## 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 (S_{G}) and insulin sensitivity (S_{I}) 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 S_{G}, S_{I},
, 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 S_{G}, because this index is a local descriptor of events taking place in the initial portion of the IVGTT. As a consequence, S_{G} 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

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 (S_{G}) and insulin sensitivity (S_{I}). S_{G} and S_{I} are composite parameters, i.e., they measure the overall effect of glucose and insulin, respectively, to enhance glucose disappearance (R_{d}) and inhibit endogenous glucose production (EGP). To segregate the effect of glucose and insulin on R_{d} 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 inde*x*es (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 S_{G} is likely to be overestimated. Saad et al. (32) have shown that S_{I} 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 S_{G} associated with decreased insulin response is an artifact of the minimal model method and that S_{G} 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 S_{G} 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

*A*_{1},*A*_{2}; ,- 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} - EGP
_{b} - 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 - g
_{b} - Plasma glucose concentration in basal state, mg/dl
- g
_{2}(*t*),g̃*_{2}(*t*) - Cold and hot glucose concentration in the second pool of the two-compartment model, mg/dl and dpm/ml, respectively
- g̃
_{2}(*t*),g̃*_{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} - GE
_{b} - 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} *k*_{21},*k*_{12},*k*_{02},*k*_{d,}- Rate parameters of the two-compartment model, min
^{−1} *k*_{22}*k*_{22}=*k*_{12}+*k*_{02}, min^{−1}*k*_{a}- Rate constant of the remote insulin compartment in the two-compartment model, min
^{−1} *k*_{bd},*k*_{bp}- Parameters describing insulin effect on glucose uptake and EGP in the two-compartment model, min
^{−2}⋅ ml ⋅ μU^{−1}, respectively *k*_{p}- Parameter describing glucose effect on EGP in the two-compartment model, min
^{−1} - i(
*t*) - Insulin concentration in plasma, μU/ml
- i
_{b} - 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 - PCR
_{b} - Plasma glucose clearance in the basal state, ml ⋅ kg
^{−1}⋅ min^{−1} *p*_{1}, p_{2}; ,- Cold and hot minimal model rate parameters, min
^{−1} *q*_{i}(*t*), (*t*)- Cold and hot glucose mass in
*i*th compartment of the two-compartment model (*i*= 1, 2), mg and dpm, respectively - R
_{d}(*t*) - Glucose disappearance rate from the accessible pool, mg ⋅ kg
^{−1}⋅ min^{−1} - R
_{d,0} - Nonzero intercept of the relationship R
_{d}vs. g, mg ⋅ kg^{−1}⋅ min^{−1} - S
_{G}, - Minimal model estimates of cold and hot glucose effectiveness, min
^{−1} - S
_{G(clamp)},S_{G,d(clamp)} - Glucose clamp measurements of cold and hot glucose effectiveness, ml ⋅ kg
^{−1}⋅ min^{−1} - S
_{I}, - Minimal model estimates of cold and hot insulin sensitivity, min
^{−1}⋅ μU ⋅ ml^{−1} - S
_{I(clamp)},S_{I,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
- V
_{1} - Volume of the accessible pool of the two-compartment model, ml/kg
- V
_{T} - 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*=*x*_{p}+*x*_{d}, min^{−1} *X̃*(*t*)- As above, with insulin-dependent removal moved to the accessible pool, i.e.,
*X̃*= x_{p}+*x̃*_{d}, min^{−1} *x*_{d}(*t*)- Two-compartment model insulin action on glucose uptake, min
^{−1} *x̃*_{d}(*t*), (*t*)- As above, with insulin-dependent removal moved to the accessible pool (the asterisk denotes tracer-based calculation), min
^{−1} *x*_{p}(*t*)- Two-compartment model insulin action on EGP, min
^{−1} - α(
*t*) - Deviation of hot glucose decay from a two-exponential function during an IVGTT at basal insulin, dpm/ml
- γ
- γ =
*k*_{21}*k*_{12,}min^{−2} - λ
_{1}, λ_{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
Equation 1where g is plasma glucose concentration (g_{b} denotes its basal end test value), i is plasma insulin concentration (i_{b} 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* = (*k*
_{4}+*k*
_{6})i′, where i′ is insulin in the remote compartment], and the*p*
_{i} values are parameters related to the *k*
_{i} values:*p*
_{1} =*k*
_{1}+*k*
_{5},*p*
_{2} =*k*
_{3},*p*
_{3} =*k*
_{2}(*k*
_{4}+*k*
_{6}).

Parameters *p*
_{1},*p*
_{2},*p*
_{3}, 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, S_{G}, and insulin sensitivity, S_{I}, as
Equation 2S_{G} and S_{I} measure the effects of glucose and insulin, respectively, on both R_{d} and EGP. In fact, because S_{G} is a function not only of*k*
_{1}, but also of*k*
_{5} (see Fig. 1), it measures the ability of glucose at basal insulin to stimulate R_{d} and to inhibit EGP. Similarly, S_{I} is a function not only of*k*
_{1},*k*
_{3},*k*
_{4}, but also of*k*
_{6}, and thus measures the ability of insulin to enhance the glucose stimulation of R_{d} and inhibition of EGP. Parameter *p*
_{2} is the rate constant of the remote insulin compartment and governs the speed of rise and decay of insulin action.

