The traditional methods for the
assessment of insulin sensitivity yield only a single index,
not the whole dose-response curve information. This curve is typically
characterized by a maximally insulin-stimulated glucose clearance
(Clmax) and an insulin concentration at half-maximal
response (EC50). We developed an approach for estimating
the whole dose-response curve with a single in vivo test,
based on the use of tracer glucose and exogenous insulin administration
(two steps of 20 and 200 mU · min
1 · m
2,
100 min each). The effect of insulin on plasma glucose clearance was
calculated from non-steady-state data by use of a circulatory model of
glucose kinetics and a model of insulin action in which glucose clearance is represented as a Michaelis-Menten
function of insulin concentration with a delay
(t1/2). In seven nondiabetic subjects, the model
predicted adequately the tracer concentration: the model residuals were
unbiased, and their coefficient of variation was similar to the
expected measurement error (~3%), indicating that the
model did not introduce significant systematic errors. Lean (n = 4) and obese (n = 3) subjects had similar half-times for
insulin action (t1/2 = 25 ± 9 vs. 25 ± 8 min)
and maximal responses (Clmax = 705 ± 46 vs. 668 ± 259 ml · min
1 · m
2,
respectively), whereas EC50 was 240 ± 84 µU/ml in the
lean vs. 364 ± 229 µU/ml in the obese (P < 0.04). EC50 and the insulin sensitivity index (ISI, initial
slope of the dose-response curve), but not Clmax, were
related to body adiposity and fat distribution with r of
0.6-0.8 (P < 0.05). Thus, despite the small number
of study subjects, we were able to reproduce information
consistent with the literature. In addition, among the lean
individuals, t1/2 was positively related to the ISI
(r = 0.72, P < 0.02). We conclude that the
test here presented, based on a more elaborate representation of
glucose kinetics and insulin action, allows a reliable quantitation of
the insulin dose-response curve for whole body glucose utilization in a
single session of relatively short duration.
 |
INTRODUCTION |
SINCE ITS INTRODUCTION into clinical investigation two
decades ago (4), the euglycemic hyperinsulinemic clamp technique has
been established as the gold standard for the measurement of insulin
sensitivity at the whole body level. The use of this steady-state
method has made a large contribution to the understanding of the
physiology and pathophysiology of insulin action (3). Thus it has been
shown that insulin sensitivity is modulated by a number of factors,
both genetic and environmental, and can be altered in many disease
states (obesity, diabetes, hypertension, and dyslipidemia, among the
most common). Nevertheless, a substantial portion of the biological
variability of insulin action in vivo remains unexplained. Clearly,
this may be due to yet-unknown factors, but the possibility that the
standard single-step insulin clamp is insufficient must be considered.
The full dose-response curve of insulin action is more informative than
the single-point estimate provided by the standard insulin clamp. In
fact, genetic or acquired factors (and therapeutic interventions) might
selectively affect different characteristics of the dose-response
curve, such as insulin sensitivity proper (i.e., the EC50),
insulin responsiveness [i.e., maximally stimulated glucose
clearance (Clmax)], or both.
Constructing the insulin dose-response curve with multiple clamp
studies presents several problems, however. At least two hours are
needed to reach steady state at each given insulin infusion rate,
resulting in either long single-day or multiple-day protocols (7, 8,
17). Either approach is demanding (for both study subject and research
staff) and not without drawbacks. Thus previous insulin exposure
influences the response to subsequent insulin infusions; this acute
carryover effect may also extend to one or more days. In the single-day
test, prolonged immobilization and the infusion of large volumes of
fluids may alter the subject's metabolic state. On the other hand,
with a multiple-day approach, the evaluation of acute changes in
insulin action is precluded. In addition, studies in which glucose
tracers have been infused during the clamp have shown that the time to
steady state is variable among individuals and probably dose dependent
and can be affected by the presence of obesity or diabetes.
