MODELING IN PHYSIOLOGY
The hot IVGTT two-compartment minimal model: indexes of glucose
effectiveness and insulin sensitivity
Paolo
Vicini1,
Andrea
Caumo2, and
Claudio
Cobelli1
1 Department of Electronics and
Informatics, University of Padova, 35131 Padua; and
2 Scientific Institute San
Raffaele, 20132 Milan, Italy
 |
ABSTRACT |
A two-compartment
minimal model (2CMM) has been proposed [A. Caumo and C. Cobelli.
Am. J. Physiol. 264 (Endocrinol. Metab. 27):
E829-E841, 1993] to describe intravenous glucose tolerance test (IVGTT) labeled (hereafter hot) glucose kinetics. This model, at
variance with the one-compartment minimal model (1CMM), allows the
estimation of a plausible profile of glucose production. The aim of
this study is to show that the 2CMM also allows the assessment of
insulin sensitivity (S2*I), glucose
effectiveness (S2*G), and plasma
clearance rate (PCR). The 2CMM was identified on
stable-isotope IVGTTs performed in normal subjects (n = 14). Results were (means ± SE) S2*G = 0.85 ± 0.14 ml · kg
1 · min
1,
PCR = 2.02 ± 0.14 ml · kg
1 · min
1,
and S2*I = 13.83 ± 2.54 × 10
2
ml · kg
1 · min
1 · µU
1 · ml.
The 1CMM was also identified; glucose effectiveness and insulin sensitivity indexes were S*GV = 1.36 ± 0.08 ml · kg
1 · min
1
and S*I V = 12.98 ± 2.21 × 10
2
ml · kg
1 · min
1 · µU
1 · ml,
respectively, where V is the 1CMM glucose distribution volume.
S*GV was lower than PCR and higher
than S2*G and did not correlate with
either [r = 0.45 (NS) and
r = 0.50 (NS),
respectively], whereas S*IV was
not different from and was correlated with
S2*I
(r = 0.95;
P < 0.001).
S*G compares well
(r = 0.78;
P < 0.001) with PCR normalized by
the 2CMM total glucose distribution volume. In conclusion, the 2CMM is
a powerful tool to assess glucose metabolism in vivo.
intravenous glucose tolerance test; tracer kinetics; plasma
clearance rate
 |
INTRODUCTION |
THE LABELED (hereafter hot) intravenous glucose
tolerance test (IVGTT) interpreted with the single-compartment minimal
model of hot glucose disappearance (1CMM) is a powerful, noninvasive tool to characterize glucose disposal in vivo (1, 7, 10). The model
provides metabolic indexes measuring glucose effectiveness (S*G) and insulin
sensitivity (S*I) in an
individual (10). However, when the model is used in conjunction with
unlabeled (hereafter cold) glucose data to estimate endogenous glucose
production during the IVGTT by deconvolution, an unphysiological time
course results due to the well-known limitations of the
monocompartmental representation of glucose kinetics in non-steady
state (7, 8). A two-compartment minimal model (2CMM) has
recently been proposed to solve this problem (8). The 2CMM provides a
physiologically plausible profile of endogenous glucose production
during the test, thus overcoming the limitations of the 1CMM.
The aim of this study is to show that the new model, in addition to
endogenous glucose release, also provides indexes of
glucose effectiveness and insulin sensitivity. In particular, the 2CMM overcomes the drawback of the 1CMM, which is unable to single out
estimates of glucose effectiveness and plasma clearance rate (they are,
in fact, the same parameter,
S*G) (7). Thus, in addition to a new index of insulin sensitivity
(S2*I), the 2CMM provides new
indexes separately measuring glucose effectiveness (S2*G) and plasma clearance rate at basal
state (PCR). Finally, the relationships between the indexes estimated
with the 1CMM and 2CMM are elucidated.
