peptide distribution spaces
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(21)
|
Therefore,
parameter f estimated from model
1 should differ from the known value
by the ratio of the differing C-peptide distribution spaces
VC1/VC.
If the known parameter is corrected to this ratio, assuming
VC1 to be
6.02% body weight and VC as 12.5% body weight, the model
1 estimated f should be 0.9659. Given the differences in model structure, C-peptide parameters
estimated using model
1 cannot be directly compared with the
known parameter values used in the original simulation. Parameter
estimates from model
2 were directly compared with the
known parameters used for the simulation by paired
t-test.
PREHEPATIC SECRETION RATES.
All secretion rates were converted to units of mass per time assuming
distribution volumes of 8,750 (12.5% body wt) and 4,214 ml (6.02%
body wt) for models
1 and
2, respectively. The two different C-peptide distribution spaces are required due to the
difference in model structures. The single-compartment representation
used for model
1 requires an estimate of the total
C-peptide distribution space, whereas only the volume of the
plasma compartment is required for
model
2.
Secretion profiles were contrasted in three ways. First, the average
estimated secretory profiles were superimposed on the known profile and
visually inspected for differences. Second, regression analysis was
performed between the known profile and the individual estimated
profiles at each error level. Although regression analysis is not
completely appropriate for comparison of time course data, it should
provide an indication of time shifts (reflected as hysteresis),
relative degree of agreement (reflected as the slope), and relative
offset (reflected as the intercept).
 |
RESULTS |
Model 1
Examination of the residuals for model
1 fits to simulated insulin and
C-peptide data revealed clear systematic patterns at all
three error levels (Fig. 4). This distinct
pattern in the residual time course is suggestive of a structural
deficiency in the model. For profile
A,
model
1 exhibits a consistent
underestimation of both the first-phase insulin and C-peptide
peaks, which are then compensated for by subsequent over- and
underestimation in the remainder of the time courses. Analysis of the
residual time courses revealed significant runs in all model fits (Runs
test; P < 0.01).
Model
1 fits to
profile
B appeared to be slightly better than
those to profile
A, in terms of relative magnitude, but
consistent runs were also observed in the model fits
(P < 0.01).

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Fig. 4.
Model 1 fits to simulated data. Average
model 1 fits (solid line) are shown along
with average simulated data (symbols) at each error level tested. Error
bars are omitted for clarity. Model 1 predictions for
profile A are shown on
left and
profile B on
right. , Insulin; ,
C-peptide. Insets show
average (means ± SE) Studentized residual time course.
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Table 2 shows the average model parameter
estimates for both profiles. Like the model fits to the data,
model
1-estimated parameters did not match
those used for the simulation. For both profiles, the model estimates
of the index of fractional hepatic insulin extraction (f) were
underestimated by ~50% (cf. Table 2). This underestimation in
fractional extraction was accompanied by an underestimation in the
fractional disappearance rate for insulin
(KI) of similar magnitude. For
profile
A,
KI was underestimated by an
average of ~40%, whereas for
profile
B the estimates were ~46% lower
than the known value. Given that model
1 uses a single-compartment C-peptide representation, it is not possible to directly
compare C-peptide parameters.
Model
1 was not able to accurately reproduce
the known prehepatic insulin secretory profiles used to generate the
simulation data (Fig. 5). For both
profiles, the model predicts a biphasic secretory pattern but
exaggerates both the first-phase response and the subsequent apparent
refractory period at all three error levels. Although neither of the
known profiles descends below the basal secretory rate during the first
30 min, in all cases model
1 incorrectly predicts a significant
period of secretory inactivity with zero secretion. The exaggeration in
the refractory period results in a loss of early second phase secretion
in both profiles. However, the model appears to correctly estimate
second phase insulin secretion from ~15 min until the end of the
"experiment" at 180 min.

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Fig. 5.
Model 1 estimated prehepatic insulin
secretory profiles. Average model 1 estimated insulin secretion rates
(dashed line) are superimposed on known rate used for simulation (solid
line) for each error level tested. A:
profile A; B:
profile B.