Because of the inherent limitations of the steady-state technique, an
attractive alternative would be to determine the dose-response curve by
also using non-steady-state data from a single euglycemic clamp
experiment in which insulin concentrations and glucose disposal rates
cover a large portion of the insulin sensitivity curve. Estimating
glucose fluxes in the nonsteady state requires the use of a glucose
tracer and a model of glucose kinetics. Although stable isotopes have
conveniently replaced radioactive tracers, the traditional approaches
to glucose modeling (one or two compartments) have intrinsic
limitations (20). In the present study, we describe a new method by
which modeling errors are minimized by the use of the specific activity
clamp format coupled with a circulatory model (10, 11), a
noncompartmental approach that relies on a physiological representation
of the glucose system (20). A simple model was then incorporated to
account for the delay in insulin action with respect to plasma
concentrations, and the relationship between glucose clearance and
insulin action was assumed to follow saturation (Michaelis-Menten) kinetics.
 |
METHODS |
Study subjects.
Seven adult male subjects, either healthy lean volunteers
(1-4) or obese nondiabetic patients attending our
metabolism clinic (5-7), were recruited (Table
1). None had a family history of diabetes
or was taking drugs known to affect glucose metabolism. Subject
5 had an unconfirmed diagnosis of essential hypertension; subject 4 was engaged in noncompetitive long-distance running. The potential risks of the study were carefully explained to each subject, who gave his informed written consent before the study. The
study protocol was approved by the Institutional Ethics Committee.
Experimental protocol.
Subjects were admitted as outpatients to the Metabolism Unit between
8:00 and 9:00 AM after an overnight (11-12 h) fast. The study was
done with the subject resting supine in a comfortable bed in a quiet
air-conditioned room. A 20-gauge catheter was inserted into an
antecubital vein (for the infusion of test substances); another
catheter was threaded retrogradely into a wrist vein of the same arm
and was used for blood sampling. The hand was kept in a heated box for
the entire duration of the study to achieve and maintain
arterialization of venous blood. The study protocol consisted of three
periods: basal (from
145 to 0 min), low-insulin infusion (at a
rate of 20 mU · min
1 · m
2,
from 0 to 100 min), and high-insulin infusion (200 mU · min
1 · m
2,
from 100 to 200 min). Each insulin infusion was primed with a bolus
designed as fourfold the constant infusion for the first 4 min. At
t
145 min, a primed (5 mg) constant (0.04 mg · min
1 · kg
1)
infusion of [6,6-2H2]glucose
(MassTrace, Woburn, MA) was started and was continued for the entire
basal period. During insulin infusion, plasma glucose concentration was
measured every 10 min and maintained at basal values by means of a
variable 20% glucose infusion according to the isoglycemic clamp
technique (4). To minimize the changes in plasma
[6,6-2H2]glucose enrichment, 2 g of
tracer were added to 500 ml of the 20% glucose solution while the
constant [6,6-2H2]glucose infusion
was turned off in a stepwise fashion (by 25% every 10 min). Blood
sampling for the assay of plasma
[6,6-2H2]glucose enrichment was
more frequent (every 2-5 min) during the first 50 min of each of
the three study periods and was spaced at 10- to 15-min intervals
thereafter. Three blood samples for plasma insulin determination were
also taken at the end of the basal period, whereas during insulin
infusion sampling for insulin was the same as for plasma
[6,6-2H2]glucose.
Analytical procedures.
Plasma glucose was assayed by the glucose oxidase method (Glucose
Analyzer, Beckman Instruments, Fullerton, CA). Specific insulin was
assayed in plasma by RIA (human insulin-specific RIA kit, Linco
Research, St. Charles, MO). Plasma
[6,6-2H2]glucose enrichment was
measured in arterialized blood samples after deproteinization with
barium hydroxide (0.3 N) and zinc sulfate (0.3 N). The supernatant was
run through columns of ion-exchange resins, evaporated, and
derivatized. The sample tracer-to-tracee ratio (TTR) was determined as
previously described (19). Briefly, isotopic enrichment was determined
on the pentaacetate derivative (1:1 acetic anhydride-pyridine) by gas
chromatography-mass spectrometry (GC-MS) (Hewlett-Packard GC 5890-MS
5972, Palo Alto, CA) by use of electronic impact ionization and
selectively monitoring ions of rounded molecular weight (rmw) 200, 201, and 202. Following the method used by Rosenblatt et al. (18), the
apparent enrichment of rmw 202 (M + 2) was corrected for the
contribution of singly labeled molecules [rmw 201, (M + 1)] by subtracting the product [TTR (M + 1) × natural abundance of (M + 2)] from the TTR (M + 2). This correction was very small and rather constant, ranging from
0.01 to 0.02% for plasma sample enrichments between 2 and 5%. All
samples from the same study were processed and assayed in the same run.
