MODELING IN PHYSIOLOGY
Undermodeling affects minimal model
indexes: insights from a two-compartment model
Andrea
Caumo1,
Paolo
Vicini2,
Jeffrey J.
Zachwieja3,
Angelo
Avogaro4,
Kevin
Yarasheski5,
Dennis M.
Bier6, and
Claudio
Cobelli7
1 San Raffaele Scientific
Institute, 20100 Milan;
4 Department of Metabolic
Diseases and 7 Department of
Electronics and Informatics, University of Padova, 35131 Padua, Italy;
2 Department of Bioengineering,
University of Washington, Seattle, Washington
98195; 3 Pennington
Biomedical Research Center, Louisiana State University, Baton Rouge,
Louisiana
70808; 5 Metabolism
Division, Washington University School of Medicine, Saint Louis,
Missouri 63110; and
6 Children's Nutrition Research
Center, Baylor College of Medicine, Houston, Texas
77030-2600
 |
ABSTRACT |
The classic (hereafter cold) and the labeled
(hereafter hot) minimal models are powerful tools to investigate
glucose metabolism. The cold model provides, from intravenous glucose
tolerance test (IVGTT) data, indexes of glucose effectiveness
(SG) and insulin sensitivity
(SI) that measure the effect of
glucose and insulin, respectively, to enhance glucose disappearance and
inhibit endogenous glucose production. The hot model provides, from hot
IVGTT data, indexes of glucose effectiveness
(
) and insulin sensitivity (
) that, respectively, measure the
effects of glucose and insulin on glucose disappearance only. Recent
reports call for a reexamination of some of the assumptions of the
minimal models. We have previously pointed out the criticality of the
single-compartment description of glucose kinetics on which both the
minimal models are founded. In this paper we evaluate the impact of
single-compartment undermodeling on
SG,
SI,
, and
by using a two-compartment model
to describe the glucose system. The relationships of the minimal model
indexes to the analogous indexes measured with the glucose clamp
technique are also examined. Theoretical analysis and simulation
studies indicate that cold indexes are more affected than hot indexes
by undermodeling. In particular, care must be exercised in the
physiological interpretation of
SG, because this index is a local
descriptor of events taking place in the initial portion of the IVGTT.
As a consequence, SG not only
reflects glucose effect on glucose uptake and production but also the
rapid exchange of glucose between the accessible and nonaccessible
glucose pools that occurs in the early part of the test.
insulin sensitivity; glucose effectiveness; mathematical model; intravenous glucose tolerance test; glucose clamp
 |
INTRODUCTION |
THE INTRAVENOUS GLUCOSE TOLERANCE TEST (IVGTT),
standard or modified with a tolbutamide or insulin injection,
interpreted with the classic minimal model of glucose disappearance
(hereafter cold minimal model) (6-10), is a powerful research tool
to investigate glucose metabolism in physiopathological and
epidemiological studies; more than 350 papers have appeared until 1998. The model provides two metabolic indexes measuring glucose
effectiveness (SG) and insulin
sensitivity (SI).
SG and
SI are composite parameters, i.e.,
they measure the overall effect of glucose and insulin, respectively,
to enhance glucose disappearance
(Rd) and inhibit endogenous
glucose production (EGP). To segregate the effect of glucose and
insulin on Rd and EGP, a labeled
(hereafter hot) IVGTT has been introduced, i.e., a glucose tracer has
been added to the glucose bolus (2, 17, 19, 23). The hot IVGTT
interpreted with a minimal model of labeled glucose disappearance
(hereafter hot minimal model) provides new indexes of glucose
effectiveness (
) and insulin
sensitivity (
) that measure the
effects of glucose and insulin, respectively, on glucose disposal only (19, 23).
Several investigators have recently reexamined some of the minimal
model assumptions (16-18, 22-24, 27, 30, 32). We have found
some unexpected relationships between the cold and hot
indexes (17, 19); in addition, we have
observed that when EGP is derived by combining the cold and hot minimal
models, its time course is physiologically absurd (17). Quon et al.
(30) have shown in a study on insulin-dependent diabetes mellitus
patients that SG is likely to be
overestimated. Saad et al. (32) have shown that
SI obtained from an
insulin-modified IVGTT is well correlated but markedly underestimated
compared with the insulin sensitivity index obtained with the glucose
clamp technique. Finegood and Tzur (24) have shown in dogs that
decreased SG associated with decreased insulin response is an artifact of the minimal model method
and that SG is poorly correlated
with the glucose effectiveness index obtained with the glucose clamp technique.
We have suggested two possible areas of model error (16, 18, 22, 23,
38): the monocompartmental structure of both the minimal models and the
description of EGP embodied in the cold minimal model. We have shown
that the monocompartmental structure is the major area responsible for
the implausible EGP profile and that a two-compartment hot minimal
model provides not only a reliable profile of EGP by deconvolution (14,
39) but also tracer-based indexes of glucose effectiveness, insulin
sensitivity, and plasma clearance rate (37). Recently, we have used the
two-compartment paradigm (18, 22, 38) to explain the findings of Quon
et al. (30) and Saad et al. (32) and the poor agreement between SG and the clamp-based index of
glucose effectiveness (24).
The aim of the present paper is to use a two-compartment model of
glucose metabolism to explain the mechanisms by which monocompartmental undermodeling affects both cold and hot minimal model indexes.
Glossary
A1, A2;
,
 |
Coefficients of two-exponential cold and hot glucose decay during an
IVGTT at basal insulin, mg/dl and dmp/ml (for a radiolabeled IVGTT)
|
D, D* |
Cold and and hot glucose IVGTT dose, mg/kg and dpm/kg, respectively
|
EGP(t) |
Endogenous glucose production,
mg · kg 1 · min 1
|
EGPb |
Endogenous glucose production in the basal state,
mg · kg 1 · min 1
|
g(t), g*(t) |
Cold and hot glucose concentration in plasma, mg/dl and dpm/ml,
respectively
|
g(0), g*(0) |
Minimal model estimates of cold and hot glucose concentration at
time 0+, mg/dl and dpm/ml, respectively
|
gb |
Plasma glucose concentration in basal state, mg/dl
|
g2(t),
*2(t) |
Cold and hot glucose concentration in the second pool of the
two-compartment model, mg/dl and dpm/ml, respectively
|
2(t),
*2(t) |
As above, with insulin-dependent removal moved to the accessible pool,
mg/dl and dpm/ml, respectively
|
GE, GE* |
Cold and hot glucose effectiveness of the two-compartment
model,
ml · kg 1 · min 1
|
GEb |
Cold glucose effectiveness measured from the area under the glucose
excursion during an IVGTT at basal insulin,
ml · kg 1 · min 1
|
GINF(t) |
Glucose infusion rate during the glucose clamp,
mg · kg 1 · min 1
|
k21, k12,
k02, kd, |
Rate parameters of the two-compartment model, min 1
|
k22 |
k22 = k12+k02,
min 1
|
ka |
Rate constant of the remote insulin compartment in the two-compartment
model, min 1
|
kbd, kbp |
Parameters describing insulin effect on glucose uptake and EGP in the
two-compartment model,
min 2 · ml · µU 1,
respectively
|
kp |
Parameter describing glucose effect on EGP in the two-compartment
model, min 1
|
i(t) |
Insulin concentration in plasma, µU/ml
|
ib |
Plasma insulin concentration in the basal state, µU/ml
|
IS,IS* |
Cold and hot insulin sensitivity of the two-compartment model,
ml · kg 1 · min 1
per µU/ml
|
PCRb |
Plasma glucose clearance in the basal state,
ml · kg 1 · min 1
|
p1, p2;
,
 |
Cold and hot minimal model rate parameters, min 1
|
qi(t),
(t) |
Cold and hot glucose mass in ith compartment of the
two-compartment model (i = 1, 2), mg and dpm, respectively
|
Rd(t) |
Glucose disappearance rate from the accessible pool,
mg · kg 1 · min 1
|
Rd,0 |
Nonzero intercept of the relationship Rd vs. g,
mg · kg 1 · min 1
|
SG,  |
Minimal model estimates of cold and hot glucose effectiveness,
min 1
|
SG(clamp),SG,d(clamp) |
Glucose clamp measurements of cold and hot glucose effectiveness,
ml · kg 1 · min 1
|
SI,  |
Minimal model estimates of cold and hot insulin sensitivity,
min 1 · µU · ml 1
|
SI(clamp),SI,d(clamp) |
Glucose clamp measurements of cold and hot insulin sensitivity,
ml · kg 1 · min 1 · µU 1 · ml
|
t |
Time, min
|
V,V* |
Cold and hot minimal model volume, ml/kg
|
V1 |
Volume of the accessible pool of the two-compartment model, ml/kg
|
VT |
Total glucose distribution volume, ml/kg
|
x(t), x*(t) |
Cold and hot minimal model insulin action, min 1
|
X(t) |
Two-compartment model insulin action, i.e., X = xp+xd,
min 1
|
(t) |
As above, with insulin-dependent removal moved to the accessible pool,
i.e., = xp + d, min 1
|
xd(t) |
Two-compartment model insulin action on glucose uptake,
min 1
|
d(t),
(t) |
As above, with insulin-dependent removal moved to the accessible pool
(the asterisk denotes tracer-based calculation),
min 1
|
xp(t) |
Two-compartment model insulin action on EGP, min 1
|
(t) |
Deviation of hot glucose decay from a two-exponential function during
an IVGTT at basal insulin, dpm/ml
|
 |
= k21k12,
min 2
|
1, 2;
,
 |
Fast and slow eigenvalues of the cold and hot glucose decay during an
IVGTT at basal insulin, min 1
|
 |
THE COLD AND HOT MINIMAL MODELS |
The Cold Model
The cold minimal model (Fig. 1) interprets
plasma glucose and insulin concentrations measured during an IVGTT
(standard, or modified with a tolbutamide or insulin injection). The
model in its uniquely identifiable parametrization (6, 8, 9, 23) is
described by
|
(1)
|
where g is plasma glucose concentration
(gb denotes its basal end test
value), i is plasma insulin concentration
(ib denotes its basal end test
value), D is the glucose dose in the bolus, V is the glucose
distribution volume, x is insulin
action [x = (k4+k6)i',
where i' is insulin in the remote compartment], and the
pi values are parameters related to
the ki values: p1 = k1+k5,
p2 = k3,
p3 = k2(k4+k6).
Parameters p1,
p2,
p3, and V can be
estimated from glucose and insulin data by use of nonlinear least
squares parameter estimation techniques (13). From them one can
calculate the cold indexes of glucose effectiveness,
SG, and insulin sensitivity,
SI, as
|
(2)
|
SG and
SI measure the effects of glucose
and insulin, respectively, on both
Rd and EGP. In fact, because
SG is a function not only of
k1, but also of
k5 (see Fig. 1),
it measures the ability of glucose at basal insulin to stimulate
Rd and to inhibit EGP. Similarly,
SI is a function not only of
k1,
k3,
k4, but also of k6, and thus
measures the ability of insulin to enhance the glucose stimulation of
Rd and inhibition of EGP.