Insets show average estimated
secretion rates plotted against known rates. Solid line in each inset
represents line of unity. Error bars have been omitted for clarity.
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The distortion in the secretory estimates is more clearly depicted when
the estimated rates are plotted against the known rate (Fig.
5A). The significant overestimation
in first-phase secretion for both profiles results in average nonunity
slopes of ~1.65 for profile
A and ~1.72 for
profile
B with a significant degree of
hysteresis. This distortion in the relationship between the known and
estimated secretory profiles is dominated by the overestimation in
first-phase secretion. When data for second-phase secretion are
examined from 16 to 180 min, the slope of the relationship between the
known and estimated secretory profiles is excellent at all three error
levels: 0.947, 0.967, and 1.04 for
profile A; 0.856, 0.875, and 0.943 for
profile
B.
Model 2
The simulated data without error were analyzed using
model
2 to determine model consistency in an
error-free environment. As shown in Table
3, the model was able to reproduce the
known parameter estimates with acceptable accuracy. Reidentification of
simulated data using model
2 with two-compartment
C-peptide kinetics resulted in significantly improved fits
compared with model
1 (Fig.
6). Model
2 showed an improved ability to
account for the rapid first-phase spike. The improvement in model fits is reflected in both the AIC index and the residual plots.
The AIC index was significantly lower with
model
2 compared with
model 1 at all three error levels
(P < 0.001), which indicates that model
2 was able to better account for the
rapid insulin and C-peptide kinetics. Furthermore, the
residual plots revealed no distinct patterns by Runs test
(P > 0.50), and overall the relative
magnitude in the residuals was significantly reduced for both secretory profiles.

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Fig. 6.
Model 2 fits to simulated data. Average
model 2 fits (solid line) are shown along
with average simulated data (symbols) at each error level tested. Error
bars are omitted for clarity. Model 2 predictions for
profile A are shown on
left and
profile B on
right. Symbols are identical to those
in Fig. 4. Insets show average (means ± SE) Studentized residual time course.
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On the basis of the excellent model fits, one would expect good
estimates of the model parameters. Comparison of the estimated model
parameters with the known values is shown in Table 3. None were
significantly different except for f, the index of hepatic insulin
extraction, which was slightly underestimated in all cases. The
accuracy with which the remaining parameters were estimated was
excellent, with the average percent difference from known values not
exceeding 12%.
As can be expected from the model fits and parameter estimates,
calculated rates of prehepatic insulin secretion were virtually identical to the known infusion profiles (Fig.
7, top).
Superimposing the average estimated secretion rates for each error
level on the known rate shows near perfect agreement. This result is
further emphasized when the known rates are plotted against the
estimated rates (Fig. 7, bottom).
For both profiles, there is no apparent hysteresis and the data fall
randomly about the line of identity. When regression analyses are
performed for each individual secretion profile, the average slopes of
the relationships were not different from unity for both profiles:
0.997 ± 0.020, 1.024 ± 0.034, and 1.00 ± 0.031 for
profile
A and 0.996 ± 0.016, 1.028 ± 0.022, and 1.074 ± 0.039 for
profile
B.

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Fig. 7.
Model 2 estimated prehepatic insulin
secretory profiles. Average model 2 estimated insulin secretion rates
(dashed line) are superimposed on known rate used for simulation (solid
line) for each error level tested. A:
profile A; B:
profile B.
Insets show average estimated
secretion rates plotted against the known rates. Solid line in each
inset represents line of unity. Error bars have been omitted for
clarity.
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 |
DISCUSSION |
The combined model was introduced as an experimentally simple method
for assessment of prehepatic insulin secretion rates (28, 30). The
method has several advantages over alternative methodology. Noteworthy
is the method's ability to provide estimates of prehepatic insulin
secretory rates without independent assessment or assumption of
parameters for C-peptide or insulin kinetics. This is in
contrast to approaches that require an independent experiment
to determine C-peptide kinetics (11) or assumption of the
kinetic parameters for C-peptide (27). Also, unlike the approach introduced by Pacini and Cobelli (20), the combined model
approach is not restricted to a single type of experimental protocol.
