1 Dipartimento di Elettronica e Informatica, Università degli Studi di Padova, 35131 Padova; and 2 Dipartimento di Informatica e Sistemistica, Università degli Studi di Pavia, 27100 Pavia, Italy
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ABSTRACT |
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The minimal model of glucose kinetics, in conjunction with an insulin-modified intravenous glucose tolerance test, is widely used to estimate insulin sensitivity (SI). Parameter estimation usually resorts to nonlinear least squares (NLS), which provides a point estimate, and its precision is expressed as a standard deviation. Applied to type 2 diabetic subjects, NLS implemented in MINMOD software often predicts SI=0 (the so-called "zero" SI problem), whereas general purpose modeling software systems, e.g., SAAM II, provide a very small SI but with a very large uncertainty, which produces unrealistic negative values in the confidence interval. To overcome these difficulties, in this article we resort to Bayesian parameter estimation implemented by a Markov chain Monte Carlo (MCMC) method. This approach provides in each individual the SI a posteriori probability density function, from which a point estimate and its confidence interval can be determined. Although NLS results are not acceptable in four out of the ten studied subjects, Bayes estimation implemented by MCMC is always able to determine a nonzero point estimate of SI together with a credible confidence interval. This Bayesian approach should prove useful in reanalyzing large databases of epidemiological studies.
insulin sensitivity; insulin resistance; mathematical model; parameter estimation; type 2 diabetes
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INTRODUCTION |
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THE MINIMAL MODEL OF GLUCOSE KINETICS, in conjunction with the insulin-modified intravenous glucose tolerance test (IM-IVGTT), is widely used to measure insulin sensitivity in subjects with impaired glucose tolerance and type 2 diabetes in both clinical and epidemiological studies (1, 14, 17, 23). Although some of its assumptions, in particular the single compartment approximation to describe glucose kinetics, have been recently reexamined (7, 8), it is important to understand the performance of the classical minimal model when it is identified with the most advanced estimation techniques. Virtually all minimal model identification strategies employed in the literature resort to nonlinear least squares (NLS) estimation. NLS provides a point estimate of each model parameter and, by means of the Fisher information matrix, a measure of its uncertainty expressed as standard deviation (SD) can be obtained, from which a confidence interval can be determined. However, this approach has difficulties in handling possible asymmetries in the probability distribution of the estimates. In providing the estimates, NLS exploits only the experimental data (e.g., plasma glucose concentration time series in our case) and the knowledge of the measurement error statistics. NLS is thus a purely data-driven approach. One reported problem with this approach in minimal model studies is that, in a number of subjects, especially those with type 2 diabetes, insulin sensitivity (SI) is calculated as SI=0 with the minimal model program MINMOD (19). For instance, Saad et al. (23) found that SI=0 occurred in 12 of 24 type 2 diabetic subjects, i.e., a prevalence of 50%. As a consequence, in population studies, histograms with a likely artificial bimodal pattern (i.e., a peak at SI=0 and another peak at a positive SI value) are obtained. At the same time, with more general purpose software packages, e.g., SAAM II or ADAPT (2, 10), SI is estimated to be nonzero, and thus physically sound, but very small and with a very large uncertainty. As a result, negative values are included in the confidence interval. The above interpretative difficulties, often referred to as the "zero" SI problem, call for the adoption of parameter estimation techniques more sophisticated than NLS.