Model of glucose kinetics.
In the circulatory model (10, 11, 15), the body is schematized as the
combination of the heart-lung block and the periphery block, which
lumps together all the remaining tissues (Fig.
1). Each block is regarded as a single
inlet-single outlet organ and can be described mathematically by an
impulse response (9). The organ impulse response is defined as the
tracer efflux observed at the outlet after a bolus injection of a unit
dose into the inlet (with assumption of no tracer recirculation). After
bolus injection into a peripheral vein, the tracer disappearance curve will be the result of the combination of the impulse responses of the
two interconnected blocks. When blood flow, i.e., cardiac output (F),
and the impulse response of the heart/lung are known, the impulse
response of the periphery can be calculated (12). Cardiac output was
calculated as follows: F (milliliters per square meter of body surface
area) = 3,200-30 × (years of age
40). This quantity
was assumed to remain constant during the test and was corrected for
the ratio of whole blood to plasma glucose concentration (= 0.84) to
obtain the actual glucose mass flux. Second, the impulse response of
the heart-lung block was assumed to be known and not affected by
insulin. It was represented by a two-exponential function starting from
zero and returning to zero after rising to an early peak. The
parameters of the heart-lung impulse response were set to match
experimentally derived curves, as detailed in Ref. 15. In particular,
the heart-lung glucose distribution volume was assumed to be 400 ml/m2, and glucose fractional extraction was assumed to be
nil. Third, the impulse response of the periphery block was represented
by a four-exponential function, starting from zero and gradually returning to zero after reaching a peak, with the fastest rising exponential term fixed. This impulse response can be conveniently represented as a convolution of a three-exponential function and a
single-exponential function,
e
t, representing the
fastest rising exponential term in which
= 10 min
1 is fixed (15).
Thus
|
|
|
(1)
|
where the symbol
is the convolution operator. The
three-exponential function in square brackets has the property that its integral from zero to infinity is one, and the parameter
wi (dimensionless) represents the relative
contribution of the exponential term of exponent
i (min
1) to the
total integral. From this property, it follows that the integral from
zero to infinity of rper(t) is 1
E,
i.e., E (dimensionless or %) is the glucose fractional extraction of
the periphery block (15), which is constant in the basal state and
varies with insulin concentration during the clamp period.

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Fig. 1.
Scheme of model used to calculate parameters of the insulin
dose-response curve. Glucose kinetics (top) is described by use
of a circulatory model (see METHODS for detailed
description). Insulin action (bottom) is modeled by allowing a
monoexponential delay between plasma insulin concentration and its
metabolic effect and by assuming a Michaelis-Menten dose-response
curve. Emax, maximal extraction; t1/2,
half-time.
|
|
Model of insulin action.
To account for saturation of glucose fluxes (as observed in most in
vivo and in vitro experimental studies), the model assumes that insulin
action on fractional glucose extraction follows Michaelis-Menten kinetics (Fig. 1). In addition, to use also the non-steady-state data,
the delay that is normally observed between changes in plasma insulin
concentration and changes in E was modeled by use of a single linear
differential equation (monoexponential delay). For this purpose, a new
variable, Z(t), representing the delayed insulin time course,
was introduced. Z(t) (in microunits per milliliter) is related
to the measured plasma insulin concentration increment from baseline by
the equation
|
(2)
|
where
I(t) (in microunits per milliliter) is the plasma insulin
concentration at time t, Ib (microunits per
milliliter) is plasma insulin concentration at baseline, and
(min
1) quantifies the delay; the
half-time of insulin action is thus calculated as
ln(2) · 
1. In this
model, therefore, it is Z(t) that determines fractional glucose
extraction E(t) according to a Michaelis-Menten relationship
|
(3)
|
where
Eb (dimensionless or %) is the basal glucose fractional
extraction, and Emax (dimensionless or %) and
EC50 (in microunits per milliliter) are the
Michaelis-Menten parameters.
