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Original Studies |
Diabetes Research Laboratory (C.M., D.T.F.), School of Kinesiology, Simon Fraser University, Burnaby, British Columbia V5A 1S6, Canada; and Brigham and Women’s Hospital (A.D.), Boston, Massachusetts 02115
Address correspondence and requests for reprints to: Diane T. Finegood, Ph.D., School of Kinesiology, Simon Fraser University, Room K9625, Burnaby, British Columbia V5A 1S6, Canada. E-mail: finegood{at}sfu.ca
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Abstract |
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 Top Abstract IntroductionSubjects and MethodsResultsDiscussionReferences |
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Introduction |
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TopAbstractIntroductionSubjects and MethodsResultsDiscussionReferences |
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The minimal-model-derived measure of insulin sensitivity, SI, has been extensively compared with model-independent measures obtained with the glucose clamp technique (e.g. Refs. 4, 5, 6, 7). Few studies, however, have considered the validity of the minimal-model-derived SG (1, 8, 9). We recently reported the first investigation of minimal-model vs. clamp-based measures of SG (10). In this experiment, SG was measured using two model-independent clamp experiments. First, a stepwise hyperglycemic clamp was performed with basal insulin maintained by infusion of SRIF and basal insulin and glucagon replacement. Second, a bolus of glucose was injected while basal insulin was maintained using the same method. In dogs treated with repeated low doses of streptozotocin who had normal fasting plasma glucose but near-zero insulin response to glucose, minimal-model estimates of SG were similar to both model-independent measures. However, in normal control dogs, the minimal-model estimates were 2-fold greater than model-independent measures (10). Quon et al. (8) also found that the minimal-model overestimated SG in subjects with Type 1 diabetes mellitus. Predictions of SG, made with an incremental insulin infusion, overestimated SG measured in the absence of an exogenous insulin infusion.
It has been suggested that overestimation of SG requires an underestimation in SI, to maintain a good least squares fit (8, 11). We agree that, if error exists in one parameter, compensatory error may occur in at least one other parameter, to accommodate the least-squares fit. It is possible, however, that overestimation of SG could largely be compensated for by incorrect estimates of both minimal-model-derived parameters (P2 and P3) that make up SI (= P3/P2) and leave SI unaffected (10). In the present study, we examined our hypothesis that overestimation of SG does not affect estimation of SI. Minimal-model parameters were determined for a series of 56 FSIGTs performed in women with varying degrees of insulin resistance and glucose intolerance. Model estimates of P2, P3, and SI were compared with estimates obtained when SG was reduced to 90%, 80%, 70%, ... , 10% of the original minimal-model determined value. Reducing SG resulted in increased values for both P2 and P3 but had only a marginal impact on the accuracy of SI.
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Subjects and Methods |
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Data from a previously published study of 27 women with polycystic ovary syndrome (PCOS) (13 nonobese and 14 obese) and 29 normal control women (14 nonobese and 15 obese) were used (12). Studies were approved by the Mount Sinai School of Medicine, and written informed consent was obtained from each subject.
Modified FSIGT
FSIGTs were performed after a 10-h overnight fast (12). An iv catheter was inserted into each arm, and then the subjects were allowed to rest for 30 min. At 0 min, glucose (0.3 g/kg) was injected over 1 min; and at 20 min, tolbutamide (Upjohn Co., Kalamazoo, MI) was injected over 20 sec. Normal women received 300 mg tolbutamide, and PCOS women received 500 mg tolbutamide, because they were expected to be more insulin resistant. Although samples were taken for 300 min after the glucose bolus in most subjects, this was found to be unnecessary (13); and only 240 min of data was used for the purposes of the present investigation. Insulin and glucose levels were assayed according to previously reported methods (12).
Minimal-model analysis
Minimal-model parameters SG,
P2, P3, and
Go were determined numerically for each study, by
applying the minimal-model of glucose kinetics, using MLAB
(Civilized Software, Inc., Bethesda, MD). The model
equations are:
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During the first 8–10 min of an FSIGT, mixing of the plasma glucose occurs. Because this mixing is not accounted for in the minimal-model, these data points are typically zero-weighted. The specific weighting scheme chosen for a data set is usually variable, depending on the quality of fit for different possible weighting schemes. However, for this experiment, a fixed weighting scheme was selected. In the least-squares fitting, different parameters contribute to the quality of fit in different regions of the curve (14). For example, in the early portion of the curve, when insulin in the remote compartment is low, glucose effectiveness contributes more strongly to the quality of fit. Therefore, varying the weighting scheme in this region could cause a decrease in the net contribution of glucose effectiveness to the net least-squares fit and, hence, cause additional variation in the parameter estimates. Accordingly, a standard weighting scheme was selected. We used a standard zero-weighting period of 0–10 min. To reduce the effect of overweighting a single outlier, the 12- and 14-min samples were both overweighted by a factor of 2.