The combined model approach can provide estimates of prehepatic insulin
secretory rates from a single protocol of the investigator's choice so
long as there are sufficient dynamics in the plasma insulin and
C-peptide time courses. The combined model achieves the
ability to derive secretory rates without separate kinetic assessment
through the simultaneous kinetic analysis of insulin and
C-peptide dynamics from the experiment of interest alone.
This integration of kinetic information provides added advantages. The
combined model provides estimates of kinetic parameters for both
insulin and C-peptide and provides an index of fractional hepatic insulin extraction. Thus one is able to obtain a near-complete kinetic profile of the insulin-C-peptide secretory system.
However, the structure of the combined model has recently been
criticized for its simplified description of C-peptide
kinetics (18, 27). The model was developed using the "minimal"
model concept, which calls for the simplest model to account for the observed data (3, 5). Thus single-compartment representation was used
under the assumption that such a description would provide an adequate,
though admittedly imprecise, description of C-peptide kinetics. It should be noted that the possible deleterious effect of a
simplification in C-peptide kinetics was not taken lightly, and validation experiments were performed, which apparently
demonstrated the ability of the combined model to faithfully reproduce
a known intraportal infusion pattern (28). However, under conditions of
slow dynamics, it is possible for a single-compartment model to
adequately account for two-compartment kinetics (K. Thomaseth, personal
communication). Thus the relatively slow infusion patterns used in the
validation study and the resultant slow plasma dynamics may have
created favorable conditions for using single-compartment C-peptide kinetics. It should be noted that
single-compartment C-peptide kinetics have been successfully
implemented for a variety of models and conditions (15, 16,
28-29).
The present study shows that, as plasma kinetics become more rapid, the
second-order nature of C-peptide becomes readily apparent and
the performance of the single-compartment model to account for such
kinetics degrades significantly. The original combined model was unable
to accurately account for insulin and C-peptide profiles
simulating plasma concentrations during an IVGTT. Use of the
single-compartment configuration for C-peptide kinetics resulted not only in poor model fits to the data, but also in grossly
inaccurate parameter estimates for the insulin component of the model
and presumably poor estimates for C-peptide parameters as
well. The poor model fits and inaccurate model parameter estimates led
to predictably poor reconstruction of the insulin secretory profile.
Thus it is clear from this study and the previous validation study (28)
that the original combined model with single-compartment insulin and
C-peptide kinetics has limited applicability.
The apparent inability of the original combined model to account for
rapid plasma insulin and C-peptide dynamics led to the consideration of an extended combined model with the more correct two-compartment C-peptide kinetics. The intent was to improve on the structure of the original combined model to allow application of
the method to a wider range of kinetic conditions. This extended model
was evaluated by analyzing the same simulation used to test the
original combined model. Although this second component of the
simulation study may appear to be circular logic, generating simulated
data using a given model and then using that same model to identify
parameters, there were several reasons such simulation testing was
warranted.
First, to our knowledge the algebraic manipulation of
model
2 equations that allowed for the
successful identification of all the model parameters independent of
the prehepatic insulin secretion rates has never been applied to a
compartmental analysis problem of this nature. Although theoretically
plausible, the robustness of the method in the face of error was
unknown. It was possible that practical application of the method would
be restricted due to an inability to extract kinetic information from
data with error. This simulation study was designed with error levels
assigned to simulate typical sampling and assay error that are likely
to be encountered for IVGTT data. Thus this simulation study provides a
good initial test of the robustness of the numerical approach used for
the extended combined model (model
2).
Second, because the method relies on one peptide profile as a known
input into the equations describing the kinetics of the other peptide,
it was unclear what effect errors in the input profile would have on
the parameter estimation. One potential possibility was the errors
associated with the plasma measurements having a deleterious effect on
the parameter identification process, leading to poor parameter
estimates. There were several options to handle this problem.