Parameter p2 is
the rate constant of the remote insulin compartment and governs the
speed of rise and decay of insulin action.
Reference values for SG and
SI have been obtained from the
analysis of insulin and cold glucose data of a hot IVGTT performed in
25 normal young adults. Values for
SG and
SI were, respectively, 0.026 ± 0.002 min
1 and 7.3 ± 1.0 × 10
4
min
1 · µU
1 · ml.
The mean precision of SG and
SI estimates was 49 and 18%, respectively. Volume V was estimated as 1.66 ± 0.05 dl/kg.
The Hot Model
The hot minimal model (Fig. 2) interprets
plasma hot glucose and insulin concentrations measured during a hot
IVGTT, that is, an IVGTT (standard, or modified with a tolbutamide or
insulin injection) in which a glucose tracer (radioactive or stable
isotope) is added to the glucose bolus. Because hot glucose
concentration only reflects Rd,
the hot model yields indexes measuring glucose and insulin effect on
Rd only. The model in its uniquely
identifiable parametrization (2, 17, 19, 23) is described
by
|
(3)
|
where the symbols are the same as in Eq. 1, with the asterisk denoting tracer-related variables
and parameters. In particular, D* is the hot glucose dose, V* is the
hot glucose distribution volume, x* is
hot insulin action (proportional to remote insulin i'*,
x* = k4i'*), and
the
values
are parameters related to the
ki values:
= k1,
= k3, and
= k2k4.
Parameters
,
,
, and V* can be
estimated from insulin and hot glucose data by using nonlinear least
squares parameter estimation techniques (13). From them one can
calculate the hot indexes of glucose effectiveness,
, and insulin sensitivity,
, as
|
(4)
|
measures the ability
of glucose at basal insulin to stimulate
Rd, and
measures the ability of insulin
to enhance glucose stimulation of
Rd. Parameter
is the rate constant
of the remote insulin compartment and governs the speed of rise and
decay of hot insulin action.
Values for
and
have been obtained in the same 25 normal young subjects from the analysis of insulin and hot glucose data
of the hot IVGTT. Data on 15 subjects have already been reported in
previous publications (2, 17). Stable isotopes ([6-2H2]glucose
and [2-2H]glucose)
were employed in 19 studies, whereas a radioactive isotope
([3-3H]glucose) was
employed in 6 studies. Values for
and
were, respectively,
0.0082 ± 0.0003 min
1 and 9.0 ± 1.2 × 10
4
min
1 · µU
1 · ml.
The mean precision of
and
estimates was 4 and 5%,
respectively. Volume V* was estimated as 1.88 ± 0.06 dl/kg.
Cold vs. Hot Indexes
The results of this study confirm previously observed trends (2, 17,
19): SG is about three times
higher than
(P < 0.001), and
SI is lower than
(P < 0.05). Of note is that these
trends are also present when the indexes are estimated from an
insulin-modified hot IVGTT (unpublished results). Thanks to the larger
data base, it is now possible to assess the degree of correlation
between SG and
and between
SI and
(Fig.
3). Whereas a strong correlation exists
between SI and
(r = 0.84, P < 0.001),
SG and
are uncorrelated
(r = 0.17, P > 0.15).

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Fig. 3.
Correlation between cold and hot indexes of glucose effectiveness
(A) and insulin sensitivity (B). See glossary
for definition of terms.
|
|
Some of the above results are unexpected and suggest the presence of
some model error. SG is higher
than
, in keeping with the
theoretical expectation, but their ratio is too high compared with that
of the analogous clamp-based indexes of cold, SG(clamp), and hot,
SG,d(clamp), glucose effectiveness
(subscript "d" denotes disappearance). In fact, whereas
SG is about three times higher
than
,
SG(clamp) is only 1.5 times higher than SG,d(clamp) (11). Also, the
complete lack of correlation between
SG and
is surprising, because
SG(clamp) and
SG,d(clamp) are presumably well
correlated, given that SG,d(clamp)
is the major determinant (~2/3) of
SG(clamp) (11).
The time courses of cold and hot insulin actions (Fig.
4) also show an unexpected trend. The cold
minimal model assumes that insulin actions on
Rd and EGP have the same timing,
but the time lag between x and
x* (caused by
p2 being lower
than
) violates this
assumption. In addition, the profile of insulin action on EGP,
calculated as the difference x
x*, is physiologically implausible
(17).

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Fig. 4.
Cold and hot insulin action during a standard hot intravenous glucose
tolerance test (IVGTT) in a representative subject.
|
|
Finally, the finding SI <
is unexpected, because
SI, which measures insulin effect
on both Rd and EGP, should be
higher than
, which measures
insulin effect on Rd only. This
incongruity is not present when insulin sensitivity is assessed with
the glucose clamp technique: in Ref. 10
SI(clamp) exceeded
SI,d(clamp) [denoted as
SI,p(clamp) in that paper]
in each subject, with SI(clamp) and SI,d(clamp) being the clamp
version analogous to SI and
, respectively.
The above inconsistencies are symptoms of model error. Two possible
areas of error are the description of glucose and insulin effect on EGP
embodied in the cold model and the single-compartment description of
glucose kinetics (17, 18, 23). In this paper we focus on the latter only.
 |
A TWO-COMPARTMENT MODEL OF THE GLUCOSE SYSTEM DURING THE IVGTT |
To investigate the mechanisms by which single-compartment undermodeling
affects the minimal model indexes, we developed a physiologically based
two-compartment model to describe the glucose system during the IVGTT.
The model, shown in Fig. 5, is described in
detail in APPENDIX A. Briefly, the
model describes both glucose kinetics and EGP during the IVGTT. The
description of glucose kinetics is the same as that of the
two-compartment minimal model proposed in Refs. 14 and 37. It is
assumed that insulin-independent glucose disposal occurs in the
accessible compartment, whereas insulin-dependent glucose disposal
occurs in the nonaccessible compartment. Consistent with known
physiology, insulin-independent glucose uptake accounts for the
inhibitory effect of hyperglycemia on glucose clearance. It consists of
two components, one constant and the other proportional to glucose concentration. Insulin-dependent glucose uptake is parametrically controlled by insulin in a remote insulin compartment. The assumption is made that, in the basal state, insulin-dependent glucose disposal is
three times insulin-independent glucose disposal. EGP is described using the same functional description embodied in the cold minimal model (8, 17, 19, 23), thus allowing us to focus on the bias due to
single-compartment undermodeling only. In fact, EGP inhibition is
assumed to be proportional to the increment of glucose concentration
above basal and to the product of glucose concentration and insulin
action. In addition, as in the minimal model, insulin action on EGP is
assumed to have the same timing as insulin action on glucose uptake.
To ascertain the ability of this model to describe satisfactorily the
glucose system during the IVGTT, we used Monte Carlo simulation
(details in APPENDIX B). Briefly,
the two-compartment model with mean parameters was used to generate
noise-free cold and hot glucose data during a hot IVGTT. The mean
insulin profile of either a standard or an insulin-modified IVGTT was
used as input to the model. Noise of appropriate characteristics was
added to the data, and the noisy IVGTT data sets were then interpreted with the minimal models. We reasoned that, if the two-compartment model
is a realistic representation of the glucose system during the IVGTT,
the minimal model parameters estimated from the simulated data should
be close to those estimated from real data and should exhibit the same
trends discussed above. In addition, the relationships between the
minimal model estimates of glucose effectiveness and insulin
sensitivity and the analogous two-compartment model indexes should be
similar to those observed experimentally between the minimal model and
clamp-based indexes. These hypotheses were all confirmed. Table
1 reports the mean results of the
identification of the two minimal models from simulated IVGTT data. The
values of SG,
SI,
, and
are similar to those reported in
the literature. In particular, SI
is close to the value found by Saad et al. (32) in normal
subjects. This similarity is
noteworthy, because the insulin sensitivity of the two-compartment model has been chosen equal to the one found by Saad et al. in normal
subjects with the clamp technique (see APPENDIX
A). Of note is that all the experimentally observed
inconsistencies between cold and hot parameters are present:
SI is lower than
,
SG is twice
, and hot insulin action is faster than cold because
> p2 (e.g., for the
simulated standard IVGTT,
= 0.069 vs.
p2 = 0.027 min
1).
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Table 1.
Monte Carlo simulation results: cold and hot minimal model indexes
estimated from standard and insulin-modified IVGTT
|
|
How do the minimal model indexes of glucose effectiveness and insulin
sensitivity compare with the "true" indexes of the
two-compartment model? To answer this question we derived indexes of
glucose effectiveness, insulin sensitivity, and basal plasma clearance
rate for the two-compartment model (details are provided in
APPENDIX C). Of note is that these
indexes are expressed in the same units as those of the corresponding
clamp-based indexes. To express also the minimal model indexes in the
same units, SG and
SI were multiplied by V, and
and
were multiplied by V*, in keeping
with the analysis reported in Vicini et al. (37). The values of the two-compartment and minimal model indexes are reported in Table 2. One can see that the cold minimal model
overestimates glucose effectiveness and underestimates insulin
sensitivity, in keeping with the experimental results (24, 32).
V* slightly underestimates basal
glucose clearance and markedly overestimates hot glucose effectiveness,
in keeping with the trend observed in Ref. 37. Specifically,
is virtually identical to the
basal fractional glucose clearance of the two-compartment model (e.g.,
from the standard IVGTT is 0.0102 min
1, and
PCR/VT = 0.0096 min
1). This is consistent
with the results of the
validation study in dogs (19).
V*
slightly underestimates the hot insulin sensitivity of the
two-compartment model, but no studies are available in the literature
comparing the hot minimal model insulin sensitivity with the analogous
clamp-based index.
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Table 2.
Cold and hot glucose effectiveness and insulin sensitivity and basal
plasma clearance rate for 2-compartment model and minimal models
during a standard and an insulin-modified IVGTT
|
|
All in all, these results support the notion that the two-compartment
model is a satisfactory representation of the glucose system during the
IVGTT. We can thus use this model with confidence to analyze the impact
of monocompartmental undermodeling on the cold and hot minimal model
indexes and elucidate their relationships with the analogous
clamp-based measures of glucose effectiveness and insulin sensitivity.
 |
COLD GLUCOSE EFFECTIVENESS |
Effects of Monocompartmental Undermodeling on
SG
To examine the effects of the monocompartmental approximation on
SG, we build on Ref. 18 and, for
the sake of clarity, we outline the reasoning followed in that paper.