During the low-dose insulin infusion, the onset of insulin action was
found to be somewhat irregular. To account for this, a further
disturbance term [Ed(t), dimensionless or
%] was introduced into Eq. 3 to add flexibility to
E(t) in the initial phase of the clamp period, i.e.
|
(4)
|
where
the disturbance term Ed(t) is nonzero only during
the first 60 min of the low-insulin period and is represented as a
generic piecewise constant function on 10-min intervals.
The dose-response curve relating glucose clearance (Cl, in milliliters
per minute per square meter) to insulin concentration at steady state
is calculated from the Michaelis-Menten parameters. Therefore, because
glucose clearance is the product of cardiac output (F) and fractional
glucose extraction (E) at the insulin concentration I (in µU/ml),
glucose clearance is given by
|
(5)
|
From Eq. 5, the maximally insulin-stimulated glucose clearance
is calculated as Clmax = F(Eb + Emax). The glucose clearance at plasma insulin
concentration of 100 µU/ml (Cl100) was calculated by
fitting I = 100. For insulin concentrations well below
EC50, the relationship between insulin concentration and
glucose clearance is nearly linear. The slope of the line, which is the
ratio of F × Emax to EC50, is a
traditional insulin sensitivity index, denoted here as ISI.
Tracer fit and parameter estimation.
The six parameters of the periphery impulse response (Eq. 1)
were estimated by fitting the circulatory model to the
[6,6-2H2]glucose concentration in
the basal state. To analyze the data of the clamp period, the
circulatory model of glucose kinetics was combined with the model of
insulin action (Eqs. 2 and 4). We assumed that in this
non-steady-state condition the major effect of insulin is on E, whereas
the other parameters of the periphery impulse response are not modified
by insulin, as supported by experimental studies using the clamp (10,
15). Thus the parameters
1,
2,
3, w1, w2
(Eq. 1), and Eb (Eq. 4), were taken from
the analysis of the basal tracer curves, whereas EC50 and
Emax (Eq. 4) and
(Eq. 2) were estimated
from [6,6-2H2]glucose concentration
during the clamp. In this analysis, plasma insulin concentration, which
is required as a continuous function of time, was smoothed and
interpolated with the use of a simplified model of insulin kinetics
(14).
Parameters were estimated by least squares with Matlab. We used equal
weights for all tracer points, because with relatively stable
[6,6-2H2]glucose concentrations the
error variance was not expected to differ substantially among time points.
The experimental data were also analyzed by use of an approach that
does not require assumptions on the mechanism of insulin action. The
circulatory model of glucose kinetics and the parameters of the basal
period were used to reconstruct the time course of the fractional
glucose extraction E(t) from the
[6,6-2H2]glucose concentration.
This time-varying E(t) was approximated as a piecewise constant
function of time on 5-min intervals. E(t) was thus represented
by a total of 40 elements in the 200 min of nonsteady state, which were
estimated by least squares fit of the
[6,6-2H2]glucose concentration
according to a scheme analogous to deconvolution (15). The time-varying
glucose clearance was then calculated as F × E(t).
After E(t) is estimated, the circulatory model makes it
possible to calculate the plasma glucose concentration that is due to
the exogenous glucose infusion. By subtracting these values from the
measured plasma glucose concentrations, the component of glucose
concentration due to endogenous glucose production (EGP) is obtained.
