The hot IVGTT two-compartment minimal model: an improved
version
Gianna
Toffolo and
Claudio
Cobelli
Department of Information Engineering, University of
Padova, 35131 Padova, Italy
 |
ABSTRACT |
The two-compartment minimal model (2CMM)
interpretation of a labeled intravenous glucose tolerance test (IVGTT)
is a powerful tool to assess glucose metabolism in a single individual.
It has been reported that a derived 2CMM parameter describing the
proportional effect of glucose on insulin-independent glucose disposal
can take physiologically unplausible negative values. In addition, precision of 2CMM parameter estimates is sometimes not satisfactory. Here we resolve the above issues by presenting an improved version of
2CMM that relies on a new assumption on the constant component Rd0 of insulin-independent glucose disposal. Here
Rd0 is not fixed to 1 mg · kg
1 · min
1
but instead is expressed as a fraction of steady-state glucose disposal. The new 2CMM is identified on the same stable labeled IVGTT
data base on which the original 2CMM was formulated. A more reliable
insulin-independent glucose disposal portrait is obtained while that of
insulin action remains unchanged. The new 2CMM also improves the
precision with which model parameters and metabolic indexes are estimated.
insulin sensitivity; glucose effectiveness; glucose production; mathematical model; parameter estimation; intravenous glucose tolerance
test
 |
INTRODUCTION |
THE TWO-COMPARTMENT
MINIMAL MODEL (2CMM) interpretation of the labeled intravenous
glucose tolerance test (IVGTT) is a powerful tool to assess glucose
metabolism in a single individual, since it allows characterization of
glucose disposal in terms of indexes of insulin sensitivity and glucose
effectiveness (10) and reconstruction of the time course
of endogenous glucose production (3, 11). The model has
been employed and is currently being used in several studies (e.g.,
Refs. 5 and 7-9). One parameter that can be calculated from the estimated model parameters is the proportional effect of glucose on insulin-independent glucose disposal, denoted as
kp (min
1). We and other
investigators (personal communication) have observed that sometimes
kp can take on negative values, a physically
unrealizable event. For instance, Vicini et al. (10) noted
that this happens in 3 of the 14 subjects studied. Another reported
finding is that sometimes the precision of the parameter estimates of
2CMM is not satisfactory.
This brief contribution aims at resolving the above issues by an
improved version of 2CMM. In particular, we first outline the
conditions under which parameter kp can take on
positive values by reconsidering the assumptions underlying the model.
Next, we formulate an improved version of the model that guarantees
positive values of all parameters. Results on the same 14 subjects
studied previously (10) are presented and compared with
those obtained with the previous version of the model.
 |
THE TWO-COMPARTMENT MINIMAL MODEL |
The 2CMM (Fig. 1) is
described by the following equations
|
(1a)
|
|
(1b)
|
|
(1c)
|
|
(1d)
|
where q1(t) and
q2(t) denote tracer glucose masses at
time t in the first (accessible) 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
total glucose mass in the accessible pool (mg/kg); g(t) is
plasma tracer glucose concentration (mg/dl); d is the tracer glucose
dose (mg/kg); V1 is the volume of the accessible pool
(ml/kg); Rd0
(mg · kg
1 · min
1)
is the constant component of glucose disposal, whereas
kp (min
1) is the proportionality
constant between glucose disposal from the accessible compartment and
glucose mass in the same compartment; 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 tracer glucose,
respectively, and overdot notation refers to time rates of change for
respective variables.
The model assumes that pools 1 and 2 represent,
respectively, plasma plus insulin-independent tissues, rapidly
equilibrating with plasma, and insulin-dependent tissues (utilization
depends on insulin in addition to glucose), slowly exchanging with
plasma. Glucose disposal from the accessible pool, Rd1, is
the sum of two components, one constant (Rd0) and the other
(kpQ1) proportional to
glucose mass Q1, thus accounting for the
inhibition of glucose clearance by glucose itself. Thus the rate
constant describing the irreversible loss of both tracer and tracee
from the accessible pool is
|
(2)
|
where rd1 is insulin-independent tracer glucose
disposal and G1 is the glucose concentration in the
accessible pool of volume V1.
Glucose disposal from the slowly exchanging pool is assumed to be
parametrically controlled by insulin in a remote compartment represented by variable x. The rate constant describing
irreversible loss of tracee and tracer from compartment 2,
Rd2 and rd2, respectively, is then
|
(3)
|
Arriving at a priori unique identifiability requires two
assumptions (3). First, in normal subjects in the basal
steady state (ss), insulin-independent glucose disposal is three times glucose disposal from insulin-dependent tissues (R
= 3R
; see Refs. 4 and 6). This
materializes in an additional relationship among the model parameters
|
(4)
|
where Gb is basal (evaluated from end test values)
glucose concentration (mg/dl). Moreover, Rd0 is fixed to
the experimentally determined value of 1 mg · kg
1 · min
1.