Usually, SG is estimated from an
IVGTT in which an insulin response is present and glucose decay depends
on both glucose and insulin. However, the effects of the
monocompartmental approximation on
SG can be more easily determined
if one first analyzes what happens during an IVGTT in which insulin is
maintained at the basal level. Under these conditions, insulin action
is identically equal to zero (Eq. 1), and the minimal model is described by a
first-order linear differential equation
|
(5)
|
Solving Eq. 5 for glucose concentration
and defining
g(t) = g(t)
gb, one has
|
(6)
|
Thus
the minimal model predicts that the decay of glucose concentration
during an IVGTT at basal insulin is monoexponential, with
SG as rate constant. The
fractional decay rate of incremental glucose concentration
(kG,
min
1), namely the
fraction of glucose concentration above basal that declines per unit
time, is constant and equal to SG
|
(7)
|
The true glucose system, however, is not monocompartmental. Using the
two-compartment model presented in the previous section, one can show
(APPENDIX D) that glucose decay
during an IVGTT at basal insulin is described by two exponentials
|
(8)
|
where
1 and
2
(min
1) are the fast and
slow components of glucose decay, respectively
(
1 >
2). Because of the presence of two time constants, the fractional decay rate of incremental glucose
concentration is no longer constant, but time varying
|
(9)
|
In
particular,
kG(t)
is higher at the beginning of the IVGTT, when the fast component of
glucose decay (
1) plays an
important role, and lower at the end of the IVGTT, when only the slow
component (
2) remains in play.
We compared the glucose decay curves and the fractional decay rates of
incremental glucose concentration predicted by the two-compartment and
the minimal models, using for the two-compartment model the parameters
of Table A1, and for the minimal model the SG and V values reported in Table
2. Figure 6 shows the glucose decay curves
(A) and the fractional decay rates
of incremental glucose concentration
(B) predicted by the two models. The
monoexponential decay curve predicted by the minimal model and the
two-exponential profile generated by the two-compartment model are
almost superimposable in the period of minutes
10-20 of the IVGTT but diverge thereafter, thus
reproducing closely the experimental observations by Quon et al. (30).
Of note is that the value of SG
(0.021 min
1) lies between
the values that
kG takes on
between 10 and 20 min [e.g.,
kG
(minute 15) = 0.023 min
1]. These results
suggest that the validity of SG as
descriptor of the effect of glucose per se is confined to the initial
portion of the IVGTT. The local validity of
SG is probably related to the fact
that, during an IVGTT with a normal insulin response, SG estimation critically depends
on the glucose data collected in the early portion of the IVGTT, when
glucose concentration is high over the baseline and insulin action,
albeit increasing, is still low (20). Because in that part of the test
both components of glucose decay are active,
SG not only reflects glucose
effects on Rd and EGP but also the
rapid exchange of glucose between the accessible and the nonaccessible
compartments occurring in the early part of the test.

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Fig. 6.
Comparison between the two-compartment and the cold minimal model
predictions of glucose concentration
(A) and fractional decay rate of
incremental glucose concentration
(kG,
B) during an IVGTT at basal insulin.
The value of SG estimated during
an IVGTT with a normal insulin response is close to the value of
kG in the initial
portion of the test, approximately between 10 and 20 min.
|
|
Validation of SG
Validation of SG entails its
comparison with the analogous index measured with the glucose clamp
method, SG(clamp). In comparing SG with SG(clamp), one
is faced with the problem that such indexes have different units:
SG is expressed in
min
1, whereas
SG(clamp) is expressed in
ml · kg
1 · min
1.
As previously suggested in Ref. 17, to convert them to a common unit
one has to multiply SG by the
minimal model volume of glucose distribution, V. The correctness of
this approach has been formally demonstrated in Refs. 16 and 37. Of
note is that V emerges from the minimal model method and can be
individualized in each subject. In addition, multiplication of
SG by V parallels the approach
used in the validation studies of
SI (10, 32).
The value of SGV found
experimentally in the present study (4.2 ml · kg
1 · min
1)
is much higher than the value of
SG(clamp) in normal subjects that
can be found in the literature (2.4 ml · kg
1 · min
1
in Ref. 11). The same trend is observed if the value of
SGV obtained from our Monte Carlo
study is compared with the glucose effectiveness index of the
two-compartment model (see Table 2). The reason for
SGV being almost twice
SG(clamp) is that
SG and SG(clamp) reflect different
combinations of the fast and slow components of glucose disappearance
at basal insulin,
1 and
2. We have shown that
SG reflects the values that
kG takes on
between 10 and 20 min. Thus, from Eq. 9, one has
|
(10)
|
where
t0 is ~15 min
in subjects with a normal insulin response. Equation 10 shows that SG
is influenced by both the fast and slow components of glucose
disappearance. To compare quantitatively SGV with
SG(clamp), it is useful to express
SG as a function of
2 only. By exploiting the fact
that
A1
1e
1t0
2A2
2e
2t0
and
A2e
2t0
6A1e
1t0
(the values of
A1,
A2,
1, and
2 reported in Table D1), one
has
|
(11)
|
SG(clamp)
is measured from a hyperglycemic glucose clamp in which somatostatin is
used to suppress the endogenous insulin release, and the baseline
insulin is replaced by an exogenous insulin infusion (11). By applying
the formal definition of glucose effectiveness reported in
Eq. C1 to a hyperglycemic clamp at
basal insulin, one finds that
SG(clamp) is defined as the ratio of
(Rd
EGP) to the
increment in plasma glucose concentration at steady state. Given that
in the hyperglycemic steady state the increment in the exogenous
glucose infusion rate,
GINF, equals
(Rd
EGP),
SG(clamp) is defined as follows
|
(12)
|
Using
the two-compartment model to describe the glucose system during the
clamp, one can express SG(clamp)
as a function of the parameters of the model and, specifically, of the
two components of glucose disappearance at basal insulin (see
derivation in APPENDIX E)
|
(13)
|
It is easy to show that
SG(clamp) is primarily determined
by the slow component of glucose disappearance. In fact,
A2/
2
18A1/
1,
and thus SG(clamp)
2/A2.
Moreover, because
1/A2 approximates
the total glucose distribution volume
VT (28), and
VT
1.3 V (see Tables 1 and
A1), we can write
|
(14)
|
By comparing Eqs. 11 and 14, one realizes why
SGV is about twice
SG(clamp). It is worth pointing
out that SGV and
SG(clamp) are not only
quantitatively, but also qualitatively different; SGV also reflects, in addition to
glucose effect on Rd and EGP [measured by
SG(clamp)], the exchange
process taking place between the two glucose compartments in the early
part of the IVGTT. As a consequence, the correlation between these two
indexes is unlikely to be strong, as suggested by the simulation
studies reported in Refs. 22 and 38.
Finegood and Tzur have compared SG
with SG(clamp) in dogs (24). To
allow the comparison, the authors divided
SG(clamp) by the total volume of
glucose distribution, VT, taken
from the literature (250 ml/kg). They found that
SG was higher than the ratio of
SG(clamp) to
VT and that such indexes were
poorly correlated. These findings seem to support the notion that
SG and
SG(clamp) reflect different aspects of glucose effect per se. However, as we pointed out in Ref.
16, using VT to convert
SG(clamp) to the same units as SG is questionable because, as we
have discussed, the minimal model yields an index of glucose
effectiveness, SGV, that has the
same units of SG(clamp) and hinges
on a volume that, in contrast to a mean value of
VT, can be individualized in each subject.
SG from an IVGTT at Basal Insulin
It is commonly believed that SG
estimated from an IVGTT at basal insulin is a reliable measure of
glucose effectiveness, because under such conditions glucose is the
only determinant of glucose decay. However, even under these optimized
conditions, the validity of SG is
uncertain because the minimal model forces a monoexponential function
to describe a two-exponential decay. To determine whether SG estimated from an IVGTT at
basal insulin is a valid measure of glucose effectiveness, it is useful
to recognize that, under such experimental conditions, a minimal
model-independent index of glucose effectiveness can be calculated
directly from the area under glucose decay. In fact in Ref. 4 we showed
that whenever insulin concentration is maintained at the basal level
and exogenous glucose forces glucose to increase and return to the
baseline, glucose effectiveness at basal insulin, denoted as
GEb in Ref. 4, is given by the
ratio between the administered amount of glucose and the area under the
curve of the glycemic excursion above baseline [AUC(
g)].
In the case of an IVGTT at basal insulin, with the assumption that
glucose decay follows the two-exponential profile of
Eq. 8,
GEb is given by
|
(15)
|
Note
that the expression of GEb in
Eq. 15 coincides with that of
SG(clamp) in Eq. 13, in keeping with the analysis carried out in Ref. 4
that ascertained the theoretical equivalence of these two measurements
of glucose effectiveness. In that study (4), insulin was maintained at
the basal level and glucose excursion was similar to that observed
during a meal. Under those circumstances,
SGV resulted in a value similar to
GEb. It is presently unknown
whether this also holds for an IVGTT at basal insulin, because during
such an experiment the glucose profile is less smooth than during a
meal, and the minimal model is unable to account for the rapid fall of
glucose immediately after the glucose bolus. Nevertheless, some
observations can be made. We have seen previously that, during an IVGTT
with a normal insulin response, glucose decay reflects both glucose
effectiveness and insulin action, and
SG is mainly estimated from the
glucose data collected in the initial part of the IVGTT, when insulin
action is still low. During an IVGTT at basal insulin, insulin action
is null throughout the test, and glucose decay is governed by glucose effectiveness only. As a result, all of the glucose data between 10 min
and the end of the test contribute to
SG estimation. Because the
contribution of the fast component of glucose disappearance,
1, soon becomes negligible
(e.g., after ~30 min in normal subjects), and most of the glucose
data are beyond that point in time,
SG will approach the slow
component of glucose disappearance,
2, and the minimal model volume
will approach the reciprocal of
A2. Therefore,
SGV is approximated by
|
(16)
|
Comparison
of Eq. 16 with Eqs.
10 and 11 sheds some
light on the reasons why the value of
SG obtained from an IVGTT at basal insulin has been found to be lower than that obtained from an insulin-modified IVGTT (24): whereas the
SG estimated during an
insulin-modified IVGTT reflects both the fast and slow components of
glucose disappearance, the SG
estimated from an IVGTT at basal insulin reflects primarily the slow
component. Comparison of Eqs. 15 and 16 indicates that, during an IVGTT at
basal insulin, SGV will be close
to GEb if
A2/
2
>>
A1/
1.
Because
A2/
2
18A1/
1, it is likely that SGV estimated
from an IVGTT at basal insulin is a reliable estimate of glucose effectiveness.