 |
THE TWO-COMPARTMENT HOT MINIMAL MODEL |
The 2CMM, proposed in Ref. 8 and shown in Fig. 1, is
described, in its uniquely identifiable parameterization, by the
following equations
|
(1a)
|
|
(1b)
|
|
(1c)
|
|
(1d)
|
where
q1 and
q2 denote hot
glucose masses in the first (accessible pool) and second (slowly
equilibrating) compartments, respectively (mg/kg for a stable-label
IVGTT);
x(t) = kcI'(t) is insulin action (min
1),
where I'(t) is the
concentration of insulin remote from plasma (µU/ml);
I(t) and
Ib are plasma insulin and basal
(end test) insulin, respectively (µU/ml);
Q1(t)
is cold glucose mass in the accessible pool (mg/kg);
g(t) is plasma hot glucose
concentration (mg/dl); d is the hot glucose dose (mg/kg);
V1 is the volume of the accessible pool (ml/kg);
Rd,0
(mg · kg
1 · min
1)
is the constant component of glucose disposal (8, 14), accounting for
the inhibition of glucose clearance by glucose itself;
kp
(min
1) is the
proportionality constant;
k21
(min
1),
k12
(min
1), and
k02
(min
1) are parameters
describing glucose kinetics; and
p2 = kb
(min
1) and
sk = kakc/kb
(ml · µU
1 · min
1)
are parameters describing insulin action. Capital and lowercase letters
are used to denote variables related to cold and hot glucose, respectively, and overdot notation refers to time rates of change for
respective variables [e.g.,
1(t),
2(t),
and
(t)].
Briefly, the model structure assumes that insulin-independent glucose
disposal takes place in the accessible pool and is the sum of two
components, one constant and the other proportional to glucose mass.
This brings us to the rate constant describing the irreversible loss
from the accessible pool
|
(2)
|
where
G(t) is the glucose concentration in the accessible pool of
volume V1. It is worth noting that
inhibition has been described by a linearized version, with nonzero
intercept, of the Rd vs. G
characteristic, which is actually of Michaelis-Menten type, where
Rd is the rate of glucose
disappearance from the accessible pool. The two are
virtually coincident in the range of IVGTT glucose concentrations.
Therefore, in the 2CMM, any changes in glucose concentration affect
glucose clearance instantaneously. Of course, this may be an
oversimplification because the dynamics of glucose's effect on its own
clearance are likely to be more complex, entailing, for instance, a
delay between a change in plasma glucose concentration and the
corresponding change in glucose clearance. However, in the model, we
preferred to keep the description of glucose's effect on its own
clearance as simple as possible because, to the best of our knowledge,
there are no data in the literature providing insight into the dynamics
of this effect under non-steady-state conditions.

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Fig. 1.
The two-compartment minimal model (2CMM). Capital and lowercase letters
denote variables related to cold and hot glucose, respectively. See
Eqs. 1a-1d for model
parameterization details.
q1 and
q2, hot glucose
masses in the first (accessible pool) and second (slowly equilibrating)
compartments, respectively; I', concentration of insulin remote
from plasma; g, plasma hot glucose concentration; G, cold glucose
concentration in the accessible pool; d, hot glucose dose;
V1, volume of the accessible pool;
Rd,0, constant
component of glucose disposal that accounts for inhibition of glucose
clearance by glucose itself;
kp,
proportionality constant;
k21,
k12, and
k02, glucose
kinetic parameters; and
ka,
kb, and
kc, insulin
action parameters.
|
|
Insulin-dependent glucose disposal occurs in the slowly exchanging pool
and is assumed to be parametrically controlled, not by plasma, but by
insulin in a remote compartment (13). The interstitial fluid has been
suggested as a possible physiological correlate of this compartment
(4). This brings us to the irreversible loss from the second
compartment
|
(3)
|
Arriving
at a priori unique identifiability requires two assumptions. First, we
assume that, in the basal steady state, insulin-independent glucose
disposal is three times insulin-dependent glucose disposal (9). This
brings us to an additional relationship among the model parameters
|
(4)
|
where
Gb is basal (end test) glucose
concentration (mg/dl). The second assumption is that
Rd,0 is fixed to
the experimentally determined value of 1 mg · kg
1 · min
1
(5).
The uniquely identifiable parameterization, as shown in Ref. 8, is
V1,
k21,
k12,
k02,
p2, and
sk.
 |
DERIVATION OF METABOLIC INDEXES |
We show below that the 2CMM provides indexes of glucose effectiveness
and insulin sensitivity. For the sake of standardization, the two
indexes will be derived in the same units as the corresponding indexes
derived using the glucose-clamp technique; i.e., we will apply their
definitions starting from the Rd
vs. G steady-state characteristic.
Glucose effectiveness.
Glucose effectiveness is defined as the ability of glucose to promote
its own disposal. If one measures glucose disposal by its rate of
disappearance from the accessible pool,
Rd(t),
glucose effectiveness is then given by the derivative of
Rd(t)
with respect to glucose concentration
G(t) at basal steady state (SS)
|
(5)
|
The
Rd(t)
predicted by the model is, by using the tracer-tracee
indistinguishability principle
|
(6)
|
Thus
(see APPENDIX) glucose effectiveness
(in
ml · kg
1 · min
1)
from the 2CMM is
|
(7)
|
Plasma clearance rate.