Reference values for S_{G} and S_{I} have been obtained from the analysis of insulin and cold glucose data of a hot IVGTT performed in 25 normal young adults. Values for S_{G} and S_{I} were, respectively, 0.026 ± 0.002 min^{−1} and 7.3 ± 1.0 × 10^{−4}min^{−1} ⋅ μU^{−1} ⋅ ml. The mean precision of S_{G} and S_{I} 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 R_{d}, the hot model yields indexes measuring glucose and insulin effect on R_{d} only. The model in its uniquely identifiable parametrization (2, 17, 19, 23) is described by
Equation 3where 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** =*k*
_{4}i′*), and the
values are parameters related to the*k _{i}
* values:
=

*k*

_{1}, =

*k*

_{3}, and =

*k*

_{2}

*k*

_{4}.

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
Equation 4
measures the ability of glucose at basal insulin to stimulate R_{d}, and
measures the ability of insulin to enhance glucose stimulation of R_{d}. 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-^{2}H_{2}]glucose and [2-^{2}H]glucose) were employed in 19 studies, whereas a radioactive isotope ([3-^{3}H]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): S_{G} is about three times higher than
(*P* < 0.001), and S_{I} 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 S_{G} and
and between S_{I} and
(Fig.3). Whereas a strong correlation exists between S_{I} and
(*r* = 0.84,*P* < 0.001), S_{G} and
are uncorrelated (*r* = 0.17,*P* > 0.15).

Some of the above results are unexpected and suggest the presence of some model error. S_{G} 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, S_{G(clamp)}, and hot, S_{G,d(clamp)}, glucose effectiveness (subscript “d” denotes disappearance). In fact, whereas S_{G} is about three times higher than
, S_{G(clamp)} is only 1.5 times higher than S_{G,d(clamp)} (11). Also, the complete lack of correlation between S_{G} and
is surprising, because S_{G(clamp)} and S_{G,d(clamp)} are presumably well correlated, given that S_{G,d(clamp)}is the major determinant (∼2/3) of S_{G(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 R_{d} and EGP have the same timing, but the time lag between *x* and*x** (caused by*p*
_{2} 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).

Finally, the finding S_{I} <
is unexpected, because S_{I}, which measures insulin effect on both R_{d} and EGP, should be higher than
, which measures insulin effect on R_{d} only. This incongruity is not present when insulin sensitivity is assessed with the glucose clamp technique: in Ref. 10S_{I(clamp)} exceeded S_{I,d(clamp)} [denoted as S_{I,p(clamp)} in that paper] in each subject, with S_{I(clamp)}and S_{I,d(clamp)} being the clamp version analogous to S_{I} 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 . 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
). 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. Table1 reports the mean results of the identification of the two minimal models from simulated IVGTT data. The values of S_{G}, S_{I},
, and
are similar to those reported in the literature. In particular, S_{I}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
). Of note is that all the experimentally observed inconsistencies between cold and hot parameters are present: S_{I} is lower than
, S_{G} is twice
, and hot insulin action is faster than cold because
>*p*
_{2} (e.g., for the simulated standard IVGTT,
= 0.069 vs.*p*
_{2} = 0.027 min^{−1}).

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 inappendix
). 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, S_{G} and S_{I} 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 Table2. 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/V_{T} = 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.

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 S_{G}

To examine the effects of the monocompartmental approximation on S_{G}, we build on Ref. 18 and, for the sake of clarity, we outline the reasoning followed in that paper. Usually, S_{G} 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 S_{G} 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
Equation 5Solving *Eq. 5
* for glucose concentration and defining Δg(*t*) = g(*t*) − g_{b}, one has
Equation 6Thus the minimal model predicts that the decay of glucose concentration during an IVGTT at basal insulin is monoexponential, with S_{G} as rate constant. The fractional decay rate of incremental glucose concentration (*k*
_{G}, min^{−1}), namely the fraction of glucose concentration above basal that declines per unit time, is constant and equal to S_{G}
Equation 7The true glucose system, however, is not monocompartmental. Using the two-compartment model presented in the previous section, one can show (appendix
) that glucose decay during an IVGTT at basal insulin is described by two exponentials
Equation 8where λ_{1} and λ_{2}(min^{−1}) are the fast and slow components of glucose decay, respectively (λ_{1} > λ_{2}). Because of the presence of two time constants, the fractional decay rate of incremental glucose concentration is no longer constant, but time varying
Equation 9In particular,*k*
_{G}(*t*) is higher at the beginning of the IVGTT, when the fast component of glucose decay (λ_{1}) plays an important role, and lower at the end of the IVGTT, when only the slow component (λ_{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 S_{G} and V values reported in Table2. 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 S_{G}(0.021 min^{−1}) lies between the values that*k*
_{G} takes on between 10 and 20 min [e.g.,*k*
_{G}(*minute 15*) = 0.023 min^{−1}]. These results suggest that the validity of S_{G} as descriptor of the effect of glucose per se is confined to the initial portion of the IVGTT. The local validity of S_{G} is probably related to the fact that, during an IVGTT with a normal insulin response, S_{G} 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, S_{G} not only reflects glucose effects on R_{d} and EGP but also the rapid exchange of glucose between the accessible and the nonaccessible compartments occurring in the early part of the test.

