INTRODUCTION

TOP ABSTRACT INTRODUCTION THE COLD AND HOT... A TWO-COMPARTMENT MODEL OF... COLD GLUCOSE EFFECTIVENESS COLD INSULIN SENSITIVITY HOT GLUCOSE EFFECTIVENESS HOT INSULIN SENSITIVITY CONCLUSIONS REFERENCES APPENDIX A APPENDIX B APPENDIX C APPENDIX D APPENDIX E APPENDIX F

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 indexes (17, 19); in addition, we have observed that when EGP is derived by combining the cold and hot minimal models, its time course is physiologically absurd (17). Quon et al. (30) have shown in a study on insulin-dependent diabetes mellitus patients that 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

A1, A2; ,	Coefficients of two-exponential cold and hot glucose decay during an IVGTT at basal insulin, mg/dl and dmp/ml (for a radiolabeled IVGTT)
D, D*	Cold and and hot glucose IVGTT dose, mg/kg and dpm/kg, respectively
EGP(t)	Endogenous glucose production, mg · kg1 · min1
EGPb	Endogenous glucose production in the basal state, mg · kg1 · min1
g(t), g*(t)	Cold and hot glucose concentration in plasma, mg/dl and dpm/ml, respectively
g(0), g*(0)	Minimal model estimates of cold and hot glucose concentration at time 0+, mg/dl and dpm/ml, respectively
gb	Plasma glucose concentration in basal state, mg/dl
g2(t), *2(t)	Cold and hot glucose concentration in the second pool of the two-compartment model, mg/dl and dpm/ml, respectively
2(t), *2(t)	As above, with insulin-dependent removal moved to the accessible pool, mg/dl and dpm/ml, respectively
GE, GE*	Cold and hot glucose effectiveness of the two-compartment model, ml · kg1 · min1
GEb	Cold glucose effectiveness measured from the area under the glucose excursion during an IVGTT at basal insulin, ml · kg1 · min1
GINF(t)	Glucose infusion rate during the glucose clamp, mg · kg1 · min1
k21, k12, k02, kd,	Rate parameters of the two-compartment model, min1
k22	k22 = k12+k02, min1
ka	Rate constant of the remote insulin compartment in the two-compartment model, min1
kbd, kbp	Parameters describing insulin effect on glucose uptake and EGP in the two-compartment model, min2 · ml · µU1, respectively
kp	Parameter describing glucose effect on EGP in the two-compartment model, min1
i(t)	Insulin concentration in plasma, µU/ml
ib	Plasma insulin concentration in the basal state, µU/ml
IS,IS*	Cold and hot insulin sensitivity of the two-compartment model, ml · kg1 · min1 per µU/ml
PCRb	Plasma glucose clearance in the basal state, ml · kg1 · min1
p1, p2; ,	Cold and hot minimal model rate parameters, min1
qi(t), (t)	Cold and hot glucose mass in ith compartment of the two-compartment model (i = 1, 2), mg and dpm, respectively
Rd(t)	Glucose disappearance rate from the accessible pool, mg · kg1 · min1
Rd,0	Nonzero intercept of the relationship Rd vs. g, mg · kg1 · min1
SG,	Minimal model estimates of cold and hot glucose effectiveness, min1
SG(clamp),SG,d(clamp)	Glucose clamp measurements of cold and hot glucose effectiveness, ml · kg1 · min1
SI,	Minimal model estimates of cold and hot insulin sensitivity, min1 · µU · ml1
SI(clamp),SI,d(clamp)	Glucose clamp measurements of cold and hot insulin sensitivity, ml · kg1 · min1 · µU1 · ml
t	Time, min
V,V*	Cold and hot minimal model volume, ml/kg
V1	Volume of the accessible pool of the two-compartment model, ml/kg
VT	Total glucose distribution volume, ml/kg
x(t), x*(t)	Cold and hot minimal model insulin action, min1
X(t)	Two-compartment model insulin action, i.e., X = xp+xd, min1
(t)	As above, with insulin-dependent removal moved to the accessible pool, i.e.,  = xp + d, min1
xd(t)	Two-compartment model insulin action on glucose uptake, min1
d(t), (t)	As above, with insulin-dependent removal moved to the accessible pool (the asterisk denotes tracer-based calculation), min1
xp(t)	Two-compartment model insulin action on EGP, min1
(t)	Deviation of hot glucose decay from a two-exponential function during an IVGTT at basal insulin, dpm/ml
	= k21k12, min2
1, 2; ,	Fast and slow eigenvalues of the cold and hot glucose decay during an IVGTT at basal insulin, min1

	THE COLD AND HOT MINIMAL MODELS

TOP ABSTRACT INTRODUCTION THE COLD AND HOT... A TWO-COMPARTMENT MODEL OF... COLD GLUCOSE EFFECTIVENESS COLD INSULIN SENSITIVITY HOT GLUCOSE EFFECTIVENESS HOT INSULIN SENSITIVITY CONCLUSIONS REFERENCES APPENDIX A APPENDIX B APPENDIX C APPENDIX D APPENDIX E APPENDIX F

The Cold Model

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View larger version (13K): [in this window] [in a new window]   Fig. 1.   The cold minimal model. See glossary for definition of terms.