Another parameter of interest that can be derived from the 2CMM is the
steady-state plasma clearance rate, which is defined as
|
(8)
|
Thus,
by using Eq. 6, one has
|
(9)
|
In
the following, PCR will be calculated at basal (end test) glucose
concentration, Gb. Note that, in
the 2CMM, PCR is different from glucose effectiveness
S2*G because the 2CMM explicitly describes the decrease of PCR when G increases, via the nonzero intercept Rd,0 of
the characteristic (Rd,G). This
condition is not true for the 1CMM (3, 7), in which
Rd,0 = 0, so that glucose effectiveness and glucose clearance coincide.
Insulin sensitivity.
Insulin sensitivity is defined as the ability of insulin to enhance
glucose effectiveness. Formally, one has
|
(10)
|
where
all derivatives are evaluated at the basal (end test) steady state.
When this definition is applied to the 2CMM, insulin sensitivity
S2*I is (see
APPENDIX)
|
(11)
|
Its
units
(ml · kg
1 · min
1 · µU
1 · ml)
are the units of a clearance
(ml · kg
1 · min
1)
per unit of insulin concentration (µU/ml).
The factor sk = kakc/kb
has an interesting interpretation: it is a measure of the insulin
sensitivity of the tissues represented by the slowly exchanging glucose
compartment, in which utilization is directly controlled by insulin. It
has the units of a fractional clearance per unit of insulin
concentration
(min
1 · µU
1 · ml).
The 2CMM also provides an estimate of total glucose distribution volume
(VD)
|
(12)
|
 |
DATABASE AND EXPERIMENTAL PROTOCOL |
This study includes 14 stable isotopically labeled IVGTTs performed on
young adults. A bolus of glucose enriched with
[6,6-2H2]glucose
was rapidly injected intravenously at time
0 in all cases except two
(subjects
2 and
13), in which
[2-2H]glucose was the
tracer. Thirty blood samples were taken at 0, 2, 3, 4, 5, 8, 10, 12, 14, 16, 18, 20, 24, 28, 32, 40, 45, 50, 60, 70, 80, 90, 100, 110, 120, 140, 160, 180, 210, and 240 min. Isotope ratios, plasma glucose, and
insulin were measured (Fig. 2). The total
glucose dose ranged from 0.25 to 0.33 g/kg, with the tracer being on
average 10% of the dose. The baseline end-test glucose and insulin
concentrations were (means ± SE) 86 ± 2 mg/dl and 9 ± 1 µU/ml, respectively, which were not different from the pretest values
of 88 ± 2 mg/dl and 11 ± 1 µU/ml, respectively.
Six of these experiments (subjects
1-6) have been previously analyzed in Ref. 1
with the 1CMM and in Ref. 8 with the 2CMM for the estimation of
endogenous glucose production. Of the remaining eight experiments, five
(subjects 7-11) were obtained at the Department of Pediatrics and Medicine, Washington University School of Medicine, St. Louis, Missouri (D. M. Bier, K. Yarasheski, and
J. J. Zachwieja, unpublished data), and the remaining three (subjects 12-14) at the Center
for Metabolic Diseases, University of Padova, Padua, Italy (A. Avogaro,
unpublished data).
 |
ESTIMATION OF METABOLIC INDEXES |
The model was identified from hot glucose concentration data,
calculated as described in Ref. 1 for subjects 1-6 and
in Ref. 2 for subjects 7-14. For a discussion of the
rationale underlying the two approaches, see Refs. 2, 11, and 12. Weighted nonlinear least squares (6) were used. The measurement error
associated with the tracer measurements was assumed to be independent,
white, and Gaussian, with zero mean and a variance generated by error
propagation from isotope ratio measurement error variance (12).
Measurement error coefficient of variation ranged from 2 to 7% on
average, with lower precision associated with lower tracer
concentrations. Weights were chosen optimally, i.e., equal to the
inverse of the measurement error variance (6). Model identification was
always performed relying on the whole data set, a method at variance
with the 1CMM, in which glucose samples between 0 and 8 min were
neglected to mitigate the single-compartment approximation. Precision
of parameter estimates (for V1,
k21, k12,
k02, p2, and
sk) was obtained from the inverse of the
Fisher information matrix, whereas that of the metabolic indexes was obtained by error propagation (6). For example, if one assumes higher
order terms can be neglected, then the following expression is valid
for the variance of PCR, defined as in Eq. 9
|
(13)
|
where
p = [V1, k21,
k12, k02,
p2, sk],
partial derivatives are evaluated at the estimated parameter values, and Cov(p) is the covariance matrix,
i.e., the inverse of the Fisher information matrix.