EGP was calculated from the endogenous glucose concentration and the
model by a deconvolution method (15). Briefly, EGP was estimated by
minimizing the sum of the squared model residuals (observed minus
model-predicted endogenous tracee concentration) plus the sum of the
squared elements of the second derivative of EGP multiplied by a
weighting factor. The latter term is necessary to eliminate the
spurious oscillations of EGP that would be observed when using ordinary
least squares, as deconvolution is an ill-conditioned problem. An
appropriate choice of the weighting factor eliminates the spurious EGP
oscillations, while fitting the glucose concentrations within the
expected experimental error.
Statistical analysis.
All fluxes were normalized per square meter of body surface area (BSA),
which was calculated according to the equation of Gehan and George, as
reported in Bailey and Briars (1). Differences between groups were
compared by the Mann-Whitney U-test. Associations between
variables were first tested by the Spearman rank correlation test; when
statistically significant (P < 0.05), linear regression was
used to obtain further information.
 |
RESULTS |
The results of one case (subject 1) are presented in Fig.
2. After the initial washout curve of the
[6,6-2H2]glucose prime, the tracer
concentration reached a steady value at the end of the basal period,
fluctuated at the beginning of the insulin infusion, and then
stabilized during the remainder of the study. The solid line of the top
panel shows the time course of the tracer concentration as predicted by
the model, i.e., following the estimation of the parameters of glucose
kinetics and insulin action in this subject. The small deviation of the
predicted from the experimental data indicates that the model described
accurately the individual characteristics of insulin action. In
response to the low- and high-dose insulin infusions (i.e., 20 and 200 mU · min
1 · m
2),
plasma insulin rapidly reached stable plateaus at ~55 and 500 µU/ml, respectively. Although slower, the exogenous glucose infusion also reached quasi-stable rates during the last 20-30 min of each insulin step, averaging 867 and 2,671 µmol · min
1 · m
2,
respectively. The insulin plot, in addition to the curve interpolating the experimental plasma insulin concentrations, shows the delayed plasma insulin concentration as reconstructed by the model. This delay
is also evident when comparing the time course of glucose clearance to
the plasma insulin concentration profile. Whole body glucose clearance
is shown as estimated by use of the Michaelis-Menten model with the
disturbance term (solid line) and the time-varying E(t)
approach (broken line curve). The visual comparison between these two
curves gives an idea of how accurate the hypothesis is that the insulin
dose-response curve follows Michaelis-Menten kinetics. Figure 2 also
depicts the time course of EGP, which, in this subject, was stable in
the basal period, rapidly halved in response to the low-dose insulin
infusion, and was completely suppressed by the high-dose insulin
infusion.

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Fig. 2.
Measured tracer and insulin concentration, glucose infusion rates,
calculated glucose clearance, and glucose production are plotted for
case 1. Solid line of first panel represents tracer
concentration as predicted by the model. Solid line of second
panel represents fitting of insulin concentration data, and dotted
line shows "delayed" insulin as calculated by the model. Glucose
clearance (fourth panel) is calculated both with (solid line)
and without (broken line) the Michaelis-Menten assumption.
|
|
The mean values of the model parameters in the basal state were:
Eb, 0.023 ± 0.002;
1, 5.6 ± 1.6 min
1;
2, 0.38 ± 0.04 min
1;
3, 0.040 ± 0.007 min
1; w1,
0.45 ± 0.04; w2, 0.48 ± 0.04. On
average, the coefficients of variation of these parameters, as given by
the least squares algorithm, were: <1% for Eb, 180% for
1, 22% for
2, 60% for
3,
and 30% for w1 and w2.
This indicates that the model parameters in the basal state are
reasonably well defined, although the estimate of the fastest
exponential of the periphery impulse response (
1) was
less precise. During the non-steady-state period, the coefficients of
variation of the Michaelis-Menten and the delay parameter
were <1% on average. On average, the disturbance
term Ed(t) was not different from zero at all time
instants from 0 to 60 min. The accuracy of model-predicted tracer
concentrations in the whole study group can also be appreciated from
Fig. 3. On average, the model-predicted
curve matched the experimental data closely. Bars with twice the SE of
the model residuals (i.e., the difference between the measured and the
predicted tracer concentration) crossed the zero line at virtually all
time points. This indicates that residuals were not different from zero
throughout. Furthermore, the average values of the standard deviation
of the model residual in the basal period and during the
non-steady-state periods were virtually identical (basal SD = 3.1, low-insulin SD = 2.6, high-insulin SD = 3.8 µmol/l). These
values correspond to a coefficient of variation of ~3%, which is of
the same magnitude as the expected error of the plasma tracer
concentration measurement. This indicates that the model error was
largely explained by the experimental error.