SG measured from an IVGTT at basal
insulin has been compared with
SG(clamp) in dogs by Finegood and
Tzur (24). They found similar values for
SG and
SG(clamp) but no correlation
between them. Whereas the agreement between the mean values of the two indexes is consistent with the above analysis, the absence of correlation between them is surprising. In fact, this would mean that
the minimal model is not able to accurately assess glucose effectiveness, even when the IVGTT is performed at basal insulin. As
pointed out in Ref. 16, one possible explanation for this finding is
the relatively narrow range of glucose effectiveness observed in the
group of dogs examined in that study. Another possible explanation is
related to the fact that, at the end of the IVGTT studies carried out
at basal insulin, glucose concentration was below the pretest level and
still declining. This outcome may be due to the difficulty of obtaining
a stable baseline for glucose concentration with the combined
somatostatin, glucagon, and insulin infusion protocol. Alternatively,
it could be the symptom of an inaccurate description of EGP in the
minimal model. In fact, the model assumes that any change in glucose
concentration is accompanied by a proportional and opposite change in
EGP. The time course of EGP during an IVGTT at basal insulin is thus
expected to mirror that of glucose concentration. However, the finding that at the end of the IVGTT glucose concentration was below the pretest level and still declining suggests that EGP was still inhibited
at that time, implying that the minimal model description is not
correct. This model inadequacy may have affected the accuracy of
SG and worsened its concordance
with SG(clamp).
 |
COLD INSULIN SENSITIVITY |
Effects of Monocompartmental Undermodeling on
SI
The monocompartmental approximation also influences the minimal model
estimates of insulin action and sensitivity. As shown in Ref. 18,
because the model has to compensate for
SG overestimation and fit the
glucose data, insulin action is underestimated approximately until
glucose returns to the baseline and is overestimated thereafter. This
bias also affects SI, because this
parameter can be expressed as the ratio between the AUCs of insulin
action and insulin concentration above basal level (18). Here we build
on that paper and analyze the bias affecting the minimal model insulin
action and SI by comparing them
with the insulin action and sensitivity of the two-compartment model.
In carrying out this comparison, one must bear in mind that the cold
model insulin action,
x(t),
represents the sum of the insulin effects on glucose uptake and
production. In the two-compartment model,
xd(t)
is insulin action on glucose uptake, and
xp(t)
is insulin action on production. Thus
X(t) = xp(t)+xd(t)
represents exactly what
x(t)
represents for the minimal model. The profiles of
X(t)
and
x(t)
during a standard IVGTT are compared in Fig.
7A.
X(t)
was generated using the two-compartment model parameters reported in
Table A1;
x(t)
was generated using the mean parameters
SI and
p2 estimated with
the Monte Carlo simulation described in APPENDIX
B (SI = 2.9 × 10
4
min
1 · µU
1 · ml
and p2 = 0.027 min
1). It can be seen
that the minimal model markedly underestimates insulin action during
the first half of the test and slightly overestimates it thereafter. It
must be recognized, however, that this bias originates not only from
the different model order (one vs. two pools) but also from the
different location of insulin action on glucose uptake (accessible vs.
nonaccessible pool). To single out the effect of monocompartmental
undermodeling per se, we calculated in APPENDIX
F the effect that
xd(t) produces on the accessible pool of the two-compartment model. We termed
this effect as
(t).
(t) = xp(t) +
d(t)
is therefore the "accessible-pool equivalent" insulin action of
the two-compartment model that produces the same effect as
X on plasma glucose concentration (i.e., the accessible-pool Rd
remains the same).
(t)
is shown in Fig. 7B plotted against
the insulin action of the minimal model. Qualitatively speaking,
(t)
is a delayed and blunted version of
X(t).

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|
Fig. 7.
A: comparison between insulin action
of the two-compartment and the cold minimal models during a standard
IVGTT. B: insulin action of the
two-compartment model, which, if applied to the accessible pool, would
produce the same effect on glucose concentration as the one taking
place in the nonaccessible pool. This "accessible-pool
equivalent" profile of the two-compartment insulin action is
contrasted with the cold minimal model insulin action.
C: difference between
insulin action profiles of B is the
effect of monocompartmental undermodeling on cold minimal model insulin
action. This difference is shown against its analytically predicted
time course.
|
|
The difference
x(t) =
(t)
x(t)
represents the effect of monocompartmental undermodeling on the minimal
model insulin action.
x(t) can be
analyzed theoretically by assuming that the minimal model fit of IVGTT
glucose concentration is perfect. In this case an analytic expression
for
x(t) can be derived (details in APPENDIX
F) that helps to single out the determinants of the
bias
|
(17)
|
where GE/V1 is the fractional
glucose effectiveness of the two-compartment model,
= k21k12
and k22 = (k02+k12),
and
2(t) is glucose concentration in the nonaccessible pool when
insulin-dependent glucose removal occurs in the accessible pool.
The question arises as to what extent the analytic expression of
x(t)
derived under the assumption of perfect minimal model fit agrees with
the profile of
x(t)
calculated by subtracting the profiles of
x(t)
and displayed in Fig. 7B [note that
x(t)
has been estimated from simulated, but realistic, IVGTT data and thus
reflects a realistic, but not perfect, minimal model fit]. The
two profiles of
x(t)
are compared in Fig. 7C. They agree
closely, indicating that Eq. 17
provides a good description of this difference. The only discrepancy
occurs in the initial portion of the IVGTT. This is not surprising,
because the analytic
x(t)
is calculated in the hypothesis of perfect minimal model fit. In
contrast, it is well-known that the minimal model is unable to fit the
initial rapid decay of glucose and that the glucose data collected in
the first 10 min of the test are not used in model identification. Thus
from 10 min on we can use with confidence Eq. 17 to gain insight into the sources of the bias
affecting the minimal model insulin action. It can be seen that
x depends not only on
SG overestimation of the
fractional glucose effectiveness of the two-compartment model
(GE/V1 = 0.0133 min
1) but also on the
dynamics of glucose in the second glucose compartment. The first
component of
x(t),
i.e., the bias affecting the estimate of glucose effectiveness
(SG
GE/V1), is weighted by the term
g(t)/g(t).
Such a function of time has a shape resembling that of plasma glucose
decay and changes its sign when glucose exhibits an undershoot below
its basal level. The second component, depending on glucose dynamics in
the nonaccessible pool, is the major determinant of
x(t).
In fact,
x(t)
becomes positive at 20 min, i.e., approximately when
2(t)
achieves its maximum, and returns negative at 140 min, just when
2(t)
achieves its minimum. This implies that the bias affecting insulin
action cannot be obviated completely by forcing
SG to assume a more accurate value.
The bias of insulin action obviously affects
SI. The difference between the
fractional insulin sensitivity of the two-compartment model
(IS/V1) and
SI is given by Eq. F7
in APPENDIX F
|
(18)
|
Equation 18 indicates that the bias of
SI depends on the whole time
course of insulin action, so that compensations may occur between
portions of the IVGTT when
x
is positive and portions when
x is negative. Thus
overestimation of SG does not
necessarily imply that SI is
underestimated, as recently pointed out (18, 24). However, because in
our simulation study the AUC of underestimation is much greater than
the AUC of overestimation, SI
underestimates the fractional insulin sensitivity of the
two-compartment model by 55% (see Tables 1, 2, and
A1). When
SI is multiplied by the minimal
model volume, V, to allow comparison with IS (10, 32), underestimation
reduces to 43% because V is higher than V1. Another observation is that
the bias affecting SI may be
scarcely influenced by the value assumed by
SG, especially during a modified IVGTT in normal subjects. In fact, under such experimental conditions and in such a group of people, the integral of
g(t)/g(t)
in Eq. 17 is very small, because the
little but prolonged undershoot of glucose below its basal level due to
the second insulin peak at 20 min gives rise to a negative AUC that
balances the positive area associated with the rapid decline of glucose
during the 1st h of the test. Because the integral of
g(t)/g(t)
measures the effect that a unit change in
SG produces on
SI, it is likely that the
sensitivity of SI to errors in
SG is small.
One may wonder whether the underestimation of
SI may be mitigated by modifying
the insulin profile during the IVGTT. This could happen, because
insulin dynamics during the IVGTT affects both the numerator and the
denominator of Eq. 18. The impact on the denominator is obvious; that on the numerator is due to the fact
that the insulin profile influences the time course of glucose concentration in both the glucose pools, thus producing an effect on
AUC[
x(t)]
as well (see Eq. 17). The Monte
Carlo simulation results of Table 1 suggest that the modified IVGTT
slightly mitigates the bias of SI.
In fact, SIV underestimation with
respect to IS reduces to 37%. Thus the modified IVGTT favors not only
a greater precision (40) but also a greater accuracy of
SI. In our simulation the
improvement of SI accuracy was
primarily due to the increment in
AUC[i(t)
ib]. In fact, although the
added burst of insulin steepened the glucose curve, thus making
x(t)
less sluggish than during the standard IVGTT,
AUC[
x(t)]
did not change much.
One final remark concerns the effect of monocompartmental undermodeling
on the estimation of SI in
non-insulin-dependent diabetes mellitus (NIDDM). Many reports have
shown that SI estimated from an
insulin-modified IVGTT in NIDDM patients is often imprecise and poorly
correlated with the index calculated with the glucose clamp (3, 32).
These problems can be interpreted, at least in part, in the light of
the above-mentioned effects of single-compartment undermodeling on
insulin action and SI. The true
insulin action in NIDDM patients is presumably very low because this
group is markedly resistant to insulin. The error due to
monocompartmental undermodeling can cause the minimal model insulin
action to become so low and slow as to degrade the precision of
SI. In addition, even when
SI can be precisely estimated, its
value will be markedly underestimated. Underestimation of
SI will further narrow the range
of the minimal model estimates of insulin sensitivity in this group,
thus worsening the correlation with the clamp-based measure of insulin sensitivity.
Validation of SI
Validation of SI entails its
comparison with SI(clamp), i.e.,
the glucose clamp measure of insulin sensitivity. By applying the
formal definition of insulin sensitivity reported in
Eq. C3 to a euglycemic,
hyperinsulinemic clamp, one finds that
SI(clamp) is the steady-state
ratio of
(Rd
EGP) to
the increment in plasma insulin concentration, normalized to the
ambient plasma glucose concentration at which the clamp is performed.
Because in the hyperinsulinemic steady state the increment in the
exogenous glucose infusion rate equals
(Rd
EGP),
SI(clamp) is defined as follows (7, 10, 32)
|
(19)
|
To
compare SI with
SI(clamp),
SI is usually multiplied by the
glucose distribution volume of the minimal model, V (10, 32).
SIV and
SI(clamp) are well correlated,
especially in normotolerant subjects, when the IVGTT is modified with
the injection of either tolbutamide or insulin (5, 10, 32). Are they
also equivalent measures of insulin sensitivity, i.e., is their
regression line indistinguishable from the unity line (slope = 1, intercept = 0)? Equivalence between
SIV and
SI(clamp) is controversial in the literature.
In the study by Beard et al. (5) in normotolerant subjects,
SI was measured with the
tolbutamide-modified IVGTT, and
SI(clamp) was measured with
sequential low-insulin-dose euglycemic clamps, bringing insulin to
plateaus of 21 and 35 µU/ml. A strong correlation between
SI and
SI(clamp) was found
(r = 0.84). However, if one calculates
the product SIV [using for V
a typical value of 170 ml/kg (10, 17)], one finds that
SIV was approximately 60% lower
than SI(clamp) (0.11 vs. 0.29 ml · kg
1 · min
1 · µU
1 · ml).