The average weighted residuals are shown in Fig.
3 (mean ± SD) and show no systematic
deviations.

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Fig. 3.
Mean weighted residuals for the 2CMM. Shaded zone between 0 and 20 min
is enlarged to better visualize model fitting of data.
|
|
In Table 1 parameter
estimates of the 2CMM are reported for the 14 subjects, together
with their precision. The calculated total glucose distribution
VD is also shown. The metabolic
indexes of the 2CMM are in Table 2. Their
mean values are S2*G = 0.85 ± 0.14 ml · kg
1 · min
1, PCR = 2.02 ± 0.14 ml · kg
1 · min
1,
and S2*I = 13.83 ± 2.54 × 10
2
ml · kg
1 · min
1 · µU
1 · ml.
Their mean precision is 21% for PCR (range 5-65%), 66% for S2*G (range 14-231%), and 76%
for S2*I (range 5-401%).
S2*I and
S2*G precision is unsatisfactory in
3 of 14 subjects.
 |
TWO- VS. SINGLE-COMPARTMENT HOT MINIMAL MODEL INDEXES |
Before the 2CMM and 1CMM indexes are compared, it is
convenient to briefly review the 1CMM and the indexes it provides.
Review of 1CMM.
The 1CMM (10), shown in Fig. 4, is
described, in its uniquely identifiable parameterization, by
|
(14a)
|
|
(14b)
|
|
(14c)
|
where
p1,
p2,
p3, and
x relate to
k1,
k2,
k3,
k4, and I'
of Fig. 4 as follows:
x(t) = k4I'(t),
p1 = k1,
p2 = k3, and
p3 = k2k4.
Note that, for the sake of comparison with the 2CMM, we did not use the
superscript asterisk for hot glucose-related variables, as was done in
the original study (10).
S*G = p1
(min
1) and
S*I = p3/p2
(min
1 · µU
1 · ml)
measure, respectively, fractional (i.e., per unit of glucose
distribution volume) glucose effectiveness at basal insulin and
fractional insulin sensitivity, i.e., the ability of insulin to enhance
fractional glucose effectiveness.
1CMM indexes of glucose effectiveness and insulin sensitivity
with the same units as those derived for the 2CMM can be calculated by
directly applying their definition. They are
|
(15a)
|
|
(15b)
|
By definition, both indexes refer to the same
volume of distribution, i.e., the volume of the accessible pool, V (and
thus of the system because the 1CMM is single compartment, and
therefore the initial and the total distribution volumes
coincide).
At this point, an important difference with the 2CMM needs to
be pointed out. In the 2CMM, glucose effectiveness
S2*G and plasma clearance rate PCR are
different because of the presence of the inhibitory effect term,
Rd,0/Q1(t),
in Rd (Eq. 6). On the other hand, the 1CMM does not account for
a nonzero Rd,0:
it assumes Rd,0 = 0 (glucose disappearance is proportional to glucose concentration). In
fact, the rate of glucose disappearance of the 1CMM is given by
|
(16)
|
When
this expression is applied to the definition of glucose effectiveness
(Eq. 5) and the definition of plasma
clearance rate in Eq. 8, it is easy to
see that both come out equal to
S*GV. Therefore, for
the 1CMM, the estimates of glucose effectiveness and plasma clearance
rate coincide, and both are (by definition) equal to
S*GV. The ability of
the 2CMM to single out glucose effectiveness and insulin sensitivity is
therefore unique to the 2CMM.
The 1CMM parameters were estimated from hot glucose concentration data
by weighted nonlinear least squares, as described in ESTIMATION OF METABOLIC
INDEXES. In model identification, glucose samples between 0 and 8 min were not considered, in order to mitigate the approximation of the single-compartment description of glucose kinetics.
The mean weighted residuals of the 1CMM are shown in Fig.
5. The 1CMM indexes
S*GV and
S*IV have been
calculated for all subjects and are reported in Table
3 together with their precision.