Parameters p₁, p₂, p₃, and V can be estimated from glucose and insulin data by use of nonlinear least squares parameter estimation techniques (13). From them one can calculate the cold indexes of glucose effectiveness, S_G, and insulin sensitivity, S_I, as

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S_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 k₁, but also of k₅ (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 k₁, k₃, k₄, but also of k₆, and thus measures the ability of insulin to enhance the glucose stimulation of R_d and inhibition of EGP. Parameter p₂ is the rate constant of the remote insulin compartment and governs the speed of rise and decay of insulin action.

Reference values for 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 min¹ and 7.3 ± 1.0 × 10⁴ min¹ · µU¹ · 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

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View larger version (13K): [in this window] [in a new window]   Fig. 2.   The hot minimal model. See glossary for definition of terms.

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

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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 ([6-²H₂]glucose and [2-²H]glucose) were employed in 19 studies, whereas a radioactive isotope ([3-³H]glucose) was employed in 6 studies. Values for and were, respectively, 0.0082 ± 0.0003 min¹ and 9.0 ± 1.2 × 10⁴ min¹ · µU¹ · 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

View larger version (11K): [in this window] [in a new window]   Fig. 3.   Correlation between cold and hot indexes of glucose effectiveness (A) and insulin sensitivity (B). See glossary for definition of terms.

Some of the above results are unexpected and suggest the presence of some model error. 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 p₂ being lower than ) violates this assumption. In addition, the profile of insulin action on EGP, calculated as the difference x  x*, is physiologically implausible (17).

View larger version (13K): [in this window] [in a new window]   Fig. 4.   Cold and hot insulin action during a standard hot intravenous glucose tolerance test (IVGTT) in a representative subject.

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. 10 S_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

TOP ABSTRACT INTRODUCTION THE COLD AND HOT... A TWO-COMPARTMENT MODEL OF... COLD GLUCOSE EFFECTIVENESS COLD INSULIN SENSITIVITY HOT GLUCOSE EFFECTIVENESS HOT INSULIN SENSITIVITY CONCLUSIONS REFERENCES APPENDIX A APPENDIX B APPENDIX C APPENDIX D APPENDIX E APPENDIX F

To investigate the mechanisms by which single-compartment undermodeling affects the minimal model indexes, we developed a physiologically based two-compartment model to describe the glucose system during the IVGTT. The model, shown in Fig. 5, is described in detail in APPENDIX A. Briefly, the model describes both glucose kinetics and EGP during the IVGTT. The description of glucose kinetics is the same as that of the two-compartment minimal model proposed in Refs. 14 and 37. It is assumed that insulin-independent glucose disposal occurs in the accessible compartment, whereas insulin-dependent glucose disposal occurs in the nonaccessible compartment. Consistent with known physiology, insulin-independent glucose uptake accounts for the inhibitory effect of hyperglycemia on glucose clearance. It consists of two components, one constant and the other proportional to glucose concentration. Insulin-dependent glucose uptake is parametrically controlled by insulin in a remote insulin compartment. The assumption is made that, in the basal state, insulin-dependent glucose disposal is three times insulin-independent glucose disposal. EGP is described using the same functional description embodied in the cold minimal model (8, 17, 19, 23), thus allowing us to focus on the bias due to single-compartment undermodeling only. In fact, EGP inhibition is assumed to be proportional to the increment of glucose concentration above basal and to the product of glucose concentration and insulin action. In addition, as in the minimal model, insulin action on EGP is assumed to have the same timing as insulin action on glucose uptake.

View larger version (19K): [in this window] [in a new window]   Fig. 5.   The two-compartment simulation model. See glossary for definitions of terms.

To ascertain the ability of this model to describe satisfactorily the glucose system during the IVGTT, we used Monte Carlo simulation (details in APPENDIX B). Briefly, the two-compartment model with mean parameters was used to generate noise-free cold and hot glucose data during a hot IVGTT. The mean insulin profile of either a standard or an insulin-modified IVGTT was used as input to the model. Noise of appropriate characteristics was added to the data, and the noisy IVGTT data sets were then interpreted with the minimal models. We reasoned that, if the two-compartment model is a realistic representation of the glucose system during the IVGTT, the minimal model parameters estimated from the simulated data should be close to those estimated from real data and should exhibit the same trends discussed above. In addition, the relationships between the minimal model estimates of glucose effectiveness and insulin sensitivity and the analogous two-compartment model indexes should be similar to those observed experimentally between the minimal model and clamp-based indexes. These hypotheses were all confirmed. Table 1 reports the mean results of the identification of the two minimal models from simulated IVGTT data. The values of 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 A). 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  > p₂ (e.g., for the simulated standard IVGTT,  = 0.069 vs. p₂ = 0.027 min¹).

