## 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 R_{d0} of insulin-independent glucose disposal. Here R_{d0} 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

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*k*
_{p} (min^{−1}). We and other investigators (personal communication) have observed that sometimes*k*
_{p} 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 *k*
_{p} 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
Equation 1a
Equation 1b
Equation 1c
Equation 1dwhere *q*
_{1}(*t*) and*q*
_{2}(*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*) =*k*
_{c}I′(*t*) is insulin action (min^{−1}), where I′(*t*) is the concentration of insulin remote from plasma (μU/ml); I(*t*) and I_{b} are plasma insulin and basal (end-test) insulin, respectively (μU/ml); *Q*
_{1}(*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); V_{1} is the volume of the accessible pool (ml/kg); R_{d0}(mg · kg^{−1} · min^{−1}) is the constant component of glucose disposal, whereas*k*
_{p} (min^{−1}) is the proportionality constant between glucose disposal from the accessible compartment and glucose mass in the same compartment; *k*
_{21}(min^{−1}), *k*
_{12} (min^{−1}), and *k*
_{02} (min^{−1}) are parameters describing glucose kinetics; and p_{2} =*k*
_{b} (min^{−1}) and s_{k}=*k*
_{a}
*k*
_{c}/*k*
_{b}(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, R_{d1}, is the sum of two components, one constant (R_{d0}) and the other (*k*
_{p}
*Q*
_{1}) proportional to glucose mass *Q*
_{1}, 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
Equation 2where r_{d1} is insulin-independent tracer glucose disposal and G_{1} is the glucose concentration in the accessible pool of volume V_{1}.

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*, R_{d2} and r_{d2}, respectively, is then
Equation 3Arriving 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
Equation 4where G_{b} is basal (evaluated from end test values) glucose concentration (mg/dl). Moreover, R_{d0} 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
Equation 5where R_{d} = R_{d1} + R_{d2}.

### Plasma Clearance Rate

Plasma clearance rate (PCR; ml · kg^{−1} · min^{−1}) measures glucose disposal at basal steady state, per unit glucose concentration
Equation 6where 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
Equation 7

## MODEL ASSUMPTIONS AND PARAMETER *K*_{P}

Model assumptions do not guarantee positive values for parameter*k*
_{p} in all circumstances. In fact, from *Eq.4
*, *k*
_{p} is the difference between the following two terms
Equation 8and assumes positive values only when
Equation 9that is, from *Eq. 6
*, when
Equation 10
Thus *k*
_{p} is positive if the constant component R_{d 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
Equation 11The 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*k*
_{p} 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 *k*
_{p}, we formulate the needed (for a priori identifiability reasons) constraint on R_{d0} in an alternative way. The idea is to relate it to total glucose disposal in steady state, by assuming that R_{d0} accounts for a fixed fraction of it
Equation 12where α is constant among individuals. Values of R_{d0}and R
measured in a group of nondiabetic subjects (2), namely R
= 21.71 and R_{d0} = 10.1 μmol · min^{−1}kg lean body mass^{−1} (R_{d0} is not far from 1 when expressed as mg · min^{−1} · kg^{−1}), suggest to fix α = 0.465. With this value for α, *Eq.10
*, which ensures positive *k*
_{p,} becomes
Equation 13Thus the new assumption on R_{d0}, *Eq. 12
*, while still ensuring a priori identifiability of the model structure, is also able to guarantee positive values of *k*
_{p}. In fact, by using *Eq. 12
* in *Eq. 8
*,*k*
_{p} becomes
Equation 14and 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*k*
_{p} appearing in *Eq. 1a
* and with the following equation, derived from *Eqs. 6
* and *
12
*, for R_{d0}/*Q*
_{1}(*t*), also appearing in *Eq. 1a
*
Equation 15Unknown model parameters *k*
_{21},*k*
_{12}, *k*
_{02},*s*
_{k}, *p*
_{2}, and V_{1} 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 *k*
_{p}(Table 2). In three subjects, R
is <1.33 and *k*
_{p} is negative, in keeping with the considerations developed above.

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 1and 2) indicate that most parameters related to both glucose kinetics (V_{1}, *k*
_{21}, *k*
_{12}, *k*
_{02}, PCR, and R
) and insulin action (*p*
_{2}, *s*
_{k}, and S
) are very similar to the previous ones. Among them, *k*
_{02}, 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 *k*
_{p}are consistently higher with the new model version, and*k*
_{p} (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 *k*
_{p} in all individuals. This goal is accomplished by introducing a different, but still physiologically sound, assumption on R_{d0}. R_{d0} 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*k*
_{p} as a slope and R_{d0} as an intercept. In the new version, R_{d0} is adjusted in every subject on the basis of his/her value of total glucose disposal in the basal state, R_{d0} = αR
= 0.78 ± 0.6 mg · min^{−1} · kg^{−1}, which is less than the value R_{d0} = 1 mg · min^{−1} · kg^{−1}assumed in the original version. The decrease in R_{d0} is balanced by an increase in the glucose-dependent component of glucose disposal, and thus by an increase of *k*
_{p} 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*k*
_{p}), 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 .

In conclusion, this improved version of 2CMM, by avoiding some inconsistencies of the original 2CMM (negative*k*
_{p}), provides a more reliable and precise parametric portrait of glucose metabolism during an IVGTT.

## Acknowledgments

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.

## Appendix

The use of 2CMM in glucose-tolerant or diabetic subjects requires reconsideration of some model assumptions. For instance, values of R_{d0} = 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 α (α = 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
Equation A1which still guarantees positive values for*k*
_{p}.

In the general case, if a ratio β/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 *k*
_{p} and R_{d0}/*Q*
_{1}(*t*)
Equation A2
Equation A3

## 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

- Copyright © 2003 the American Physiological Society