S*GV was 1.37 ± 0.15 ml · kg
1 · min
1
(S*G = 0.74 ± 0.04 × 10
2
min
1), and
S*IV was 12.98 ± 2.21 × 10
2
ml · kg
1 · min
1 · µU
1 · ml
(S*I = 6.82 ± 0.99 × 10
4
min
1 · µU
1 · ml).
Mean precision of the metabolic indexes, calculated via error
propagation (6), was 5% for
S*GV (range
2-12%) and 5% for
S*IV (range
3-9%).

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Fig. 5.
Mean weighted residuals for the 1CMM. In model identification, data
between 0 and 8 min are excluded.
|
|
At this point, a natural question remains, How do the 1CMM indexes
compare with the 2CMM indexes?
Comparison of hot 1CMM and 2CMM indexes.
To compare the 1CMM and 2CMM indexes, we compared the mean of
S*GV vs. either PCR or
S2*G and
S*IV vs.
S2*I. S*GV is significantly
lower (P < 0.001) than PCR but
greater (P < 0.001) than
S2*G.
S*IV is not different
from S2*I (NS). If we now compare
S*GV vs.
S2*G,
S*GV vs. PCR, and
S*IV vs.
S2*I by linear regression, we find that S*GV correlates weakly
with S2*G
(r = 0.50, NS) and with PCR
(r = 0.45, NS), whereas
S*IV correlates very
well with S2*I (Fig.
6B)
(r = 0.95, P < 0.001). Notably, the regression
line between S*IV and
S2*I is not different from the identity
line (slope 0.83 ± 0.08, P = 0.05, intercept 1.55 ± 1.31, NS). It is worth noting that, given this
near-perfect correlation and the remarkable precision with which the
1CMM index is estimated, in case the two-compartment
S2*I is estimated with unsatisfactory
precision, one can use the single-compartment value,
S*IV, instead.

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Fig. 6.
Scatterplot and regression line of
S*G vs.
PCR/VD
(A) and of
S*IV vs.
S2*I
(B).
S*G, 1CMM fractional
glucose effectiveness (i.e., per unit of glucose distribution volume);
PCR, 2CMM plasma clearance rate;
VD, 2CMM total glucose
distribution volume;
S*IV, 1CMM insulin
sensitivity; S2*I, 2CMM insulin
sensitivity.
|
|
To elucidate the reasons that the comparison of
S*GV with
S2*G and PCR is unsatisfactory, we may
usefully analyze the two components of the product
S*GV separately. Let us
examine S*G, first.
S*G is estimated in the
final portion of the tracer disappearance curve and thus measures the
fractional disappearance rate of glucose when insulin and (cold)
glucose concentrations have almost reached steady state. Under these
circumstances, the contribution of the fast component of glucose
kinetics and the inhibitory effect of hyperglycemia on glucose
clearance have both become negligible. As a result, the disappearance
of hot glucose is governed by only the slow component of glucose
kinetics, and the single-pool description of glucose kinetics given by
the 1CMM is in all likelihood accurate. In other words, in this portion of the IVGTT, the glucose system behaves like a single-pool system, characterized by a fractional disappearance rate equal to
S*G and a volume of
glucose distribution close to the entire glucose distribution space.
Thus the product of S*G
and the total volume of glucose distribution should provide an estimate
of basal PCR (not of S2*G). Because the
1CMM uses a single compartment to represent the entire glucose system,
the volume V should coincide with the total volume of glucose
distribution and the product S*GV should measure
basal PCR. This, however, does not occur
(S*GV < PCR, as shown
above) because V = 185 ± 6 ml/kg is markedly lower than the total
glucose distribution volume (usually 260 ml/kg). When we multiply
S*G by the literature
value VD = 260 ml/kg, the mean
value of S*GV (1.92 ml · kg
1 · min
1)
becomes quite close to PCR. However, the correlation remains poor as
before (r = 0.55, P = 0.04). A likely explanation for this outcome is the need for individual estimates of
VD. By normalizing the individual
values of PCR using the corresponding estimate of
VD provided by the 2CMM in each
subject (278 ± 14 ml/kg on average), one has
PCR/VD = 0.73 ± 0.03 × 10
2
min
1, which is not
different (NS) from S*G = 0.74 ± 0.04 × 10
2
min
1. The correlation (Fig.
6A) is also good
(r = 0.78;
P < 0.001), with the regression line
not different from the identity line (slope 0.88 ± 0.21, NS;
intercept 0.0010 ± 0.0015, NS). These results again indicate that
S*G has the unequivocal
meaning of a fractional clearance rate.