Bergman et al. (10) compared SIV
and SI(clamp) in a group of
normotolerant and obese subjects.
SIV was measured with the tolbutamide-modified IVGTT, and
SI(clamp) was measured with low- and high-insulin-dose euglycemic clamps carried out on different days.
Insulin levels were 41 and 114 µU/ml, thus higher than in Ref. 5.
Although SIV was slightly lower
than SI(clamp) in all subjects but
one, SIV was
equivalent to SI(clamp) (0.046 vs.
0.052 dl · min
1 · µU
1 · ml, respectively).
Saad et al. (32) compared SIV with
SI(clamp) in normal controls,
subjects with impaired glucose tolerance (IGT), and patients with
NIDDM. SI was measured with the
insulin-modified IVGTT (insulin bolus of 0.03 U/kg at 20 min), while
SI(clamp) was measured with a
single high-insulin-dose euglycemic clamp, bringing insulin levels to
83 µU/ml. Although results show
SIV and
SI(clamp) well correlated in
normal and IGT subjects, SIV was
>50% lower than SI(clamp).
More recently, Saad et al. (33) measured
SIV in normal controls with both
the tolbutamide- and the insulin-modified IVGTT, while
SI(clamp) was measured with the
same insulin infusion used in Ref. 32.
SIV from the tolbutamide-modified
IVGTT was only 13% lower than
SI(clamp) (0.045 vs. 0.054 dl · min
1 · µU
1 · ml),
whereas SIV from the
insulin-modified IVGTT (0.030 dl · min
1 · µU
1 · ml)
was 44% lower than SI(clamp).
Comparison of the results obtained in the above-mentioned studies
suggests that equivalence of SIV
and SI(clamp) depends on how the
IVGTT and the clamp are performed. For instance, the marked difference
between SIV and
SI(clamp) that is present when
either SIV is estimated from an
insulin-modified IVGTT (32) or
SI(clamp) is estimated from
low-dose insulin clamps (5) vanishes when SIV is estimated from a
tolbutamide-boosted IVGTT and
SI(clamp) is derived from
high-dose insulin clamps (10, 33). To understand the reasons for this
protocol dependency, it is useful to examine the hypotheses governing
the assessment of SI and
SI(clamp). For SI to be equivalent to
SI(clamp), a number of conditions
must be met, the most important of which are that
1) the minimal model single-pool
description of glucose kinetics is adequate;
2) insulin effect on the aggregation
of Rd and EGP increases linearly
with insulin concentration across the insulin range experienced during the IVGTT and the clamp; 3) insulin
sensitivity is independent from the route of insulin delivery (portal
vs. peripheral); and 4) tolbutamide
has no effects per se on glucose metabolism. We have already shown that
monocompartmental undermodeling leads to
SI underestimation. We now analyze
how the other factors can influence the estimation of
SI and
SI(clamp).
Linearity of insulin effect.
Both SI and
SI(clamp) measure the ability of
insulin not only to increase Rd
but also to inhibit EGP. Therefore, both such aspects of insulin action
are important in determining overall insulin sensitivity. In glucose
clamp studies, insulin levels are brought to ~40 and 100 µU/ml
during either low- or high-dose insulin clamps. Whereas the
steady-state relationship between Rd and insulin concentration is
approximately linear in the physiological range (10-100 µU/ml)
(7), the relationship between EGP and insulin concentration is highly
nonlinear in the same range, because EGP achieves nearly complete
suppression at insulin levels of ~40 µU/ml (26), so that any
further increment in insulin concentration is not accompanied by a
proportional decrement in EGP. Evidence that the nonlinearity of the
relationship between insulin concentration and EGP suppression is
likely to affect the measurement of
SI(clamp) can be derived from the
study of Katz et al. (26). The data reported in that dose-response
study allow calculation of
SI(clamp) at three insulin levels:
25, 43, and 123 µU/ml. SI(clamp)
shows results of 0.19, 0.21, and 0.10 ml · kg
1 · min
1 · µU
1 · ml,
respectively. These values indicate that
SI(clamp) is independent of the
insulin level until EGP reaches nearly complete suppression at ~40
µU/ml. As the insulin level increases beyond that point, any further
increase in insulin action will depend solely on an increase in
Rd. That
SI(clamp) depends on the insulin
level at which the clamp is performed can also be inferred by comparing the values of SI(clamp) in normal
subjects obtained in the studies by Beard et al. (5) and Saad et al.
(32). SI(clamp) was 0.29 ml · kg
1 · min
1 · µU
1 · ml
in Ref. 5, in which insulin levels were 21 and 35 µU/ml, but 0.10 ml · kg
1 · min
1 · µU
1 · ml
in Ref. 32, in which insulin level was 83 µU/ml. All in all, these
data suggest that when SI(clamp)
is derived from a high-dose insulin clamp, it will tend to
underestimate insulin effect on EGP.
During an IVGTT, because of the dynamic nature of the test, what really
matters is the insulin level attained in the remote insulin compartment
from which insulin action is exerted. During a standard or a
tolbutamide-boosted IVGTT, plasma insulin levels rarely exceed 200 µU/ml, and insulin action is likely to remain within the quasi-linear
range. The risk of entering into the nonlinear range of insulin action
increases during an insulin-modified IVGTT in which peak insulin levels
as high as 400-600 µU/ml are elicited by a bolus or a short
infusion of exogenous insulin. As a matter of fact, a recent report by
Vicini et al. (39) suggests that saturation of insulin effect on EGP is
likely to take place during an insulin-modified IVGTT. In that study
(39), the time course of EGP during an insulin-modified IVGTT was
accurately assessed by using the tracer-to-tracee (specific activity)
clamp. EGP achieved almost complete suppression at 20 min and remained
suppressed for another 20 min after the exogenous insulin
administration. We speculate that, because between 20 and 40 min the
high insulin levels due to the exogenous insulin injection cannot
produce any further inhibition in EGP, insulin effect on EGP is
underestimated in that interval. It seems, however, that this transient
saturation of insulin effect on EGP (and possibly on
Rd) is unable to influence SI appreciably (comprising both
the effects of insulin on Rd and EGP). In fact, recent results provided by Saad et al. (34), contradicting a previous report by Prigeon et al. (29), indicate that
the estimate of SI is not
appreciably influenced by the peak insulin level achieved during an
insulin-modified IVGTT in which insulin is administered either as a
bolus or as an equidose short infusion.
Portal vs. peripheral route of insulin delivery.
In comparing SI with
SI(clamp), we assume that the
peripheral and portal routes of insulin delivery are equally effective in inhibiting EGP. Unlike
SI(clamp), which measures insulin
ability to suppress EGP and elevate
Rd in response to peripheral
insulin delivery, SI is estimated
from insulin and glucose profiles in response to either portally
delivered insulin (during a standard or a tolbutamide-modified IVGTT)
or to a mixture of portally and peripherally appearing insulin (during
an insulin-modified IVGTT). Recently, Steil et al. (35) studied the
contribution of portal insulin to the assessment of
SI by performing paired
insulin-modified IVGTTs in dogs in which insulin was infused either
portally or peripherally with matched peripheral insulin levels. They
found that portal insulin delivery does not significantly affect
insulin's ability to normalize plasma glucose after the glucose bolus
and that the route of insulin delivery does not appreciably affect SI.
Effects of tolbutamide.
Saad and colleagues (33, 34) have shown that the tolbutamide-boosted
protocol provides higher SI
estimates than the insulin-modified protocol, regardless of the method
of insulin administration (bolus or 5- or 10-min infusion). The higher
SI values from the tolbutamide protocol cannot be explained by differences in peripheral insulinemia, because giving insulin as a 10-min infusion results in peripheral insulin levels similar to those measured after tolbutamide (34). In
addition, the aforementioned study by Steil et al. (35) seems to rule
out the possibility that this difference is due to the effect of higher
portal concentrations seen after tolbutamide but not insulin injection.
Saad et al. (34) hypothesized that tolbutamide-induced proinsulin
release during the IVGTT could play a role in elevating the estimates
of SI with the tolbutamide protocol with respect to those obtained with the insulin protocol. Alternatively, differences in SI
between tolbutamide- and insulin-modified IVGTTs could be due to some
extrapancreatic effect of tolbutamide, as recently suggested (31).
Whatever the case, the injection of tolbutamide contributes to elevate
SI with respect to
SI(clamp).
 |
HOT GLUCOSE EFFECTIVENESS |
Effect of Monocompartmental Undermodeling on
To study the effect of the monocompartmental approximation on
, we will use the same rationale
previously used for SG, i.e., we
will first analyze the decay of hot glucose during a hot IVGTT in which
insulin is maintained at the basal level. The minimal model predicts
that the decay of hot glucose concentration is monoexponential, with
as rate constant
|
(20)
|
Thus
the fractional decay rate of the tracer will be constant and equal to
|
(21)
|
In contrast, the two-compartment model predicts that the hot
glucose decay is described by an almost two-exponential profile (APPENDIX D)
|
(22)
|
where
and
(
>
,
min
1) are the fast and
slow rate constants of glucose kinetics in the basal state,
respectively, and the term
(t)
accounts for the effect of hyperglycemia on glucose clearance. The
corresponding fractional decay rate is no longer constant, and its time
course is shown in Fig. 8.
is high at the
beginning of the IVGTT, when the fast component of glucose kinetics
plays a relevant role, shows a rapid decline followed by a slight
undershoot due to
(t), and then
increases slowly, getting closer and closer to
.

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Fig. 8.
Fractional decay rate of hot glucose
( ) during a hot IVGTT
at basal insulin. The value of
estimated during a hot IVGTT with a normal insulin response is close to
the value of in the
final portion of the test.
|
|
During a hot IVGTT,
is primarily
estimated in the final portion of the test, when both insulin and
glucose concentrations have almost returned to the baseline. At that
point in time, both the contribution of the fast component of glucose
kinetics and the inhibitory effect of hyperglycemia on glucose
clearance have become negligible, and hot glucose decay is governed
only by the slow component of glucose kinetics,
. Thus, in the final part of the
IVGTT, a single-pool description of glucose kinetics is adequate to
describe hot glucose decay, and
approximates
. As a matter of
fact, by comparing Tables 1 and D1, one can see that the value of
estimated with the Monte Carlo
simulation is close to
(
= 0.0102 and 0.0098 min
1 from the
standard and modified IVGTT, respectively, and
= 0.0093 min
1). This agreement
suggests that the domain of validity of
is confined to the final portion
of the IVGTT. To clarify the physiological meaning of
, it is of interest to recall that
the basal plasma clearance rate of glucose,
PCRb, is the inverse of the area
under the hot glucose impulse response at basal insulin. The area can
be expressed as a function of the eigenvalues of the two-exponential
impulse response, as follows (13)
|
(23)
|
Given
that
/
~ 14
/
(see Table D1), Eq. 23 reduces to
|
(24)
|
Because
approximates
and
1/
approximates
VT (28), one has
|
(25)
|
Therefore,
measures the ratio of glucose
clearance and total distribution volume, i.e., basal fractional glucose clearance rate. In fact, if the
found experimentally in this study (0.0082 min
1) is multiplied by a
total volume of glucose distribution taken from the literature (260 ml/kg), one obtains 2.03 ml · kg
1 · min
1,
which agrees with the values of the basal plasma clearance rate found
in the literature. The same trend is observed if
estimated from the Monte Carlo
study is compared with the ratio between the plasma clearance rate and
the total distribution volume of the two-compartment model
(
= 0.0102 and 0.0098 min
1 from the standard and
modified IVGTT, respectively, and
PCR/VT = 0.0096 min
1). The reliability of
as a descriptor of the basal
fractional glucose clearance has been assessed in dogs by comparing
with the basal glucose clearance
rate independently measured by the arteriovenous technique (19).