### Validation of S_{G}

Validation of S_{G} entails its comparison with the analogous index measured with the glucose clamp method, S_{G(clamp)}. In comparing S_{G} with S_{G(clamp)}, one is faced with the problem that such indexes have different units: S_{G} is expressed in min^{−1}, whereas S_{G(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 S_{G} 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 S_{G} by V parallels the approach used in the validation studies of S_{I} (10, 32).

The value of S_{G}V found experimentally in the present study (4.2 ml ⋅ kg^{−1} ⋅ min^{−1}) is much higher than the value of S_{G(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 S_{G}V obtained from our Monte Carlo study is compared with the glucose effectiveness index of the two-compartment model (see Table 2). The reason for S_{G}V being almost twice S_{G(clamp)} is that S_{G} and S_{G(clamp)} reflect different combinations of the fast and slow components of glucose disappearance at basal insulin, λ_{1} and λ_{2}. We have shown that S_{G} reflects the values that*k*
_{G} takes on between 10 and 20 min. Thus, from *Eq.9
*, one has
Equation 10where*t*
_{0} is ∼15 min in subjects with a normal insulin response. *Equation10
* shows that S_{G}is influenced by both the fast and slow components of glucose disappearance. To compare quantitatively S_{G}V with S_{G(clamp)}, it is useful to express S_{G} as a function of λ_{2} only. By exploiting the fact that*A*
_{1}λ_{1}
≈ 2*A*
_{2}λ_{2}
and*A*
_{2}
≈ 6*A*
_{1}
(the values of*A*
_{1},*A*
_{2}, λ_{1}, and λ_{2} reported in Table D1), one has
Equation 11S_{G(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. EC1
* to a hyperglycemic clamp at basal insulin, one finds that S_{G(clamp)} is defined as the ratio of Δ(R_{d} − 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, ΔGINF, equals Δ(R_{d} − EGP), S_{G(clamp)} is defined as follows
Equation 12 Using the two-compartment model to describe the glucose system during the clamp, one can express S_{G(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
)
Equation 13It is easy to show that S_{G(clamp)} is primarily determined by the slow component of glucose disappearance. In fact,*A*
_{2}/λ_{2}≈ 18*A*
_{1}/λ_{1}, and thus S_{G(clamp)} ≈ λ_{2}/*A*
_{2}. Moreover, because 1/*A*
_{2} approximates the total glucose distribution volume V_{T} (28), and V_{T} ≈ 1.3 V (see Tables 1 andTA1), we can write
Equation 14By comparing *Eqs. 11
* and *
14
*, one realizes why S_{G}V is about twice S_{G(clamp)}. It is worth pointing out that S_{G}V and S_{G(clamp)} are not only quantitatively, but also qualitatively different; S_{G}V also reflects, in addition to glucose effect on R_{d} and EGP [measured by S_{G(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 S_{G}with S_{G(clamp)} in dogs (24). To allow the comparison, the authors divided S_{G(clamp)} by the total volume of glucose distribution, V_{T}, taken from the literature (250 ml/kg). They found that S_{G} was higher than the ratio of S_{G(clamp)} to V_{T} and that such indexes were poorly correlated. These findings seem to support the notion that S_{G} and S_{G(clamp)} reflect different aspects of glucose effect per se. However, as we pointed out in Ref.16, using V_{T} to convert S_{G(clamp)} to the same units as S_{G} is questionable because, as we have discussed, the minimal model yields an index of glucose effectiveness, S_{G}V, that has the same units of S_{G(clamp)} and hinges on a volume that, in contrast to a mean value of V_{T}, can be individualized in each subject.