How do the minimal model indexes of glucose effectiveness and insulin sensitivity compare with the "true" indexes of the two-compartment model? To answer this question we derived indexes of glucose effectiveness, insulin sensitivity, and basal plasma clearance rate for the two-compartment model (details are provided in APPENDIX C). Of note is that these indexes are expressed in the same units as those of the corresponding clamp-based indexes. To express also the minimal model indexes in the same units, 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 Table 2. One can see that the cold minimal model overestimates glucose effectiveness and underestimates insulin sensitivity, in keeping with the experimental results (24, 32). V* slightly underestimates basal glucose clearance and markedly overestimates hot glucose effectiveness, in keeping with the trend observed in Ref. 37. Specifically, is virtually identical to the basal fractional glucose clearance of the two-compartment model (e.g., from the standard IVGTT is 0.0102 min¹, and PCR/V_T = 0.0096 min¹). 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

TOP ABSTRACT INTRODUCTION THE COLD AND HOT... A TWO-COMPARTMENT MODEL OF... COLD GLUCOSE EFFECTIVENESS COLD INSULIN SENSITIVITY HOT GLUCOSE EFFECTIVENESS HOT INSULIN SENSITIVITY CONCLUSIONS REFERENCES APPENDIX A APPENDIX B APPENDIX C APPENDIX D APPENDIX E APPENDIX F

Effects of Monocompartmental Undermodeling on S_G

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1

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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 Table 2. Figure 6 shows the glucose decay curves (A) and the fractional decay rates of incremental glucose concentration (B) predicted by the two models. The monoexponential decay curve predicted by the minimal model and the two-exponential profile generated by the two-compartment model are almost superimposable in the period of minutes 10-20 of the IVGTT but diverge thereafter, thus reproducing closely the experimental observations by Quon et al. (30). Of note is that the value of S_G (0.021 min¹) lies between the values that k_G takes on between 10 and 20 min [e.g., k_G (minute 15) = 0.023 min¹]. 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.

View larger version (19K): [in this window] [in a new window]   Fig. 6.   Comparison between the two-compartment and the cold minimal model predictions of glucose concentration (A) and fractional decay rate of incremental glucose concentration (kG, B) during an IVGTT at basal insulin. The value of SG estimated during an IVGTT with a normal insulin response is close to the value of kG in the initial portion of the test, approximately between 10 and 20 min.

Validation of S_G

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The value of S_GV found experimentally in the present study (4.2 ml · kg¹ · min¹) is much higher than the value of S_G(clamp) in normal subjects that can be found in the literature (2.4 ml · kg¹ · min¹ in Ref. 11). The same trend is observed if the value of S_GV 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_GV 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, ₁ and 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

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where t₀ is ~15 min in subjects with a normal insulin response. Equation 10 shows that S_G is influenced by both the fast and slow components of glucose disappearance. To compare quantitatively S_GV with S_G(clamp), it is useful to express S_G as a function of ₂ only. By exploiting the fact that A₁₁e^₁t₀  2A₂₂e^₂t₀ and A₂e^₂t₀  6A₁e^₁t₀ (the values of A₁, A₂, ₁, and ₂ reported in Table D1), one has

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S_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. C1 to a hyperglycemic clamp at basal insulin, one finds that S_G(clamp) is defined as the ratio of (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, GINF, equals (R_d  EGP), S_G(clamp) is defined as follows

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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 E)

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It is easy to show that S_G(clamp) is primarily determined by the slow component of glucose disappearance. In fact, A₂/₂  18A₁/₁, and thus S_{G(clamp)  2}/A₂. Moreover, because 1/A₂ approximates the total glucose distribution volume V_T (28), and V_T  1.3 V (see Tables 1 and A1), we can write

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By comparing Eqs. 11 and 14, one realizes why S_GV is about twice S_G(clamp). It is worth pointing out that S_GV and S_G(clamp) are not only quantitatively, but also qualitatively different; S_GV 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_GV, 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

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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

TOP ABSTRACT INTRODUCTION THE COLD AND HOT... A TWO-COMPARTMENT MODEL OF... COLD GLUCOSE EFFECTIVENESS COLD INSULIN SENSITIVITY HOT GLUCOSE EFFECTIVENESS HOT INSULIN SENSITIVITY CONCLUSIONS REFERENCES APPENDIX A APPENDIX B APPENDIX C APPENDIX D APPENDIX E APPENDIX F

Effects of Monocompartmental Undermodeling on S_I

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View larger version (14K): [in this window] [in a new window]   Fig. 7.   A: comparison between insulin action of the two-compartment and the cold minimal models during a standard IVGTT. B: insulin action of the two-compartment model, which, if applied to the accessible pool, would produce the same effect on glucose concentration as the one taking place in the nonaccessible pool. This "accessible-pool equivalent" profile of the two-compartment insulin action is contrasted with the cold minimal model insulin action. C: difference between insulin action profiles of B is the effect of monocompartmental undermodeling on cold minimal model i