Validation of
In the hot minimal model, it is assumed that glucose uptake is
proportional to glucose concentration. As a result, glucose clearance
and glucose effectiveness on glucose disposal coincide in the model
(see Table C1). However, it is a well-established notion that glucose
uptake is not proportional to glucose concentration and that, in the
range of interest, the Rd vs. g
relationship can be approximated by a line that has a nonzero intercept
with the Rd axis (6, 11, 12, 36).
As a consequence, glucose clearance and glucose effectiveness on
glucose disposal do not coincide. In this section we clarify the
relationship existing between
and
the clamp estimate of glucose effectiveness on glucose disposal,
SG,d(clamp).
SG,d(clamp) is measured from hyperglycemic glucose clamp studies at basal insulin in which exogenous
glucose is used to progressively increase glucose concentration, g, at
various steady state levels, and a tracer is concurrently infused to
measure Rd (12). By applying the
definition of hot glucose effectiveness reported in
Eq. C2 to a hyperglycemic clamp at
basal insulin, one finds that
SG,d(clamp) is the slope of the linear relationship between the Rd
and g
|
(26)
|
where
Rd,0 is the nonzero intercept.
Recalling that
measures
fractional basal glucose clearance (Eq. 25), and using the definition of basal glucose
clearance (Eq. C5), one has
|
(27)
|
Combining
Eqs. 26 and 27 yields
|
(28)
|
Equation 28 confirms that
does not coincide with SG,d(clamp)
(apart from the volume factor) because of the presence of the nonzero
intercept Rd,0. Thus
cannot be used as an index of
glucose effectiveness unless the presence of
Rd,0 is explicitly taken into
account in the hot minimal model. In a recent paper investigating
glucose effectiveness during a meal-like study (4), the hot minimal
model was modified to allow for the presence of
Rd,0. Both
and
Rd,0 were estimated from the data,
because the basal plasma clearance rate was available in each subject,
thanks to a pretest tracer equilibration experiment. It is worth noting
that this pretest tracer experiment is not commonly performed before
the IVGTT, and this makes the simultaneous estimation of
and
Rd,0 from hot IVGTT data extremely
difficult (17).
SG vs.
The analysis of the relationships
SG vs.
SG(clamp) and
vs.
SG,d(clamp) suggests a possible
explanation for the lack of correlation between
SG and
found in this study. One would
expect to find a good correlation between
SG and
because glucose clamp studies
have shown that glucose effectiveness on
Rd is the major determinant
(~2/3) of overall glucose effectiveness, with the remainder accounted
for by the effect of glucose to inhibit EGP (11). However, as we have
shown, SG and
are not equivalent to
SG(clamp) and
SG,d(clamp), respectively. In
fact, SG and
SG(clamp) measure related, but not
identical, physiological processes, because
SG, at variance with
SG(clamp), is markedly influenced
by the rapid exchange of glucose that takes place between the
accessible and nonaccessible compartments after the glucose bolus.
Likewise,
measures the fractional
basal glucose clearance rate, but not glucose effectiveness on
Rd, because the hot model does not
account for the the inhibitory effect of glucose on its own clearance.
 |
HOT INSULIN SENSITIVITY |
Effects of Monocompartmental Undermodeling on
The estimation of
suffers from
problems that, to some extent, are opposite to those affecting
SI. Because
approximates the slow time
constant of glucose kinetics at basal insulin, the minimal model tends
to underestimate the rate of hot glucose decay per se (independently of
insulin) during the early portion of the IVGTT, when the fast component
of glucose kinetics plays an important role. To compensate for this
underestimation and to fit hot glucose data, the hot minimal model
insulin action is probably overestimated in the initial part of the
IVGTT. To verify this hypothesis, we compared the insulin action on
glucose disposal of the two-compartment model,
xd(t),
with the hot minimal model insulin action,
x*(t)
(Fig.
9A). The
latter profile is generated using the mean parameters
and
estimated by Monte
Carlo simulation (
= 3.2 × 10
4
dl · kg
1 · min
1 · µU
1 · ml
and
= 0.069 min
1). As argued in
comments concerning cold insulin action, the difference between these
profiles does not reflect only the different model order but also the
different location of insulin action in the two models. Thus, to single
out the effect of monocompartmental undermodeling per se, we calculated
in APPENDIX F the effect that
xd(t)
produces on hot glucose concentration in the accessible pool of the
two-compartment model. This "accessible-pool equivalent" profile
of insulin action on glucose disposal,
(t) (the asterisk denotes that it is derived from tracer data), is compared
with
x*(t)
in Fig. 9B. One can see that the hot
minimal model overestimates insulin action until ~30 min and
underestimates it thereafter. The difference
=
x* represents the effect of
monocompartmental undermodeling on the insulin action of the hot
minimal model.
can
be given the following analytic expression in the hypothesis that the
minimal model fit of hot glucose data is perfect
(APPENDIX F)
|
(29)
|
where GE*/V1 is the fractional
hot glucose effectiveness of the two-compartment model;
*2 is hot glucose concentration in the
nonaccessible glucose pool when insulin-dependent glucose removal
occurs in the accessible pool.
obtained from
Eq. 29 is plotted in Fig.
9C against the profile of
obtained from the
difference between the profiles of
and
x* displayed in Fig.
9B. The two profiles agree closely
except in the initial part of the test, because the hot model is unable to fit the initial rapid decay of hot glucose that follows the glucose
bolus. The structure of
resembles that of
x
(Eq. 17). In fact,
depends on the difference between the fractional glucose effectiveness of the two models and on
the dynamics of hot glucose concentration in the nonaccessible pool;
also shows a term proportional
to Rd,0 that accounts for the fact
that the hot minimal model does not comprise a description of the
inhibitory effect of glucose on its own clearance.

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|
Fig. 9.
A: comparison between insulin action
on glucose disposal of two-compartment model and insulin action of hot
minimal model during a standard hot IVGTT.
B: insulin action of two-compartment
model, which, if applied to the accessible pool, would produce the same
effect on hot glucose concentration as the one taking place in the
nonaccessible pool. This "accessible-pool equivalent" profile of
two-compartment insulin action is contrasted with the hot minimal model
insulin action. C: difference between
insulin action profiles of B is the
effect of monocompartmental undermodeling on hot minimal model insulin
action. This difference is shown against its analytically predicted
time course.
|
|
The bias affecting
can be
calculated as previously done for
SI
|
(30)
|
As
for SI, the bias of
SI depends on the whole time
course of insulin action, so that compensations may occur between
portions of the IVGTT when
*x is
positive and others when
*x is negative. It can
be calculated from Table 1 that
underestimates the fractional hot insulin sensitivity of the
two-compartment model by 36%. When
is multiplied by the hot minimal
model volume, V*, to allow comparison with IS*, underestimation reduces to ~15%.
underestimation can
be mitigated by enhancing the insulin signal during the hot IVGTT. In
fact, the Monte Carlo simulation results of Table 1 indicate that
V* calculated from an
insulin-modified IVGTT is almost identical to IS*.
One final remark concerns the ability of the hot minimal model to
overcome the problems encountered with the cold minimal model in
assessing insulin sensitivity in NIDDM patients. In Ref. 3 we have
shown and explained why
can be
precisely estimated even in those NIDDM patients in whom
SI cannot be estimated or is
estimated with poor precision.
Validation of
has not yet been validated
against the analogous clamp-based index
SI,d(clamp). However,
has been compared with the
insulin sensitivity index yielded by the two-compartment minimal model
developed in Ref. 14. Such a model, which explicitly accounts for
glucose inhibition of glucose clearance, yields an insulin sensitivity
index,
, that is expressed in the
same units as SI,d(clamp)(37).
V* has been found to be similar
and well correlated to
(37). Although this agreement does not prove
accuracy, it suggests that
monocompartmental undermodeling and the inadequate description of
glucose effect on its own clearance do not bias
V* appreciably, presumably because
of a reasonably good compensation between the initial overestimation
and the subsequent underestimation of hot insulin action.
SI vs.
The error analysis of cold and hot insulin action and sensitivity
explains the paradoxical finding
SI <
. In fact, whereas the
underestimation affecting SI is
large, that affecting
is modest.
Also, the unexpected time lag between cold and hot insulin action may
be due, at least in part, to a different effect of the monocompartmental approximation on the insulin action of the two models. By comparing Figs. 7B and
9B, one can see that
x is delayed with respect to the
insulin action of the two-compartment model, whereas
x* is anticipated. This would provide
a possible explanation of why parameter
p2 of the cold
minimal model, which governs the speed of rise and decay of
x, is systematically lower than the analogous hot parameter
.
 |
CONCLUSIONS |
The findings of the present paper can be summarized as follows.
SG reflects the rate of glucose
decay per se, independent of increased insulin, in the initial portion
of the IVGTT, approximately between 10 and 20 min. It reflects not only
the ability of glucose to promote
Rd and inhibit EGP but also the
rapid exchange of glucose that occurs between the two glucose
compartments after the glucose bolus. Because the latter component of
SG is nonnegligible, as suggested
by the simulation studies in Refs. 22 and 38, the reliability of
SG as descriptor of glucose
effectiveness is uncertain.
The effects of single-compartment undermodeling on
SG determine undesired
compensations on cold insulin action and
SI. Insulin action is markedly
underestimated for a considerable portion of the IVGTT.
SI, which is proportional to the
integral of insulin action, is also markedly underestimated. Although
the SI estimated from an
insulin-modified or a tolbutamide-boosted IVGTT strongly correlates
with SI(clamp), it must be pointed
out that both SI and
SI(clamp) are protocol dependent
and, in general, are not equivalent. In fact, besides the description
of glucose kinetics (one vs. two compartments), other factors like
nonlinearity of insulin action on EGP and effects of tolbutamide can
influence their estimation.
cannot be used as a descriptor of
glucose effectiveness on glucose disposal unless an explicit
description of the inhibitory effect of glucose on its own clearance is
included in the hot model. Nevertheless,
has a clear-cut physiological interpretation, because it measures basal fractional glucose clearance.