### S_{G} from an IVGTT at Basal Insulin

It is commonly believed that S_{G}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 S_{G} is uncertain because the minimal model forces a monoexponential function to describe a two-exponential decay. To determine whether S_{G} 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 GE_{b} 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(Δg)]. In the case of an IVGTT at basal insulin, with the assumption that glucose decay follows the two-exponential profile of*Eq. 8
*, GE_{b} is given by
Equation 15 Note that the expression of GE_{b} in*Eq. 15
* coincides with that of S_{G(clamp)} in *Eq.13
*, in keeping with the analysis carried out in Ref. 4that 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, S_{G}V resulted in a value similar to GE_{b}. 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 S_{G} 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 S_{G} estimation. Because the contribution of the fast component of glucose disappearance, λ_{1}, soon becomes negligible (e.g., after ∼30 min in normal subjects), and most of the glucose data are beyond that point in time, S_{G} will approach the slow component of glucose disappearance, λ_{2}, and the minimal model volume will approach the reciprocal of*A*
_{2}. Therefore, S_{G}V is approximated by
Equation 16 Comparison of *Eq. 16
* with *Eqs.10
* and *
11
* sheds some light on the reasons why the value of S_{G} obtained from an IVGTT at basal insulin has been found to be lower than that obtained from an insulin-modified IVGTT (24): whereas the S_{G} estimated during an insulin-modified IVGTT reflects both the fast and slow components of glucose disappearance, the S_{G}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, S_{G}V will be close to GE_{b} if*A*
_{2}/λ_{2}>>*A*
_{1}/λ_{1}. Because*A*
_{2}/λ_{2}≈ 18*A*
_{1}/λ_{1}, it is likely that S_{G}V estimated from an IVGTT at basal insulin is a reliable estimate of glucose effectiveness.

S_{G} measured from an IVGTT at basal insulin has been compared with S_{G(clamp)} in dogs by Finegood and Tzur (24). They found similar values for S_{G} and S_{G(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 S_{G} and worsened its concordance with S_{G(clamp)}.

## COLD INSULIN SENSITIVITY

### Effects of Monocompartmental Undermodeling on S_{I}

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 S_{G} 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 S_{I}, 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 S_{I} 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,*x*
_{d}(*t*) is insulin action on glucose uptake, and*x*
_{p}(*t*) is insulin action on production. Thus*X*(*t*) =*x*
_{p}(*t*)+*x*
_{d}(*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.7
*A*.*X*(*t*) was generated using the two-compartment model parameters reported in Table A1;*x*(*t*) was generated using the mean parameters S_{I} and*p*
_{2} estimated with the Monte Carlo simulation described in appendix
(S_{I} = 2.9 × 10^{−4}min^{−1} ⋅ μU^{−1} ⋅ ml and *p*
_{2} = 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
the effect that*x*
_{d}(*t*) produces on the accessible pool of the two-compartment model. We termed this effect as*x̃*(*t*).*X̃*(*t*) =*x*
_{p}(*t*) +*x̃*
_{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 R_{d}remains the same).*X̃*(*t*) is shown in Fig. 7
*B* plotted against the insulin action of the minimal model. Qualitatively speaking,*X̃*(*t*) is a delayed and blunted version of*X*(*t*).

The difference Δ_{x}(*t*) =*X̃*(*t*) −*x*(*t*) represents the effect of monocompartmental undermodeling on the minimal model insulin action. Δ_{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 Δ_{x}(*t*) can be derived (details in appendix
) that helps to single out the determinants of the bias
Equation 17where GE/V_{1} is the fractional glucose effectiveness of the two-compartment model, γ =*k*
_{21}
*k*
_{12}and *k*
_{22} = (*k*
_{02}+*k*
_{12}), andg̃_{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 Δ_{x}(*t*) derived under the assumption of perfect minimal model fit agrees with the profile of Δ_{x}(*t*) calculated by subtracting the profiles of*x*(*t*) and displayed in Fig. 7
*B* [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 Δ_{x}(*t*) are compared in Fig. 7
*C*. 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 Δ_{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 Δ_{x} depends not only on S_{G} overestimation of the fractional glucose effectiveness of the two-compartment model (GE/V_{1} = 0.0133 min^{−1}) but also on the dynamics of glucose in the second glucose compartment. The first component of Δ_{x}(*t*), i.e., the bias affecting the estimate of glucose effectiveness (S_{G} − GE/V_{1}), is weighted by the term Δg(*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 Δ_{x}(*t*). In fact, Δ_{x}(*t*) becomes positive at 20 min, i.e., approximately wheng̃_{2}(*t*) achieves its maximum, and returns negative at 140 min, just wheng̃_{2}(*t*) achieves its minimum. This implies that the bias affecting insulin action cannot be obviated completely by forcing S_{G} to assume a more accurate value.

The bias of insulin action obviously affects S_{I}. The difference between the fractional insulin sensitivity of the two-compartment model (IS/V_{1}) and S_{I} is given by *Eq. EF7
*in appendix
Equation 18 *Equation18
* indicates that the bias of S_{I} depends on the whole time course of insulin action, so that compensations may occur between portions of the IVGTT when Δ_{x}is positive and portions when Δ_{x} is negative. Thus overestimation of S_{G} does not necessarily imply that S_{I} 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, S_{I}underestimates the fractional insulin sensitivity of the two-compartment model by 55% (see Tables 1, 2, andTA1)**.** When S_{I} is multiplied by the minimal model volume, V, to allow comparison with IS (10, 32), underestimation reduces to 43% because V is higher than V_{1}. Another observation is that the bias affecting S_{I} may be scarcely influenced by the value assumed by S_{G}, especially during a modified IVGTT in normal subjects. In fact, under such experimental conditions and in such a group of people, the integral of Δg(*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 Δg(*t*)/g(*t*) measures the effect that a unit change in S_{G} produces on S_{I}, it is likely that the sensitivity of S_{I} to errors in S_{G} is small.