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Fig. 3.
A: mean tracer concentration ± SE (circles) and mean
model-predicted tracer concentration (solid line) in 7 subjects.
B: mean ± 2SE of model residuals, i.e., difference between
measured and model-predicted tracer concentrations at each time point.
For this analysis, the Michaelis-Menten model included disturbance term
Ed(t) (see METHODS).
|
|
Figure 4 shows the time course of plasma
glucose clearance (separately for lean and obese subjects) as predicted
by the model with (solid line) or without (broken line) the
Michaelis-Menten assumption. In this calculation, the disturbance term
was not included to show that the Michaelis-Menten approximation, even without Ed(t), is equivalent to the prediction
obtained with the time-varying E(t) approach. The obese
subjects showed a lower activation of whole body glucose clearance at
both insulin infusion steps.

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Fig. 4.
Time course of whole body glucose clearance during low- and
high-insulin infusions, as calculated by the model with (solid line)
and without (broken line) the Michaelis-Menten assumption. For this
analysis, disturbance term Ed(t) (see
METHODS) was not included in individual calculation. Error
bars are SE shown at 10-min intervals.
|
|
Basal EGP averaged 303 ± 65 µmol · min
1 · m
2
in the seven subjects; the low-insulin infusion suppressed this flux
more in lean than in obese subjects (78 ± 7 vs. 44 ± 7%, P < 0.05), whereas the high-insulin infusion suppressed EGP to a
similar extent in both [85 ± 19% vs. 83 ± 8%, P = not significant (NS)].
The individual values of plasma glucose and insulin concentrations,
glucose clearance, and the model-derived parameters are given Table
2 for lean and obese subjects, whereas the
individual Michaelis-Menten functions are plotted in Fig.
5. The EC50 was lower
(P < 0.04) in lean than in obese subjects. In the latter, a
blunted response to insulin stimulation was particularly pronounced at
physiological insulin values. Despite the small number of subjects studied, indexes of insulin sensitivity, such as ISI or the glucose clearance at a plasma insulin of 100 µU/ml, were clearly correlated with basal insulin concentration (Ib), body mass index
(BMI), and waist-to-hip ratio (WHR), as expected. For instance, ISI was inversely correlated with Ib (r = 0.87, P < 0.02), BMI (r = 0.80, P < 0.05) and WHR
(r = 0.86, P < 0.02) (all correlations are performed on logarithmic transformed values). The correlation of ISI with Ib, BMI, and WHR was due to EC50
(r values of 0.6-0.8, P < 0.05), because Clmax was not correlated with these variables
(P = ~0.7 or greater).

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Fig. 5.
Individual insulin dose-response curves of the 4 lean (solid lines) and
3 obese (broken lines) subjects.
|
|
Moreover, in the nonobese subjects, the t1/2 and
ISI were reciprocally related to one another (r = 0.92, P < 0.05), indicating that the higher the insulin
sensitivity, the longer the delay in insulin action. In accordance with
this prediction, when measured at the presumed steady state (i.e.,
after 80 min of each insulin infusion), glucose clearance
underestimated the true steady-state clearance (as calculated by the
model) by an amount that was directly related to
t1/2 (r = 0.72, P < 0.02).
 |
DISCUSSION |
The present study was prompted by the need for a single test providing
a more complete description of the whole body response to insulin
stimulation. By combining a circulatory (noncompartmental) model of
glucose kinetics with a simple model of insulin action (exponential
delay), we were able to make use of non-steady-state data to determine
in vivo the full dose response of insulin in the form of a
Michaelis-Menten function.