Hot insulin action is influenced by monocompartmental undermodeling as
well as by the hot model assumption that glucose has no effect on its
own clearance. Hot insulin action is markedly overestimated in the
initial portion of the IVGTT and is underestimated thereafter.
, which is proportional to the
integral of hot insulin action, is only slightly underestimated, thus
giving results more accurate than
SI. Although
has not yet been compared with
the analogous clamp-based estimate of peripheral insulin sensitivity,
it is well correlated to the estimate of insulin sensitivity provided
by a physiological two-compartment minimal model (37).
The cold indexes SG and
SI, but not the hot indexes
and
, suffer from an additional
problem: their accuracy depends not only on the description of glucose
kinetics (one vs. two compartments) but also on the reliability of the
description of glucose and insulin control on EGP embodied in the cold
model. This issue has not been examined in this article. Recently, the tracer-to-tracee (specific activity) clamp has been used to estimate EGP in a model-independent fashion during an IVGTT (39). The resulting
profile is not in agreement with the assumptions of the minimal model.
However, it is difficult to single out the extent to which
SG and
SI are affected by this problem
until a more reliable description of glucose and insulin control on EGP becomes available.
A two-compartment structure is the obvious way to go to anticipate and
prevent monocompartmental undermodeling. As far as the hot minimal
model is concerned, a two-compartment hot minimal model has been
proposed that not only allows estimation of EGP by deconvolution (14,
39) but also yields metabolic indexes of glucose effectiveness, insulin
sensitivity, and glucose clearance (37). The relationships between the
indexes provided by the two-compartment and the single-compartment hot
minimal models have been thoroughly examined in Ref. 37. The
formulation of a cold two-compartment minimal model is far more
difficult, because one is faced with a priori identifiability problems.
Preliminary results (15) indicate that a two-compartment model can be
resolved from cold IVGTT data if the available knowledge on the
exchange kinetics between the accessible and nonaccessible glucose
pools is incorporated in the model by a Bayesian approach. A possible alternative to mitigate the impact of monocompartmental undermodeling is to design experimental protocols that, at variance with the IVGTT,
are characterized by more physiological glucose and insulin profiles,
i.e., they are smoother than those observed during the IVGTT. This
strategy has been pursued with success in Ref. 4, where the single-pool
minimal models have interpreted cold and hot glucose data during an
experiment in which insulin remained basal and glucose exhibited a
prandial profile. Further studies are warranted to more fully explore
both of these approaches.
 |
APPENDIX A. TWO-COMPARTMENT SIMULATION MODEL OF THE IVGTT |
Here we describe the two-compartment model used for simulating cold and
hot glucose data during the hot IVGTT. Because cold glucose
concentration is the result of the balance between
Rd and EGP, both of these
processes are described by the model. To describe
Rd, we use a two-compartment
model, which has been shown to provide a physiological description of
glucose kinetics during the IVGTT (14, 37, 39). This model, shown in
Fig. 5, has already been described in Refs. 14, 37, and 39; it builds on the two-compartment structure extensively analyzed in Ref. 21.
Briefly, the accessible pool comprehends tissues that are in rapid
equilibrium with plasma, like red blood cells, central nervous system,
kidneys, and liver. These tissues consume glucose largely in an
insulin-dependent way. The second pool comprehends tissues that
equilibrate more slowly, with plasma, like muscle and fat. These
tissues are mainly insulin dependent. This is the reason why in the
model insulin-independent glucose disposal is assumed to take place in
the accessible pool (pool 1),
whereas insulin-dependent glucose disposal is assumed to occur in the nonaccessible pool (pool 2).
Insulin-independent glucose uptake has two components, one constant and
the other proportional to glucose concentration. Thus the fractional
disappearance rate of the accessible pool is
|
(A1)
|
where
kd accounts for
the proportional term, and Rd,0 is
the nonzero intercept of the steady-state relationship
Rd vs. g. Because
k01 decreases as
glucose concentration increases, the model accounts for the well-known
inhibitory effect of hyperglycemia on its own clearance (11, 12, 36).
Insulin-dependent glucose disposal is described by a parametric control
on k02
|
(A2)
|
where
xd(t)
is insulin action on glucose uptake originating from an insulin
compartment remote from plasma. The dynamics of
xd(t)
is governed by parameters
kbd and
ka, which
describe the transport of plasma insulin into the compartment and the
removal of remote insulin from the compartment, respectively.
Physiological knowledge indicates that, in the basal state, one-fourth
of glucose uptake is due to insulin-dependent glucose tissues and
three-fourths to insulin-independent glucose tissues (21). This yields
the following relationship among the parameters of the model
|
(A3)
|
The model description of EGP during the IVGTT employs the same
functional description embodied in the cold minimal model (8, 17, 19,
23). In fact, inhibition of EGP is assumed to be proportional to the
glucose excursion above basal and to the product of glucose
concentration and insulin action on EGP, xp(t),
assumed to occur with the same timing as insulin action on
Rd. This is the reason why the
remote insulin compartments from which
xp(t)
and
xd(t)
originate have the same rate constant, ka. In contrast,
parameters kbd
and kbp are
different, with
kbd being higher
than kbp, because
it is known from clamp studies that insulin effect is greater on
Rd than on EGP.
The model equations for cold and hot glucose are
|
(A4)
|
where
qi(t)
and
(t)
(i = 1, 2) are, respectively, the cold
and hot glucose masses in the ith
compartment of the model.
The values of the model parameters are reported in Table
A1. The values of
V1,
k21,
k12, and
k02 were taken
from Ref. 21; Rd,0 was
chosen to be equal to the value experimentally determined in Ref.
12; kd was
determined from Eq. A3;
kp
was calculated with the assumption that glucose effectiveness on
EGP is one-third of overall glucose effectiveness (11); parameter
ka was
taken from Ref. 14; parameters
kbd and
kbp were
chosen in such a way that the two-compartment model values of
insulin sensitivity related to
Rd + EGP and to
Rd only were equal to the
analogous clampbased values reported by Saad et al. (32) in
normal subjects.
 |
APPENDIX B. MONTE CARLO SIMULATION OF A HOT IVGTT |
To obtain minimal model indexes that could be compared with the
corresponding indexes of the two-compartment model, we resorted to
Monte Carlo simulation. The two-compartment model equations were used
to generate noise-free cold and hot glucose data during a hot IVGTT.
The mean insulin profile of either a standard or an insulin-modified
IVGTT was used as input to the model (Fig. B1).

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Fig. B1. Insulin profiles used in Monte Carlo simulation to assess the effect of
insulin dynamics during the IVGTT on cold and hot minimal model
indexes. Insulin profiles A and
B are typical of a standard and an
insulin-modified hot IVGTT, respectively.
|
|
It is of interest that the profiles
of EGP generated in both occasions always assumed positive
values, implying that it was not necessary to force EGP to
be nonnegative. Subsequently, 200 realizations of noisy cold and
hot glucose data were obtained by adding measurement noise of
appropriate statistical characteristics to the noise-free data.
Specifically, for cold (g, mg/dl) and hot (g*, dpm/ml) glucose
concentration, measurement noise was assumed to be independent,
gaussian, and with a zero mean and standard deviations of
0.02 g and 30+0.018 g*, respectively. Each noisy data set was
analyzed with the cold and hot minimal models. The cold minimal
model was identified from cold glucose and insulin data and the
hot minimal model from hot glucose and insulin data via nonlinear
weighted least squares (13), with weights chosen optimally, i.e., equal
to the inverse of the variance of the measurement error. As is
usually done in practice, to mitigate the error of the
single-compartment assumption, cold and hot glucose data up to 10 min
after the glucose bolus were ignored in model identification. For the
cold model, the value of gb
was chosen to be equal to the mean of the last two glucose data (at 210 and 240 min). Each noisy data set yielded an estimate of fractional
indexes SG,
SI,
, and
, and volumes V and V*.
Results for the standard and insulin-modified IVGTT have been reported
in Table 1.
 |
APPENDIX C. TWO-COMPARTMENT MODEL INDEXES OF GLUCOSE EFFECTIVENESS
AND INSULIN SENSITIVITY |
Here we outline how indexes of cold and hot glucose effectiveness and
insulin sensitivity can be derived from the parameters of the
two-compartment model. For the sake of clarity, we first provide their definitions.
Cold glucose effectiveness is defined as the ability of glucose to
promote its own disappearance by stimulating
Rd and inhibiting EGP. It can be
expressed mathematically as the derivative of
Rd+EGP with respect to glucose
concentration at basal steady state
|
(C1)
|
Hot
glucose effectiveness measures glucose effect on
Rd only and is defined as
|
(C2)
|
Insulin sensitivity is defined as the ability of insulin to enhance
glucose effectiveness. Whereas cold insulin sensitivity measures
insulin effect on both Rd and EGP,
hot insulin sensitivity refers to
Rd only
|
(C3)
|
|
(C4)
|
Plasma clearance rate is the ratio between Rd and
plasma glucose concentration at steady state
|
(C5)
|
Applying
the above definitions to the two-compartment model described in
APPENDIX A, one obtains indexes
measuring cold and hot glucose effectiveness and insulin sensitivity
and basal plasma clearance rate (GE, GE*, IS, IS*, and PCR). Details on
the formal derivation of the indexes can be found in Vicini et al.
(37). The expressions of the two-compartment model indexes are reported
in Table C1.
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Table C1.
Glucose effectiveness, insulin sensitivity, and basal plasma clearance
rate for two-compartment model, minimal models, and glucose clamp
technique
|
|
In comparing the minimal model indexes obtained from the Monte Carlo
simulation (Table 1) with the corresponding indexes of the
two-compartment model, one is faced with the problem that the units are
different. For instance, SG is
expressed in min
1, whereas
the two-compartment model glucose effectiveness, GE, is expressed in
milligrams per kilogram per minute. As a matter of fact, the minimal
model indexes SG,
SI,
, and
are all fractional indexes, i.e.,
they refer to a unit of glucose distribution volume. As shown in Ref.
37, to convert the minimal model indexes to the same units of the
two-compartment model indexes, the cold and hot indexes must be
multiplied by the cold and hot minimal model volumes V and V*,
respectively. The expressions of the minimal model indexes multiplied
by the respective volumes are reported in Table C1. It is worth
mentioning that, for the two-compartment model, glucose effectiveness
and basal plasma clearance rate are different because
Rd is not proportional to g, thus
determining the presence of the nonzero intercept
Rd,0. In contrast, because the hot
minimal model assumes that Rd is proportional to g, one has Rd,0 = 0, and glucose effectiveness and clearance rate coincide.