The 2CMM allows the estimation of glucose effectiveness, insulin
sensitivity, and plasma clearance rate.
Glucose Effectiveness
Glucose effectiveness
(S
*;
ml · kg
1 · min
1)
quantifies the ability of glucose to promote its own disposal at steady
state
|
(5)
|
where Rd = Rd1 + Rd2.
Plasma Clearance Rate
Plasma clearance rate (PCR;
ml · kg
1 · min
1)
measures glucose disposal at basal steady state, per unit glucose
concentration
|
(6)
|
where the last equality follows from Eq. 4.
Insulin Sensitivity
Insulin sensitivity (S
*;
ml · kg
1 · min
1
per µU/ml) quantifies the ability of insulin to enhance glucose effectiveness
|
(7)
|
 |
MODEL ASSUMPTIONS AND PARAMETER KP |
Model assumptions do not guarantee positive values for parameter
kp in all circumstances. In fact, from Eq. 4, kp is the difference between the
following two terms
|
(8)
|
and assumes positive values only when
|
(9)
|
that is, from Eq. 6, when
|
(10)
|
Thus kp is positive if the constant
component Rd 0, which is fixed equal to 1 mg · min
1 · kg
1
in all subjects, is less than the steady-state value of glucose disposal from the accessible compartment, which accounts for
three-fourths of total glucose disposal. This condition can also be
read as a lower bound for total glucose disposal at steady state
|
(11)
|
The assumption of a constant component of glucose disposal equal
to 1 mg · min
1 · kg
1
is thus critical because it leads to a negative value of the kp parameter in those subjects having a total
glucose disposal in the basal state <1.33
mg · min
1 · kg
1.
 |
AN IMPROVED 2CMM |
To ensure positive values of kp, we
formulate the needed (for a priori identifiability reasons) constraint
on Rd0 in an alternative way. The idea is to relate it to
total glucose disposal in steady state, by assuming that
Rd0 accounts for a fixed fraction of it
|
(12)
|
where
is constant among individuals. Values of Rd0
and R
measured in a group of nondiabetic subjects (2), namely R
= 21.71 and
Rd0 = 10.1 µmol · min
1kg lean body
mass
1 (Rd0 is not far from 1 when expressed
as
mg · min
1 · kg
1),
suggest to fix
= 0.465. With this value for
, Eq. 10, which ensures positive kp, becomes
|
(13)
|
Thus the new assumption on Rd0, Eq. 12,
while still ensuring a priori identifiability of the model structure,
is also able to guarantee positive values of kp.
In fact, by using Eq. 12 in Eq. 8,
kp becomes
|
(14)
|
and always assumes positive values.
Metabolic indexes S
, PCR, and S
are still defined as before and can be evaluated from model parameters
by using the same expressions (Eqs. 5-7).
 |
MODEL IDENTIFICATION |
The new 2CMM equations to be used in normal subjects are
Eqs. 1a-1d, coupled with Eq. 14 for parameter
kp appearing in Eq. 1a and with the
following equation, derived from Eqs. 6 and 12, for Rd0/Q1(t), also
appearing in Eq. 1a
|
(15)
|
Unknown model parameters k21,
k12, k02,
sk, p2, and
V1 were estimated in each individual by using SAAMII
software (1). Weights were chosen as described previously
(10).
 |
RESULTS |
In a previous study (10), the 2CMM was identified on
stable labeled IVGTT data performed in 14 young adults. For the
individual parameter estimates and metabolic indexes, we refer to
Tables 1 and 2 of the original paper; here, their average values are reported (Table 1) along with the
individual values of R
and kp
(Table 2). In three subjects,
R
is <1.33 and kp is
negative, in keeping with the considerations developed above.
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Table 1.
Estimated parameters of the 2CMM in 14 normal subjects both in its
original and improved version. Precision of parameter estimates is also
shown as mean CV in parenthesis
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Table 2.
Total glucose disposal at steady state and kp
parameter evaluated from parameters of the 2CMM both in its original
and improved version
|
|
The improved version of 2CMM was then identified on the same data set.
Its ability to describe the data is virtually the same as that of the
original 2CMM, i.e., since the plots of their average residuals are
virtually superimposable (data not shown).