To predict initial values, a logarithmic fit of the first 20 min of
glucose samples was performed. The initial estimates for
SG and Go were calculated
as the slope and the y-intercept of this fit, respectively. The fixed
initial estimate for P2 was .02; and for
P3, it was .00001. SG,
P2, and P3 were constrained
to be greater than 0. Results from this fit are referred to as the
unconstrained (U) minimal-model fit. SI
ranged from 0.2–22.6 x
10-4
min-1/(µU/mL),
and SG ranged from 0.8–3.8 x
10-2
min-1 for the
unconstrained fits (Fig. 1
).
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Eleven further model fits were performed for each FSIGT data set, also using MLAB. These fits were executed to determine the minimal-model estimation for Go, P2, and P3 (and therefore SI) if SG were reduced. SG was constrained by the relation SG = C·SGU, where SGU is the SG, as determined by the unconstrained fit, and C is a constant set to C = 1, 0.9, 0.8, ... , 0.1. During the fitting process, SG was held at this value, and parameters Go, P2, and P3 were estimated. These fits are referred to as the constrained minimal-model fits.
Handling of unacceptable fits
Although the standard weighting scheme was optimized, to be
appropriate for as many data sets as possible, it was impossible to
select a weighting scheme that yielded acceptable fits for all data
sets. Further, some fits yielded parameter values that were at the
extreme maximum of our system’s calculating ability, indicating
numerical instability had likely occurred. We applied a test for
detection of several outliers in multivariate data with 
= 0.1
(15). The variables considered in this test were
P2, P3, and
Go. Table 1
indicates the number of experiments eliminated for each value of C. In
total, 6% of the fits were removed as outliers. These studies were
distributed throughout the range of SI and
SG investigated, and the distributions of
SG and SI were
unaffected.
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The numerical algorithm predicts the parameter values by minimizing the sum of squared error (ESOS). ESOS was calculated by multiplying the square of the difference between the glucose data (Gidata) and the model fit (Gifit) by each data point’s weight. The unconstrained model fits result in the absolute minimum ESOS; and by constraining SG, the quality of fit degrades, as reflected in the value of ESOS.
A repeated-measures ANOVA was used to assess the effects of constraining SG on the estimates of each of the parameters and other variables, such as ESOS (PROC GLM, SAS System for Windows Ver 6.12, SAS Institute, Inc., Cary, NC). Differences between the unconstrained fit and the value of a parameter at a given level of C was deemed significant for P < 0.05. Results are displayed as mean ± SEM unless otherwise noted.
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Results |
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Overall, as C was changed, SI remained
constant (P = 0.2726, Fig. 2, top). Because
SI varied widely between subjects, we eliminated
the between-subjects component of the variation in
SI by first normalizing the constrained values of
SI to the unconstrained estimate (Fig. 2, bottom). When normalized, the between-subjects variability
is reduced, but there remains no overall significant change in
SI with C (P = 0.1087).
Contrasting the values of SI(normalized), at
specific levels of C vs. the unconstrained fit, indicated
that, from C = 0.5–0.8, SI was
significantly reduced, although not by more than 4%. With C =
0.2, which corresponds to constraining SG to 20%
of its default minimal-model value, SI was only
reduced to 97.5 ± 2.1% of the unconstrained value.
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Although SI was independent of the
constrained value of SG, P2
and P3 were found to change with C
(P < 0.003, Fig. 3).
When the constrained estimates for P2 and
P3 were normalized, with respect to the
unconstrained estimate, variability between subjects was reduced.
Clearly, as C was reduced, the proportional change in
P2 and P3 was similar. With
SG reduced to 40% of the unconstrained level
(C = 0.4), P2 and P3
increased 3.9 ± 0.6- and 3.7 ± 0.5-fold, respectively
(P < 0.0001). As C was further reduced,
P2 and P3 increased
exponentially but always to a similar degree. This is consistent with
the result that SI, the ratio of
P3 and P2, is insensitive
to changes in C.
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The majority of the model fits were acceptable, because the
ESOS did not increase dramatically with a
decrease in C (Fig. 4). Even with a near
doubling of ESOS at C = 0.2, the
acceptability of the fits did not deteriorate beyond what is often
deemed visually acceptable (Fig. 5). In
fact, the mean residual difference between the model fit and the actual
data in the early portion of the curve is only mildly affected by
constraining C (Fig. 6).
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Discussion |
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From equation (1a) we can see that to approximately maintain dG(t)/dt
[and therefore G(t)] when SG is changed, X(t),
but not necessarily SI, must change in some
compensatory way. By integrating equation (1b) from zero to infinity,
it can be shown that:
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Given that I(t) is model-independent, if X(t) is underestimated during part of the experiment and overestimated later in the experiment, then the integral might be unchanged, allowing SI to be unaffected by the error in SG. We examined the X(t) for the condition C = 0.4 (data not shown) and found the above relationship holds. From 0–30 min after the glucose bolus, X(t) increased and increased more rapidly in the constrained, as compared with the unconstrained, condition. From 40 min to the end of the study, X(t) fell and fell more rapidly in the constrained, as compared with the unconstrained, case. Because P2 and P3 control the rate of rise and fall of X(t), a more rapid rise and fall is consistent with an increase in both P2 and P3. Yet, despite this change in the dynamics of X(t), the area under the X(t) curves was unaffected by the value of C (data not shown). Although this analysis indicates why SI remains constant, it does not specifically explain why P2 and P3 change by equivalent degrees.