One would have been to ignore the errors and hope that the weighted
least squares would be immune to the effects of the input error. This
was an undesirable choice, since the least-squares algorithm is not
designed to handle errors of this type. Another possibility would have
been to smooth the input profile before the identification. With the
assumption of an unbiased smoothing algorithm being used, this option
would clearly minimize errors in the input profile. However, this
option has the negative effect of possible removal of physiological
fluctuations in the secretion profile by the smoothing routine. Thus a
certain amount of information loss is to be expected with this
approach.
The approach finally chosen was to perform the simultaneous
identification of both forms of the model: C-peptide-given
insulin and insulin-given C-peptide. This approach has the
advantage that the least-squares algorithm must provide best fits to
both the insulin and C-peptide data, given the errors in both
input profiles. Thus information from both the insulin and
C-peptide model fit and both input profiles are contributing
to the cost function in the overall identification process. The net
result of this dual identification approach would be a dampening of the
effect of measurement error associated with the profile used as the
known input. This approach assumes that the errors in insulin and
C-peptide were random Gaussian and not biased in a correlated
fashion.
The extended combined model provided reasonably accurate and precise
estimates of all model parameters. However, there was a small, but
systematic, underestimation of f, the index of fractional hepatic
insulin extraction. There is no clear explanation for this
inconsistency in the model. One possibility is that, because the model
showed a modest inability to precisely fit the first-phase plasma
responses (cf. Fig. 6), this translated into an inaccurate estimate of
the f parameter. Although modifications to the numerics of the problem
may relieve this problem, it should be noted that the relatively minor
deviation in the model fits during the first phase and the
underestimation in parameter f appear to have no negative effects on
the model's ability to estimate the other parameters or the final
calculation of insulin secretion. The results clearly indicate an
ability of the model to provide accurate and precise assessment of the
parameters of insulin and C-peptide kinetics and the final
outcome of interest, the prehepatic insulin secretion rates under the
controlled conditions of the computer environment.
In summary, the current computer simulation study evaluated the ability
of two different models to account for plasma insulin and
C-peptide data simulating an IVGTT. The original combined model using single-compartment kinetics for both insulin and
C-peptide was unable to account accurately for the simulated
IVGTT plasma dynamics, resulting in poor parameter estimates and
inaccurate reconstruction of the insulin secretory profiles. However,
because previous work has shown that the original combined model was
capable of accounting for alternative plasma dynamics (28), it would be
biased to reject the original model outright. The original combined
model can be applied under certain conditions but only after careful
consideration of the type of data to be analyzed. Analysis of the
identical set of simulated data using an extended combined model, which
incorporates two-compartmental C-peptide kinetics, resulted
in near-perfect reproduction of the known insulin secretory profiles.
These results indicate that the novel numerical approach used for the
extended combined model has no negative impact on the model's ability
to identify model parameters and estimate prehepatic insulin secretion.
Thus we conclude that this new combined model with two-compartmental
C-peptide kinetics provides accurate estimates of
"prehepatic" insulin secretion without separate assessment of
C-peptide kinetics or assumption of parameter values.
The ability of the extended combined model to easily provide a complete
kinetic profile of the insulin secretory system from a single
experimental protocol gives clinical investigators a powerful tool for
the assessment of
-cell function in vivo. Furthermore, integrated
application of the extended combined model and the Minimal Model of
glucose kinetics (2, 5, 6) to intravenous glucose tolerance test data
provides clinicians the ability to obtain a near complete metabolic
profile for a given subject. This ability should allow for improved
characterization of pathogenesis of NIDDM. However, the "acid
test" of the efficacy of the extended combined model will require
the demonstration of the ability to reconstruct known secretion rates
in an in vivo model.
 |
APPENDIX |
The identification of kinetic parameters for
model
2 is based on algebraic manipulation
to factor out the insulin secretion rate
R(t) from the model equations. The
principle behind this manipulation is based on the fact that the plasma
time courses of insulin and C-peptide are determined by the
identical secretory profile. Thus it should be possible to use one
peptide concentration profile as a surrogate representation of
prehepatic insulin secretion to characterize the kinetics of the other
peptide. In other words, it should be possible to use insulin as a
known input representative of prehepatic secretion and fit the plasma
C-peptide data to estimate the various kinetic parameters.