Average values of its model parameters and metabolic indexes (Table 1
and 2) indicate that most parameters related to both glucose kinetics
(V1, k21, k12
, k02, PCR, and R
) and
insulin action (p2, sk,
and S
) are very similar to the previous ones. Among
them, k02, R
, and PCR were
statistically different (P < 0.05). This is because of
small systematic differences (<3%) in most subjects. Conversely, average values of S
and kp
are consistently higher with the new model version, and
kp (Table 2) is now positive in all subjects.
The improved 2CMM also has a better performance in terms of precision
of parameter estimates, since the coefficients of variation are
markedly lower for all parameters (Table 1).
 |
DISCUSSION |
We have presented a new version of 2CMM that guarantees
positive values of the derived parameter kp in
all individuals. This goal is accomplished by introducing a different,
but still physiologically sound, assumption on Rd0.
Rd0 is a nonphysiological parameter that represents the
nonzero intercept of the linear approximation, in the experimental
glucose range, of the relationship between insulin-independent glucose
disposal and glucose concentration. In all likelihood, this
relationship is a sigmoidal-shaped curve that, starting at zero
(glucose utilization is 0 at 0 glucose concentration), saturates at a
plateau. It is also often described by a Michaelis-Menten relationship,
but the range of glucose concentrations spanned during an IVGTT does
not allow for reliable estimation of the two parameters of this model.
The relationship is then approximated by a straight line, having
kp as a slope and Rd0 as an
intercept. In the new version, Rd0 is adjusted in every subject on the basis of his/her value of total glucose disposal in the
basal state, Rd0 =
R
= 0.78 ± 0.6 mg · min
1 · kg
1,
which is less than the value Rd0 = 1 mg · min
1 · kg
1
assumed in the original version. The decrease in Rd0 is
balanced by an increase in the glucose-dependent component of glucose
disposal, and thus by an increase of kp and
S
, because their sum, which gives
insulin-independent glucose disposal, is similar in the two model
versions. All of the remaining model indexes are also similar. It is
not possible to prove that the new model provides more accurate
estimates of S
, since we do not have a
model-independent reference for it. However, we can argue that, because
the new 2CMM avoids some inconsistencies of the original 2CMM (negative
kp), it provides a more reliable description of
the system and thus a more reliable value for S
. Similarly, we can also argue that the new model should provide more
reliable estimates of endogenous glucose production. Finally, with the
new assumption, precision of parameter estimates considerably improves.
The 2CMM, developed here for application in normal subjects, can be
extended to impaired glucose-tolerant or diabetic subjects. This,
however, requires reconsideration of some model assumptions, as
discussed in the APPENDIX.
In conclusion, this improved version of 2CMM, by avoiding some
inconsistencies of the original 2CMM (negative
kp), provides a more reliable and precise
parametric portrait of glucose metabolism during an IVGTT.
 |
APPENDIX |
The use of 2CMM in glucose-tolerant or diabetic subjects
requires reconsideration of some model assumptions. For instance, values of Rd0 = 14.6 µmol · min
1 · kg
lean body mass
1 and R
= 19.2 µmol · min
1 · kg
lean body mass
1 measured in diabetic subjects
(2) result in a different
(
= 0.759). Also,
the proportion between glucose disposal from insulin-independent and
insulin-dependent tissues in the basal state is in all likelihood
different from the 3:1 ratio assumed in normal subjects and moves to a
higher value. For instance, if in diabetics a 5:1 ratio is assumed,
i.e., a glucose disposal from insulin-independent tissues accounts for
83.5% of total glucose disposal in the basal state, then Eq. 14 becomes
|
(A1)
|
which still guarantees positive values for
kp.
In the general case, if a ratio
/1 is assumed between glucose
disposal from insulin-independent and insulin-dependent tissues under
basal conditions, model equations are Eqs. 1a-1d, coupled with the following equations for kp and
Rd0/Q1(t)
|
(A2)
|
|
(A3)
|
 |
ACKNOWLEDGEMENTS |
This work was supported in part by a MIUR COFIN Grant on "Stima
di parametri non accessibili in sistemi fisiologici" and by Division
of Research Resources Grant RR-12609.
 |
FOOTNOTES |
Address for reprint requests and other correspondence:
C. Cobelli, Dipartimento di Ingegneria dell'Informazione,
Università degli Studi di Padova, Via Gradenigo, 6a-35131 Padova,
Italy (E-mail: cobelli{at}dei.unipd.it).
The costs of publication of this
article were defrayed in part by the
payment of page charges. The article
must therefore be hereby marked
"advertisement"
in accordance with 18 U.S.C. Section 1734 solely to indicate this fact.
First published October 1, 2002;10.1152/ajpendo.00499.2001
Received 6 November 2001; accepted in final form 27 September 2002.
 |
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