One may wonder whether the underestimation of S_{I} 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[Δ_{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 S_{I}. In fact, S_{I}V 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 S_{I}. In our simulation the improvement of S_{I} accuracy was primarily due to the increment in AUC[i(*t*) − i_{b}]. In fact, although the added burst of insulin steepened the glucose curve, thus making Δ_{x}(*t*) less sluggish than during the standard IVGTT, AUC[Δ_{x}(*t*)] did not change much.

One final remark concerns the effect of monocompartmental undermodeling on the estimation of S_{I} in non-insulin-dependent diabetes mellitus (NIDDM). Many reports have shown that S_{I} 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 S_{I}. 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 S_{I}. In addition, even when S_{I} can be precisely estimated, its value will be markedly underestimated. Underestimation of S_{I} 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 S_{I}

Validation of S_{I} entails its comparison with S_{I(clamp)}, i.e., the glucose clamp measure of insulin sensitivity. By applying the formal definition of insulin sensitivity reported in*Eq. EC3
* to a euglycemic, hyperinsulinemic clamp, one finds that S_{I(clamp)} is the steady-state ratio of Δ(R_{d} − 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 Δ(R_{d} − EGP), S_{I(clamp)} is defined as follows (7, 10, 32)
Equation 19 To compare S_{I} with S_{I(clamp)}, S_{I} is usually multiplied by the glucose distribution volume of the minimal model, V (10, 32). S_{I}V and S_{I(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 S_{I}V and S_{I(clamp)} is controversial in the literature.

In the study by Beard et al. (5) in normotolerant subjects, S_{I} was measured with the tolbutamide-modified IVGTT, and S_{I(clamp)} was measured with sequential low-insulin-dose euglycemic clamps, bringing insulin to plateaus of 21 and 35 μU/ml. A strong correlation between S_{I} and S_{I(clamp)} was found (*r* = 0.84). However, if one calculates the product S_{I}V [using for V a typical value of 170 ml/kg (10, 17)], one finds that S_{I}V was approximately 60% lower than S_{I(clamp)} (0.11 vs. 0.29 ml ⋅ kg^{−1} ⋅ min^{−1} ⋅ μU^{−1} ⋅ ml).

Bergman et al. (10) compared S_{I}V and S_{I(clamp)} in a group of normotolerant and obese subjects. S_{I}V was measured with the tolbutamide-modified IVGTT, and S_{I(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 S_{I}V was slightly lower than S_{I(clamp)} in all subjects but one, S_{I}V was equivalent to S_{I(clamp)} (0.046 vs. 0.052 dl ⋅ min^{−1} ⋅ μU^{−1} ⋅ ml, respectively).

Saad et al. (32) compared S_{I}V with S_{I(clamp)} in normal controls, subjects with impaired glucose tolerance (IGT), and patients with NIDDM. S_{I} was measured with the insulin-modified IVGTT (insulin bolus of 0.03 U/kg at 20 min), while S_{I(clamp)} was measured with a single high-insulin-dose euglycemic clamp, bringing insulin levels to 83 μU/ml. Although results show S_{I}V and S_{I(clamp)} well correlated in normal and IGT subjects, S_{I}V was >50% lower than S_{I(clamp)}.

More recently, Saad et al. (33) measured S_{I}V in normal controls with both the tolbutamide- and the insulin-modified IVGTT, while S_{I(clamp)} was measured with the same insulin infusion used in Ref. 32. S_{I}V from the tolbutamide-modified IVGTT was only 13% lower than S_{I(clamp)} (0.045 vs. 0.054 dl ⋅ min^{−1} ⋅ μU^{−1} ⋅ ml), whereas S_{I}V from the insulin-modified IVGTT (0.030 dl ⋅ min^{−1} ⋅ μU^{−1} ⋅ ml) was 44% lower than S_{I(clamp)}.

Comparison of the results obtained in the above-mentioned studies suggests that equivalence of S_{I}V and S_{I(clamp)} depends on how the IVGTT and the clamp are performed. For instance, the marked difference between S_{I}V and S_{I(clamp)} that is present when either S_{I}V is estimated from an insulin-modified IVGTT (32) or S_{I(clamp)} is estimated from low-dose insulin clamps (5) vanishes when S_{I}V is estimated from a tolbutamide-boosted IVGTT and S_{I(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 S_{I} and S_{I(clamp)}. For S_{I} to be equivalent to S_{I(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 R_{d} 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 S_{I} underestimation. We now analyze how the other factors can influence the estimation of S_{I} and S_{I(clamp)}.