This approach has several advantages. First, it shortens the duration
of the test, even in comparison with the single-day sequential clamp
protocol. In our experiment, insulin stimulation lasted <3.5 h,
whereas
6 h are required to create three hyperinsulinemic steady
states, i.e., the minimum necessary to define a
Michaelis-Menten curve. By using only the steady-state data of
the present experiments, the generation of a Michaelis-Menten
function failed in two insulin-resistant subjects, yielding negative
EC50 values (results not shown). Second, the test provides
additional parameters (Table 2) that may be of interest in the study of
the pathophysiology of glucose homeostasis. Third, the test is not
strictly dependent on the experimental design, provided that large
enough insulin gradients are created and that quasi-steady-state
periods are included.
These advantages rely on the accuracy of the model of glucose kinetics
and insulin action. The three main assumptions involved in the
circulatory model of glucose kinetics (constant cardiac output,
constant glucose mean transit time, and characteristics of heart and
lung kinetics) appear to be rather robust, as recently discussed (14).
In brief, whereas fractional clearance and mean transit time would be
strongly affected by errors in cardiac output, volume and clearance
(and thus the insulin sensitivity indexes) are less sensitive, and this
dependence is progressively lost as steady state is approached. The
heart and lung kinetics assumptions are even less important, because
the contribution of this block to overall glucose kinetics is small and
short-lived. Although experimental data are scarce, the assumption that
insulin enhances whole body glucose disposal by increasing fractional
extraction without affecting transit times is supported by direct
evidence (15). Furthermore, with a protocol that minimizes the changes in glucose specific activity, the calculation of glucose fluxes is
almost model independent. More critical is the model of insulin action.
The monoexponential delay used to represent the lag phase of insulin
action has been successfully employed in other models (6, 13). The
Michaelis-Menten function has been previously shown to suitably
approximate the experimental dose-response data. In our experiments,
this model succesfully fit the observed tracer concentration profile
during the entire test. As shown in Fig. 3, the mean of the differences
between the predicted and the measured tracer concentrations (i.e., the
residuals) at each time point was not different from zero. Furthermore,
on average, the standard deviation of the residuals of the basal period
(when the model error is presumably negligible) was virtually identical
to that of the non-steady-state period and similar to the expected
measurement error. This indicates that the deviation of the
model-predicted tracer concentrations from the measured values is due
mainly to measurement and not to modeling error. Another finding
supporting the model of insulin action is that the time course of
glucose clearance predicted by the model and that calculated without
assuming a specific model of insulin action [time-varying
E(t) approach] were very similar (Fig. 4).
The only data that were unable to fit properly in the plain
Michaelis-Menten model in some subjects were at the start
of the insulin infusion (0-60 min). From the analysis with the
time-varying approach, we know that this failure is due to an irregular
onset of insulin action during this phase. This is why the disturbance term (see METHODS) was used only in those first 60 min.
Although there could be some physiological explanation for this
transient perturbation (in some subjects, standing up to void), other
reasons support the conclusion that neither the presence of
irregularities nor the disturbance term invalidates the
Michaelis-Menten model. First, the bulk of the experimental information
for determining the parameters of the Michaelis-Menten model comes from
the two quasi-steady-state periods (for both Emax and
EC50) and from the rise in glucose clearance at the
beginning of the high-insulin period (for
); during these periods,
the disturbance is clearly irrelevant. In other words, the data points
in the initial 60-min period, in which the disturbance is present, do
not affect the estimation of Emax and EC50.
Second, although the disturbance term was included in all subjects for
uniformity of analysis, in some of them (e.g., the subject of Fig. 2)
it could be omitted without a significant loss of fit, i.e., the plain
Michaelis-Menten model was an adequate representation of the system in
some cases. Third, on average, the disturbance term was not different
from zero, and the time course of glucose clearance as predicted by the
Michaelis-Menten model without the disturbance was very similar to that
calculated by the time-varying fractional extraction approach (Fig. 4).
This indicates that the deviations of the Michaelis-Menten model from
the reality were not systematic and that even without Ed(t) this model is on average a good predictor of
glucose clearance.