Validation studies have demonstrated that SI is well correlated with clamp-derived measures of insulin sensitivity under various clinical conditions: in normal subjects (12, 17, 18), in lean (4, 5) and obese subjects (5), and in cirrhotic (18), PCOS (12), and heart failure patients (19). The range of R-values for these studies was 0.74–0.91; so, although the accuracy of the actual value of SI is in dispute (20, 21), it seems to be a valid index of insulin sensitivity. The present results suggest that estimation of SI is independent of error in the estimation of SG. The simulation studies of Ni and colleagues (16) also suggest that errors in SG do not affect estimation of SI. These investigators used a two-compartment model of glucose kinetics to generate simulated data for both glucose clamp and FSIGT experiments. The minimal-model was applied to the simulated FSIGTs, and model parameters were compared with those obtained from the simulated clamps. They found that SG was overestimated, as compared with a simulated glucose clamp experiment, when SG was below 0.03 min-1 and underestimated when SG was above 0.03 min-1 (16). Despite some conditions where SG was overestimated and others where SG was underestimated, they found that estimation of SI was relatively insensitive to the error in SG. Thus, using a different approach, these studies confirm that estimation of SI with the unconstrained minimal-model is independent of error in SG.
Ni et al. (16) and Cobelli and colleagues (20) suggest that estimates of both SI and SG are inaccurate because the one-compartment minimal-model structure is inadequate to describe actual glucose kinetics. It has been shown that two or three compartments are needed for complete description of radiolabeled glucose kinetics in humans and dogs, even in the absence of changes in plasma insulin (22, 23). Whether a two-compartment model is also necessary for unlabeled glucose kinetics is less certain.
Our previous results indicate that, when the first 10-min postglucose injections are not included, a one-compartment representation is generally adequate for FSIGTs performed at basal insulin (10). On the other hand, simulation studies by Cobelli and colleagues (24) seem to support the idea that the monocompartment structure leads to the bias in SG. In these studies, the minimal model was used to analyze data simulated, using a two-compartment model and varying insulin profiles, revealing a relationship between insulin profile and predicted SG. Despite these results, whether the single-compartment assumption explains why SG is biased by the plasma insulin response remains to be determined.
Even if the reason for inaccuracies in SI and SG is the lack of a two-compartment description of glucose kinetics, a solution to this problem remains difficult. Because of the limited information contained in a simple FSIGT, all of the parameters of a two-compartment model are not identifiable. Alternatively, Vicini et al. (25) suggested that a two-compartment model be used with a tracer labeled FSIGT protocol. Though such an approach can be used under some limited circumstances, it will not be feasible and will be too expensive for large population studies or clinical trials. An alternate solution to the problem may exist, however, through modification of the insulin compartment. As variations in insulin profile seem to lead to bias in a supposedly insulin-independent measure (10, 24), perhaps investigations of the model description of the effect of insulin would be appropriate. An alternative model structure that still uses the single-compartment assumption (in combination with zero weighting of early data points, to largely overcome the inadequacy), but uses a different effect-of-insulin structure, may lead to a decoupling of glucose effectiveness estimates and the insulin profile.
The minimal-model method has been used in well over 350 papers published since its introduction in 1979 (13, 26). Although not all of these studies have reported and interpreted the value of SG, many studies have concluded that SG is reduced, unchanged, or elevated. Our previous studies suggest that, if the reduction of SG is concomitant with a decreased insulin secretory response, then the reduction in SG may be an artifact of the minimal-model method. The present studies suggest, however, that even if SG is reduced, the investigator can maintain confidence in their estimate of insulin sensitivity, SI. This suggests that we do not have to reconsider conclusions of insulin resistance in a given study, even if we lack confidence in the model-derived estimate of glucose effectiveness.
In conclusion, these results suggest that application of the minimal-model method to FSIGTs results in an estimate of insulin sensitivity, SI, which is insensitive to the value of glucose effectiveness, SG. Overestimation of SG does not lead to underestimation of SI, although other factors, such as inadequacy of a one-compartment representation of glucose kinetics, may affect the accuracy of SI, as compared with a glucose-clamp-based index. The optimal solution to correct for error in estimation of SG remains to be determined.
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Footnotes |
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Received July 22, 1999.
Revised March 21, 2000.
Accepted March 21, 2000.
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References |
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TopAbstractIntroductionSubjects and MethodsResultsDiscussionReferences |
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