#### Linearity of insulin effect.

Both S_{I} and S_{I(clamp)} measure the ability of insulin not only to increase R_{d}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 R_{d} 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 S_{I(clamp)} can be derived from the study of Katz et al. (26). The data reported in that dose-response study allow calculation of S_{I(clamp)} at three insulin levels: 25, 43, and 123 μU/ml. S_{I(clamp)}shows results of 0.19, 0.21, and 0.10 ml ⋅ kg^{−1} ⋅ min^{−1} ⋅ μU^{−1} ⋅ ml, respectively. These values indicate that S_{I(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 R_{d}. That S_{I(clamp)} depends on the insulin level at which the clamp is performed can also be inferred by comparing the values of S_{I(clamp)} in normal subjects obtained in the studies by Beard et al. (5) and Saad et al. (32). S_{I(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 S_{I(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 R_{d}) is unable to influence S_{I} appreciably (comprising both the effects of insulin on R_{d} 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 S_{I} 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 S_{I} with S_{I(clamp),} we assume that the peripheral and portal routes of insulin delivery are equally effective in inhibiting EGP. Unlike S_{I(clamp)}, which measures insulin ability to suppress EGP and elevate R_{d} in response to peripheral insulin delivery, S_{I} 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 S_{I} 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 S_{I}.

#### Effects of tolbutamide.

Saad and colleagues (33, 34) have shown that the tolbutamide-boosted protocol provides higher S_{I}estimates than the insulin-modified protocol, regardless of the method of insulin administration (bolus or 5- or 10-min infusion). The higher S_{I} 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 S_{I} with the tolbutamide protocol with respect to those obtained with the insulin protocol. Alternatively, differences in S_{I}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 S_{I} with respect to S_{I(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 S_{G}, 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
Equation 20 Thus the fractional decay rate of the tracer will be constant and equal to
Equation 21In contrast, the two-compartment model predicts that the hot glucose decay is described by an almost two-exponential profile (appendix
)
Equation 22 where
and
(
>
, min^{−1}) are the fast and slow rate constants of glucose kinetics in the basal state, respectively, and the term α(*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 α(*t*), and then increases slowly, getting closer and closer to
.

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, PCR_{b}, 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)
Equation 23 Given that
/
∼ 14
/
(see Table D1), *Eq. 23
* reduces to
Equation 24 Because
approximates
and 1/
approximates V_{T} (28), one has
Equation 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/V_{T} = 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 R_{d} vs. g relationship can be approximated by a line that has a nonzero intercept with the R_{d} 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, S_{G,d(clamp)}. S_{G,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 R_{d} (12). By applying the definition of hot glucose effectiveness reported in*Eq. EC2
* to a hyperglycemic clamp at basal insulin, one finds that S_{G,d(clamp)} is the slope of the linear relationship between the R_{d}and g
Equation 26 where R_{d,0} is the nonzero intercept. Recalling that
measures fractional basal glucose clearance (*Eq.25
*), and using the definition of basal glucose clearance (*Eq. EC5
*), one has
Equation 27 Combining*Eqs. 26
* and *
27
* yields
Equation 28 *Equation28
* confirms that
does not coincide with S_{G,d(clamp)}(apart from the volume factor) because of the presence of the nonzero intercept R_{d,0}. Thus
cannot be used as an index of glucose effectiveness unless the presence of R_{d,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 R_{d,0}. Both
and R_{d,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 R_{d,0} from hot IVGTT data extremely difficult (17).

### S_{G} vs.

The analysis of the relationships S_{G} vs. S_{G(clamp)} and
vs. S_{G,d(clamp)} suggests a possible explanation for the lack of correlation between S_{G} and
found in this study. One would expect to find a good correlation between S_{G} and
because glucose clamp studies have shown that glucose effectiveness on R_{d} 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, S_{G} and
are not equivalent to S_{G(clamp)} and S_{G,d(clamp)}, respectively. In fact, S_{G} and S_{G(clamp)} measure related, but not identical, physiological processes, because S_{G}, at variance with S_{G(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 R_{d}, 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 S_{I}. 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,*x*
_{d}(*t*), with the hot minimal model insulin action,*x**(*t*) (Fig.9
*A*). 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
the effect that*x*
_{d}(*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. 9
*B*. 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
)
Equation 29where GE*/V_{1} is the fractional hot glucose effectiveness of the two-compartment model;g̃*_{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.9
*C* against the profile of
obtained from the difference between the profiles of
and*x** displayed in Fig.9
*B*. 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 Δ_{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 R_{d,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.

The bias affecting
can be calculated as previously done for S_{I}
Equation 30 As for S_{I}, the bias of S_{I} depends on the whole time course of insulin action, so that compensations may occur between portions of the IVGTT when Δ*_{x} is positive and others when Δ*_{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 S_{I} cannot be estimated or is estimated with poor precision.

### Validation of

has not yet been validated against the analogous clamp-based index S_{I,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 S_{I,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.

### S_{I} vs.

The error analysis of cold and hot insulin action and sensitivity explains the paradoxical finding S_{I} <
. In fact, whereas the underestimation affecting S_{I} 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. 7
*B* and9
*B*, 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*p*
_{2} 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.

S_{G} 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 R_{d} 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 S_{G} is nonnegligible, as suggested by the simulation studies in Refs. 22 and 38, the reliability of S_{G} as descriptor of glucose effectiveness is uncertain.