Cold and hot glucose effectiveness and insulin sensitivity can also be
assessed under steady-state conditions by use of the glucose clamp
technique (7, 11). Glucose effectiveness is measured from hyperglycemic
clamps at basal insulin, whereas insulin sensitivity is measured from
hyperinsulinemic, euglycemic clamps. The clamp-based indexes are thus
derived on the basis of finite (
), rather than differential (
),
increments of glucose and insulin. Their expressions are reported in
Table C1. As one can see, although the cold indexes reflect changes in
the glucose infusion rate, GINF, and thus in both
Rd and EGP, the hot indexes
reflect changes in Rd only.
 |
APPENDIX D. TWO-COMPARTMENT MODEL SIMULATION OF AN IVGTT AT BASAL
INSULIN |
The purpose here is to use the two-compartment model to describe the
time courses of cold and hot glucose during a hot IVGTT in which
insulin is maintained at the basal level.
Cold Glucose Decay
When insulin remains at the basal level throughout the IVGTT, insulin
action is identically null and the two-compartment model equations for
cold glucose become
|
(D1)
|
In the basal steady state, EGP equals
Rd and thus
|
(D2)
|
Substitution of EGPb given by
Eq. D2 into Eq. D1 permits elimination of the term containing
Rd,0. Using the position
qi(t) = qi(t)
qib for
i = 1, 2, one obtains
|
(D3)
|
where
(t) is the Dirac impulse
function. Equation D3 indicates that,
during an IVGTT at basal insulin, glucose decay above the baseline is
the impulse response of a linear, second-order system with constant
parameters. As a result, glucose decay is two-exponential
|
(D4)
|
where
1 and
2 are the fast and slow
eigenvalues, respectively. Of note is that
A1,
A2,
1, and
2 reflect the ability of glucose not only to promote Rd but
also to inhibit EGP. Their values, reported in Table
D1, have been derived from the
two-compartment model parameters (Table A1) by calculat
ing the transfer function of the system described by
Eq. D3 and equating it to the Laplace
transform of the two-exponential decay of Eq. D4 (13).
The fractional decay rate of incremental glucose concentration,
kG, which
measures the fraction of glucose concentration above basal that
declines per unit time, is given by
|
(D5)
|
Hot Glucose Decay
During a hot IVGTT at basal insulin, the two-compartment model
equations for hot glucose become
|
(D6)
|
If Rd,0 were equal to
zero, the system described by Eq. D6
would be linear, and the decay of hot glucose concentration after the
(cold + hot) glucose injection would coincide with the two-exponential impulse response of glucose kinetics in the basal steady state
|
(D7)
|
where
and
are the fast and slow
eigenvalues of glucose kinetics in the basal state. Of note is that parameters
,
,
, and
reflect
Rd only. Their values, reported in
Table D1, have been derived from the two-compartment model parameters
by calculating the transfer function of the linear system described by Eqs. D6 with
Rd,0 = 0 and equating it to the
Laplace transform of the two-exponential decay of Eq. D7 (13).
However, because Rd,0 is greater
than zero, hot glucose decay is not two-exponential
|
(D8)
|
where the term
(t) is the
deviation of the true hot glucose decay from the two-exponential
function of Eq. D7. The fractional decay rate of hot glucose during the IVGTT at basal insulin,
(t), is the ratio
g(
)*/g(t).
The time course of
(t) has been shown in Fig. 8.
 |
APPENDIX E. SG(CLAMP) FROM THE PARAMETERS OF THE
TWO-COMPARTMENT MODEL |
Here we show that the clamp-based measure of glucose effectiveness,
SG(clamp), can be expressed as a
function of the parameters of the two-compartment model and is related
to the area under the glucose decay curve during an IVGTT at basal insulin.
SG(clamp) and the Two-Compartment Model
SG(clamp) is measured from
hyperglycemic clamp studies in which glucose concentration is elevated
via an exogenous glucose infusion rate, and insulin concentration is
maintained at the basal level (11). By applying the formal definition
of glucose effectiveness reported in Eq. C1 to a hyperglycemic clamp at basal insulin, one finds
that SG(clamp) corresponds to the
slope of the steady-state relationship between the exogenous glucose
infusion rate, GINF, and plasma glucose concentration
|
(E1)
|
To express SG(clamp) as a
function of the parameters of the two-compartment model, one must write
the two-compartment model equations describing glucose dynamics during
a hyperglycemic glucose clamp at basal insulin. They are the same as
those derived in APPENDIX D for an
IVGTT at basal insulin except that the exogenous glucose input is a
variable glucose infusion, GINF(t), instead of a bolus injection
|
(E2)
|
When an elevated steady state for glucose is achieved, the
derivatives of
q1 and
q2 become
null, and Eq. E2 yields the following relationship between
GINF and
g
|
(E3)
|
By substituting Eq. E3 into
Eq. E1, one obtains
|
(E4)
|
It is of interest that the expression of
SG(clamp) in Eq. E4 is identical to the index of cold glucose
effectiveness, GE, of the two-compartment model (see Table C1).
SG(clamp) and the IVGTT at Basal Insulin
SG(clamp) can also be expressed as
a function of the impulse response parameters
A1,
A2,
1, and
2, which describe glucose decay
during an IVGTT at basal insulin. To do so one has to integrate from
zero to infinity Eq. D3, which
describes the glucose glucose system during an IVGTT at basal insulin
|
(E5)
|
Because
g(t) is a two-exponential impulse
response, AUC[
g(t)]
can also be expressed as a function of the impulse response parameters
A1, A2,
1,
and
2
|
(E6)
|
By equating Eqs. E5 and E6 and remembering that the
denominator of Eq. E5 is equal to
SG(clamp) (see
Eq. E4), one obtains
|
(E7)
|
 |
APPENDIX F. INSULIN ACTION AND INSULIN SENSITIVITY OF THE
MINIMAL MODELS |
The purpose of this appendix is to investigate the relationships
between insulin action and sensitivity of the two-compartment model and
the minimal models.
In the two-compartment model,
xd(t)
is insulin action on glucose disposal, and
xp(t)
is insulin action on glucose production. The overall insulin action of
the two-compartment model is thus X(t) = xp(t)+xd(t).
The insulin action of the cold minimal model, x(t),
differs from
X(t)
for two reasons: the different model order (1 vs. 2 compartments) and
the compartment where insulin action takes place (accessible vs.
nonaccessible). To single out the effect of the monocompartmental
approximation on
x(t),
an "accessible-pool equivalent" profile of insulin action,
denoted as
(t),
has been derived.
(t) = xp(t) +
d(t),
where
d(t)
is the profile of insulin action on glucose uptake that, placed in the
accessible pool of the two-compartment model, produces the same effect
as xd(t)
on plasma glucose concentration. Application of this definition to the
two-compartment model (Eq. A4) leads
to the following expression for
d(t)
|
(F1)
|
where
2(t)
is glucose mass in the second compartment when the insulin-dependent
glucose removal is moved to the accessible pool. Calculation of
2(t)
requires solution of the mass-balance equation of the nonaccessible
glucose pool (second equation in A4) without the insulin-dependent
term
|
(F2)
|
Eq. F1 shows that
d(t)
is such that the increase in the flux irreversibly leaving the
accessible compartment,
d(t)q1(t), exactly compensates the increase in the flux coming from the
nonaccessible compartment. As a result, the time course of glucose
concentration in the accessible pool remains unchanged.
The difference
x(t) =
(t)
x(t)
is the bias affecting the minimal model insulin action due to
monocompartmental undermodeling.
x(t)
can be given an analytic expression by assuming that the minimal model
is able to perfectly describe the glucose decay generated by the
two-compartment model. In this case,
x(t)
can be derived by equating the expressions of the glucose fractional
decay rate yielded by the two models. The fractional decay rate of
glucose concentration of the minimal model is made up of two
components, one depending on insulin action and the other on glucose
effectiveness (18)
|
(F3)
|
The fractional decay rate of the two-compartment model can be
calculated from Eq. A1
|
(F4)
|
where
= k21k12,
k22 = (k02+k12),
and
2(t)
is glucose concentration in the second nonaccessible pool. Its
structure is similar to the one of the minimal model except for an
additional term proportional to the ratio between the derivative of
glucose concentration in the nonaccessible pool and glucose
concentration in the accessible pool. By equating Eqs.
F3 and F4 one obtains
an expression for
x(t)
|
(F5)
|
Equation F5 shows that the bias
affecting the minimal model insulin action has two components: the
first is proportional to the minimal model underestimation of
fractional glucose effectiveness, whereas the second is modulated by
the rate of change of glucose concentration in the nonaccessible pool,
normalized to glucose concentration in the accessible pool.
The bias affecting
x(t)
influences the accuracy of SI. In
Ref. 18 we have shown that SI is
proportional to the integral of insulin action
|
(F6)
|
The fractional insulin sensitivity of the two-compartment model,
IS/V1, is well approximated by the
integral of
(t)
normalized to the area under the insulin concentration curve (6.46 vs.
6.52 min
1 · µmol
1 · ml).
The agreement is not perfect because
(t),
being the accessible-pool equivalent of the two-compartment model
insulin action, does not satisfy, like the minimal model insulin action x(t)
(Eq. 1), the equation of a remote
insulin compartment. The difference between the two-compartment and the
cold minimal model estimates of insulin sensitivity is thus well
approximated by
|
(F7)
|
Analogous expressions for the difference between the two-compartment
and the hot minimal model insulin action and insulin sensitivity can be
derived by following an approach similar to that just outlined for the
cold model.
, which is the difference between the two-compartment insulin action on
Rd (applied to the accessible
pool) and the hot minimal model insulin action, is given
by
|
(F8)
|
where
(t)
is the "accessible-pool equivalent" profile of two-compartment insulin action on glucose disposal calculated from tracer data.
(t)
is calculated from the two-compartment model (Eq. A1) so as to produce the same effect on hot glucose
concentration as
xd(t);
GE*/V1 is fractional hot glucose
effectiveness of the two-compartment model;
*2(t) is
hot glucose concentration in the nonaccessible glucose pool when
insulin-dependent glucose removal takes place in the accessible pool.
The expression of
(t)
is similar to that of
x(t) but includes an additional term proportional to
Rd,0 that accounts for the fact
that the hot minimal model does not describe the inhibitory effect of
glucose on its own clearance.
, i.e., the difference
between the (fractional) insulin sensitivity on
Rd of the two-compartment model
and
, can be calculated as
previously done for SI
|
(F9)
|
where the hot fractional insulin sensitivity of the two-compartment
model, IS*/V1, is well
approximated by the integral of
(t)
normalized to the area under the insulin concentration curve (5.02 vs.
5.78 min
1 · µmol
1 · ml).
The agreement is not perfect because
(t), the tracer-based accessible-pool equivalent of the two-compartment model insulin action on glucose disposal, does not satisfy, as the
minimal model insulin action
x*(t)
(Eq. 2), the equation of a remote
insulin compartment.
 |
FOOTNOTES |
Address for correspondence and reprint requests: C. Cobelli, Dept.
of Electronics and Informatics, Univ. of Padova, 35131 Padova, Italy.
(E-mail: cobelli{at}dei.unipd.it).
Received 26 March 1996; accepted in final form 19 February 1999.
 |
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