 |
CONCLUSIONS |
The hot 2CMM of glucose kinetics provides three new metabolic indexes:
glucose effectiveness (S2*G), insulin
sensitivity (S2*I), and plasma clearance
rate PCR. These indexes take into account the fact that glucose
kinetics is more accurately described by a two-compartment model. When
they are compared with the 1CMM indexes, i.e., glucose effectiveness
(or plasma clearance rate)
(S*GV) and insulin
sensitivity (S*IV), one
has that, although S*IV
is very well correlated with S2*I,
S*GV underestimates PCR
and overestimates S2*G and is
uncorrelated with both. This result is due to the fact that the 1CMM
assumes that glucose disappearance is proportional to glucose
concentration, whereas the 2CMM properly takes into account the
inhibitory effect of glucose on its own clearance. In conclusion, the
hot IVGTT 2CMM, by allowing the derivation, in addition to endogenous
glucose production, of a rich parametric portrait of glucose disposal, constitutes a powerful tool to assess glucose metabolism in various physiopathological conditions.
 |
APPENDIX |
Glucose effectiveness.
By applying the definition of glucose effectiveness to the rate of
disappearance of glucose measured by the 2CMM, one
has
|
(A1)
|
Because, from Eq. 1b
|
(A2)
|
by
taking the first derivative of
2(t)
with respect to
Q1(t),
we have [because
x(t)/
Q1(t) = 0]
|
(A3)
|
At steady state
|
(A4)
|
and
|
(A5)
|
Therefore
|
(A6)
|
Rearranging
|
(A7)
|
Substituting
Eq. A7 into Eq. A1, we have the expression for
S2*G
|
(A8)
|
Note
that for the 2CMM, at variance with the 1CMM, the accessible pool does
not coincide with the system. Thus, in addition to the glucose
disappearance rate from the accessible pool, it is also possible to
calculate the rate of glucose disappearing from the system, i.e.,
glucose utilization,
U
|
(A9)
|
It is easy to verify that, if one derived glucose
effectiveness from the expression of glucose utilization, thus without having to refer to the accessible pool only
|
(A10)
|
then
the resulting expression of S2*G would be
the same as in Eq. A8.
Insulin sensitivity.
By applying the definition of insulin sensitivity to the 2CMM, one has
|
(A11)
|
Because,
from Eq. 1b
|
(A12)
|
by
taking the first derivative of
2(t)
with respect to
Q1(t),
we have [because
x(t)/
Q1(t) = 0]
|
(A13)
|
Rearranging
|
(A14)
|
If
we take now the derivative of both members of this equation with
respect to I(t), we
have
|
(A15)
|
and
|
(A16)
|
Because, in steady state,
2 = 0 and x =
= 0, we have
|
(A17)
|
and substituting in Eq. A11, we have the expression for the 2CMM insulin
sensitivity
|
(A18)
|
This
formulation suggests an interesting interpretation of
S2*I. Provided that
sk is the
(fractional) insulin sensitivity of the tissues represented by the
slowly exchanging glucose compartment, in which utilization is directly
controlled by insulin, then S2*I
corresponds to
sk multiplied by
the volume of the second compartment "as seen from" the
accessible compartment. This representation is better understood by
rearranging Eq. A18 to the form
|
(A19)
|
in
which the term in parentheses is the volume of the second compartment
and the last factor is a partition coefficient.
 |
ACKNOWLEDGEMENTS |
This work was partially supported (40%) by a grant from the
Italian Ministero della Università e della Ricerca Scientifica e
Tecnologica (on Bioingegneria dei Sistemi Metabolici e Cellulari), by
Italian Consiglio Nazionale delle Ricerche Grant 9300457PF40 (on
Aging), and by National Institutes of Health Grant
RR-02176.
 |
FOOTNOTES |
Address for reprint requests: C. Cobelli, Dept. of Electronics and
Informatics, Via Gradenigo 6/A, Univ. of Padova, 35131 Padua, Italy.
Received 29 January 1996; accepted in final form 15 July 1997.
 |
REFERENCES |
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Avogaro, A.,
J. D. Bristow,
D. M. Bier,
C. Cobelli,
and
G. Toffolo.
Stable-label intravenous glucose tolerance test minimal model.
Diabetes
38:
1048-1055,
1989[Abstract].
2.
Avogaro, A.,
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AJP Endocrinol Metab 273(5):E1024-E1032
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