The effects of single-compartment undermodeling on S_{G} determine undesired compensations on cold insulin action and S_{I}. Insulin action is markedly underestimated for a considerable portion of the IVGTT. S_{I}, which is proportional to the integral of insulin action, is also markedly underestimated. Although the S_{I} estimated from an insulin-modified or a tolbutamide-boosted IVGTT strongly correlates with S_{I(clamp)}, it must be pointed out that both S_{I} and S_{I(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 S_{I}. 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 S_{G} and S_{I}, 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 S_{G} and S_{I} 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.

## 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 R_{d} and EGP, both of these processes are described by the model. To describe R_{d}, 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
Equation A1 where*k*
_{d} accounts for the proportional term, and R_{d,0} is the nonzero intercept of the steady-state relationship R_{d} vs. g. Because*k*
_{01} 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 *k*
_{02}
Equation A2 where*x*
_{d}(*t*) is insulin action on glucose uptake originating from an insulin compartment remote from plasma. The dynamics of*x*
_{d}(*t*) is governed by parameters*k*
_{bd} and*k*
_{a}, 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
Equation A3The 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,*x*
_{p}(*t*), assumed to occur with the same timing as insulin action on R_{d.} This is the reason why the remote insulin compartments from which*x*
_{p}(*t*) and*x*
_{d}(*t*) originate have the same rate constant,*k*
_{a}. In contrast, parameters *k*
_{bd}and *k*
_{bp} are different, with*k*
_{bd} being higher than *k*
_{bp}, because it is known from clamp studies that insulin effect is greater on R_{d} than on EGP.

The model equations for cold and hot glucose are

where*q _{i}
*(

*t*) and (

*t*) (

*i*= 1, 2) are, respectively, the cold and hot glucose masses in the

*i*th compartment of the model.

The values of the model parameters are reported in Table A1. The values of V_{1},*k*
_{21},*k*
_{12}, and*k*
_{02} were taken from Ref. 21; R_{d,0} was chosen to be equal to the value experimentally determined in Ref.12; *k*
_{d} was determined from *Eq. EA3
*;*k*
_{p}was calculated with the assumption that glucose effectiveness on EGP is one-third of overall glucose effectiveness (11); parameter*k*
_{a} was taken from Ref. 14; parameters*k*
_{bd} and*k*
_{bp} were chosen in such a way that the two-compartment model values of insulin sensitivity related to R_{d} + EGP and to R_{d} only were equal to the analogous clampbased values reported by Saad et al. (32) in normal subjects.

## 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). 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 g_{b}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 S_{G}, S_{I},
, and
, and volumes V and V*. Results for the standard and insulin-modified IVGTT have been reported in Table 1.

## 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 R_{d} and inhibiting EGP. It can be expressed mathematically as the derivative of R_{d}+EGP with respect to glucose concentration at basal steady state
Equation C1 Hot glucose effectiveness measures glucose effect on R_{d} only and is defined as
Equation C2Insulin sensitivity is defined as the ability of insulin to enhance glucose effectiveness. Whereas cold insulin sensitivity measures insulin effect on both R_{d} and EGP, hot insulin sensitivity refers to R_{d} only
Equation C3
Equation C4Plasma clearance rate is the ratio between R_{d} and plasma glucose concentration at steady state
Equation C5Applying the above definitions to the two-compartment model described inappendix
, 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.

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, S_{G} 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 S_{G}, S_{I},
, 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 R_{d} is not proportional to g, thus determining the presence of the nonzero intercept R_{d,0}. In contrast, because the hot minimal model assumes that R_{d} is proportional to g, one has R_{d,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 (Δ), rather than differential (δ), 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 R_{d} and EGP, the hot indexes reflect changes in R_{d} only.

## 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

In the basal steady state, EGP equals R_{d} and thus
Equation D2Substitution of EGP_{b} given by*Eq. ED2
* into *Eq.ED1
* permits elimination of the term containing R_{d,0}. Using the position Δ*q _{i}
*(

*t*) =

*q*(

_{i}*t*) −

*q*for

_{ib}*i*= 1, 2, one obtains

Equation D3
where δ(*t*) is the Dirac impulse function. *Equation ED3
* 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
Equation D4where λ_{1} and λ_{2} are the fast and slow eigenvalues, respectively. Of note is that*A*
_{1},*A*
_{2}, λ_{1}, and λ_{2} reflect the ability of glucose not only to promote R_{d} 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. ED3
* and equating it to the Laplace transform of the two-exponential decay of *Eq.ED4
* (13).

The fractional decay rate of incremental glucose concentration,*k*
_{G}, which measures the fraction of glucose concentration above basal that declines per unit time, is given by
Equation D5

### Hot Glucose Decay

During a hot IVGTT at basal insulin, the two-compartment model equations for hot glucose become

Equation D6
If R_{d,0} were equal to zero, the system described by *Eq. ED6
*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
Equation D7where
and
are the fast and slow eigenvalues of glucose kinetics in the basal state. Of note is that parameters
,
,
, and
reflect R_{d} 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. ED6
* with R_{d,0} = 0 and equating it to the Laplace transform of the two-exponential decay of *Eq.ED7
* (13).