Although to minimize the errors in the estimation of the dose-response
curves we have used an elaborate model of glucose kinetics, simplifications, such as the use of Steele's model, could also be
satisfactory, particularly with an experimental design that reduces the
modeling error by minimizing the changes in specific activity. However,
the use of Steele's model would still require simulation of nonlinear
differential equations and data fitting, as with the circulatory model,
because of the coupling with the model of insulin action. Therefore,
the potential loss of accuracy of the simplified models would not be
accompanied by a significant increase in simplicity.
The physiological significance of the current results is obviously
limited by the small number of subjects studied; nevertheless, comparison with existing information is possible. In agreement with
previous studies, in our subjects obesity was associated with a twofold
higher EC50 and similar Clmax. In absolute
terms, the estimates of Clmax and EC50
generated by our model are higher than those previously reported by
other laboratories (2, 7, 16, 17). However, already at the end of the
second step of insulin infusion, we measured a mean steady-state (i.e.,
model-independent) glucose clearance rate of 480 ml · min
1 · m
2
(Table 2), a value similar to previous steady-state estimates of
Clmax obtained at much higher plasma insulin
concentrations. Similarly, Groop et al. (5) reported values of whole
body glucose clearance of ~400
ml · min
1 · m
2
in normal subjects at a plasma insulin concentration of 200 µU/ml above baseline. Therefore, differences in the estimated maximal insulin-stimulated glucose clearance may reflect the variability inherent in small groups of study subjects. Alternatively, because in
our data the t1/2 of insulin action in normal
subjects was longer when insulin was higher, it is possible that true
Clmax may have been underestimated in previous studies.
Available estimates of EC50 vary widely (2, 7, 16, 17).
This variability is, at least in part, related to experimental problems
such as the use of protocols not optimized with regard to changes in
glucose specific activity, actual attainment of steady state, or number of insulin steps. By modeling non-steady-state data, our approach makes
use of a continuum of increasing glucose uptake values from basal to
near-maximal and not just the few values that would be generated by the
standard steady-state approach.
Of interest is the finding that the t1/2 of insulin
action was not different between obese and lean subjects, although they had clearly different indexes of insulin sensitivity (Table 2). In
contrast, previous analyses of insulin dose-response curves in lean and
obese subjects have led to the conclusion that the half-time of insulin
activation is prolonged in the obese (16). Although we cannot deny that
our small study groups are not a representative sample of the
respective populations, the discrepancy could be due to the method used
by Prager et al. (16) to estimate the t1/2 of
insulin action. When glucose uptake functions are expressed as a
percentage of maximal glucose uptake, the closer glucose uptake is
driven to saturation, the shorter is its apparent t1/2. Because in obese subjects glucose uptake
saturates at higher insulin concentrations than in lean subjects
(higher EC50), at the same insulin concentration, glucose
uptake is less saturated in the obese than in the lean. Thus linear
scaling of a nonlinear (saturable) process artificially shortens the
t1/2 of insulin-sensitive subjects more than that
of insulin-resistant individuals. In contrast, our estimate of
t1/2 is obtained directly by fitting a model of insulin action to the data, thereby providing an unbiased estimate of
insulin's half-time of activation.
In conclusion, the test here presented exploits a more elaborate
representation of insulin sensitivity than the standard euglycemic insulin clamp to characterize the full dose-response curve of insulin
action in a single session of relatively short duration. It has the
potential to prove useful in the clinical investigation of disordered
carbohydrate tolerance.
This work has been supported in part by a grant from the project
"Mathematical methods and models for the study of biological phenomena" of the Italian National Research Council and by the 1996 Glaxo/European Association for the Study of Diabetes
"Burden of Diabetes Research Fellowship" received by Andrea Natali.
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. §1734 solely to indicate this fact.
Address for reprint requests and other correspondence: A. Natali,
C.N.R. Institute of Clinical Physiology, Via Savi, 8, 56126 Pisa, Italy
(E-mail: anatali{at}ifc.pi.cnr.it).