The derivation of new model equations is described herein.
Consider the extended combined model with two-compartment
C-peptide kinetics (cf.,
model
2, Fig.
1B) in incremental form, i.e., the
increment in insulin secretion from basal. Initially focusing on the
C-peptide component of the model independent of the insulin
equation, differential equations describing the kinetics of
C-peptide can be written as
|
(A1)
|
|
(A2)
|
Equations
A1 and A2 can be rewritten in matrix form by
combining terms, yielding
|
(A3)
|
Application
of the Laplace operator converts Eq.
A3 from the time domain to the Laplace
domain and yields the following transfer function
|
(A4)
|
A similar conversion can be made for the insulin component of the
model. The single differential equation for insulin can be written as
|
(A5)
|
and
the final transfer function in the Laplace domain is
|
(A6)
|
Two
items should be noted. First, the two-differential equation description
for C-peptide kinetics has now been reduced to a single
equation. Second, the insulin secretion rate in the Laplace domain
[R(s)] now appears in both the C-peptide and
insulin transfer functions (cf., Eqs.
A4 and A6). It is now possible to solve for R(s) in one equation and substitute into the other. In this manner, the
prehepatic secretion rate falls out of the calculation after algebraic
manipulation as shown in Eq.
A7, when R(s) was solved for in the
insulin transfer function and substituted into the C-peptide
transfer function.
|
(A7)
|
This
equation can be simplified to the single transfer function
|
(A8)
|
where
1 and
2 are the quadratic solutions
to the denominator in Eq.
A7 and characterize the kinetics of
C-peptide in plasma. The inverse of these two parameters
represents the fast and slow time constants for C-peptide
disappearance. More importantly, note that the insulin secretion rate
[R(s)] has now disappeared from the equation, with plasma
insulin now acting as a known input to the equation describing
C-peptide kinetics. Equation A8 can now be converted back into the
time domain by defining two new "state variables"
Y1 and
Y2
|
(A9)
|
|
(A10)
|
|
(A11)
|
In this formulation of the model, the plasma insulin profile is used as
a surrogate representation of the insulin secretion rate. With the use
of the plasma insulin profile as a "known" input, the plasma
C-peptide data can be fit to
Eqs.
A9-A11
to estimate the individual model parameters. It should be noted that
the alternative formulation of using C-peptide as a known
input to fit the plasma insulin data can also be derived as
|
(A12)
|
|
(A13)
|
|
(A14)
|
As noted earlier, this new modification to the parameter estimation and
calculation of the prehepatic insulin secretion rates is based on the
increment in prehepatic insulin secretion. The basal prehepatic insulin
secretion rate (rb) was
determined as
|
(A15)
|
 |
ACKNOWLEDGEMENTS |
The authors thank Dr. Jang-Hyun Youn for helpful suggestions
during the performance of these studies.
 |
FOOTNOTES |
R. M. Watanabe was a predoctoral trainee supported by the National
Institutes of Aging (Grant AG-00093) when these studies were performed
and is currently a postdoctoral fellow supported in part by the
National Center for Human Genome Research (Grant T32-HG-00042) and the
National Institute of Diabetes and Digestive and Kidney Diseases
(NIDDK) (Grant F32-DK-09525).
Support for these studies was provided by NIDDK Grant DK-29867 awarded
to R. N. Bergman.
Current address of R. M. Watanabe: Univ. of Michigan School of Public
Health, Dept. of Biostatistics, 1420 Washington Heights, Ann Arbor, MI
48109-2029.
Current address of G. M. Steil: Joslin Diabetes Center, One Joslin
Place, Boston, MA 02215.
Address for reprint requests: R. N. Bergman, Chairman, Univ. of
Southern California School of Medicine, Dept. of Physiology & Biophysics, 1333 San Pablo St., MMR-626, Los Angeles, CA 90033.
Received 4 April 1997; accepted in final form 18 September 1997.
 |
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