However, because R_{d,0} is greater than zero, hot glucose decay is not two-exponential
Equation D8where the term α(*t*) is the deviation of the true hot glucose decay from the two-exponential function of *Eq. ED7
*. The fractional decay rate of hot glucose during the IVGTT at basal insulin,
(*t*), is the ratio g(*t˙*)*/g(*t*). The time course of
(*t*) has been shown in Fig. 8.

## S_{G(CLAMP)} FROM THE PARAMETERS OF THE TWO-COMPARTMENT MODEL

Here we show that the clamp-based measure of glucose effectiveness, S_{G(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.

### S_{G(clamp) }and the Two-Compartment Model

S_{G(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.EC1
* to a hyperglycemic clamp at basal insulin, one finds that S_{G(clamp)} corresponds to the slope of the steady-state relationship between the exogenous glucose infusion rate, GINF, and plasma glucose concentration
Equation E1To express S_{G(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
for an IVGTT at basal insulin except that the exogenous glucose input is a variable glucose infusion, GINF(*t*), instead of a bolus injection
Equation E2
When an elevated steady state for glucose is achieved, the derivatives of Δ*q*
_{1} and Δ*q*
_{2} become null, and *Eq. EE2
* yields the following relationship between ΔGINF and Δg
Equation E3By substituting *Eq. EE3
* into*Eq. EE1
*, one obtains
Equation E4It is of interest that the expression of S_{G(clamp)} in *Eq.EE4
* is identical to the index of cold glucose effectiveness, GE, of the two-compartment model (see Table C1).

### S_{G(clamp)} and the IVGTT at Basal Insulin

S_{G(clamp)} can also be expressed as a function of the impulse response parameters*A*
_{1},*A*
_{2}, λ_{1}, and λ_{2}, which describe glucose decay during an IVGTT at basal insulin. To do so one has to integrate from zero to infinity *Eq. ED3
*, which describes the glucose glucose system during an IVGTT at basal insulin
Equation E5Because Δg(*t*) is a two-exponential impulse response, AUC[Δg(*t*)] can also be expressed as a function of the impulse response parameters*A*
_{1}, *A*
_{2}, λ_{1}, and λ_{2}
Equation E6By equating *Eqs. EE5
* and *
EE6
* and remembering that the denominator of *Eq. EE5
* is equal to S_{G(clamp)} (see*Eq. EE4
*), one obtains
Equation E7

## 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,*x*
_{d}(*t*) is insulin action on glucose disposal, and*x*
_{p}(*t*) is insulin action on glucose production. The overall insulin action of the two-compartment model is thus*X*(*t*) =*x*
_{p}(*t*)+*x*
_{d}(*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*X̃*(*t*), has been derived.*X̃*(*t*) =*x*
_{p}(*t*) +*x̃*
_{d}(*t*), where*x̃*
_{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*x*
_{d}(*t*) on plasma glucose concentration. Application of this definition to the two-compartment model (*Eq. EA4
*) leads to the following expression for*x̃*
_{d}(*t*)
Equation F1where*q̃*
_{2}(*t*) is glucose mass in the second compartment when the insulin-dependent glucose removal is moved to the accessible pool. Calculation of*q̃*
_{2}(*t*) requires solution of the mass-balance equation of the nonaccessible glucose pool (second equation in A4) without the insulin-dependent term
Equation F2 *Eq. EF1
* shows that*x̃*
_{d}(*t*) is such that the increase in the flux irreversibly leaving the accessible compartment,*x̃*
_{d}(*t*)*q*
_{1}(*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 Δ_{x}(*t*) = X̃(*t*)−*x*(*t*) is the bias affecting the minimal model insulin action due to monocompartmental undermodeling. Δ_{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, Δ_{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)
Equation F3The fractional decay rate of the two-compartment model can be calculated from *Eq. EA1
*
Equation F4where γ =*k*
_{21}
*k*
_{12},*k*
_{22} = (*k*
_{02}+*k*
_{12}), andg̃_{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.EF3
* and *
EF4
* one obtains an expression for Δ_{x}(*t*)
Equation F5 *Equation EF5
* 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 S_{I}. In Ref. 18 we have shown that S_{I} is proportional to the integral of insulin action
Equation F6The fractional insulin sensitivity of the two-compartment model, IS/V_{1}, is well approximated by the integral of X̃(*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*X̃*(*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
Equation F7Analogous 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 R_{d} (applied to the accessible pool) and the hot minimal model insulin action, is given by
Equation F8where
(*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.EA1
*) so as to produce the same effect on hot glucose concentration as*x*
_{d}(*t*); GE*/V_{1} is fractional hot glucose effectiveness of the two-compartment model;g̃*_{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 Δ_{x}(*t*) but includes an additional term proportional to R_{d,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 R_{d} of the two-compartment model and
, can be calculated as previously done for S_{I}
Equation F9where the hot fractional insulin sensitivity of the two-compartment model, IS*/V_{1}, 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.

- Copyright © 1999 the American Physiological Society