The so-called Bayes approach is a methodology that can be employed to attack the parameter estimation problem as an alternative to NLS, but it is less adopted in practice. Bayesian estimation techniques exploit not only the experimental data but also the a priori information (also denoted in the following by the term "prior," as is usual in the statistics literature) on the unknown parameters of the model (27). In physiological modeling, this a priori information, e.g., nonnegativity and range of variability, can typically be made available from physical considerations and population studies (see Refs. 8 and 24 for two recent applications of Bayesian estimation in metabolism). The key advantage of Bayes estimation over NLS is that the former returns the entire a posteriori probability distribution of the model parameters, from which the marginal probability distribution of each parameter can be obtained. From this probabilistic quantity, a point estimate and its confidence interval can be computed. Notably, with use of Bayes estimation, confidence intervals are in general allowed to be asymmetric with respect to the point parameter estimates, in contrast to NLS, where the implicit assumption of parameter estimates to be Gaussian leads to confidence intervals symmetric by construction. The limited popularity of Bayes estimation in physiological modeling is likely due to its heavy computational burden, because often the a posteriori probability distribution of the model parameters must be numerically computed by the Markov chain Monte Carlo (MCMC) simulation approach (13).
In this article we develop a Bayesian parameter estimation strategy to identify the minimal model of glucose kinetics in diabetic subjects and implement it by MCMC. First, we only supply to the estimator a rough prior, i.e., the unknown model parameters are positive. Then, because more refined a priori information on SI can be made available from the literature, we reidentify the model and show improved results. The database consists of IM-IVGTT performed in ten subjects. Whereas NLS provides SI estimates and confidence intervals that are not acceptable in four cases, Bayes estimation always determines a nonzero estimate of SI and, when resorting to the refined prior model, a credible confidence interval. The Bayesian approach we propose should prove useful to reanalyze large databases of epidemiological studies, such as GENNID (20), the Finland-US investigation of NIDDM genetics (FUSION) (26), and the Insulin Resistance Atherosclerosis Study (IRAS) (17), where the numbers of subjects are in the range of thousands.
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MATERIALS AND METHODS |
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Subjects, Protocol, and Data
An IM-IVGTT (glucose dose of 300 mg/kg at time 0 plus 0.05 U/kg of insulin given as a square wave between 20 and 25 min) was performed in 10 type 2 diabetic subjects. The data of 7 subjects have already been published in Ref. 1, to which we refer the reader for details on protocol and measurement. Plasma glucose and insulin concentrations were frequently measured for 4 h.Minimal Model of Glucose Kinetics
The minimal model of glucose kinetics (4) during an IVGTT (Fig. 1) is given by
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(1) |
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(2) |
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NLS Identification
From the model of Eqs. 1 and 2, glucose concentration at time ti, i.e., G(ti), is predicted by a function of the unknown parameter vector
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(3) |
As in Ref. 1, we estimate, in each of the 10 subjects of
our database, the unknown parameter vector from the data vector y = [y1,
y2,...,
yN]T by weighted NLS, i.e.
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(4) |
Bayesian Identification
Bayes estimation is based on the concept of a priori information on the unknown parameter vectorThe a posteriori probability function of contains
extremely insightful information. For instance, a point estimate, called the posterior mean of the unknown parameter vector, can be
determined as the expected value of the vector
|y
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(5) |
The a posteriori probability density function of the parameter vector
can be obtained from the Bayes theorem as
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(6) |
A nonnegativity prior on the four parameters.
Prior information is a key additional ingredient of Bayes estimation,
i.e., one needs to specify the a priori probability density function
f(
) of Eq. 6, which
summarizes our beliefs on model parameters "before having seen the
data." From Eq. 6 one can note that the Bayesian approach
coincides with maximum likelihood, if a uniform distribution from
and +
is assumed as a prior for all of the unknown parameters. In
our minimal model studies, we know in advance that all the parameters
(SG, p2, Go, and
SI) are intrinsically nonnegative. Therefore, it is reasonable to choose a prior
f
(
) that excludes negative parameter values. For SG, p2, and
Go, nonnegativity is the only information embedded into the
prior distribution, i.e., an expected range of variability is not
declared. In formal terms, we state that each value of SG,
p2, and Go is a priori equally
probable in [0, a] with a
. The same
poorly informative prior could obviously also be applied to
SI. One can note from Eq. 6 that, in this way, the posterior density function coincides, except for a constant factor,
with the likelihood function in the domain of interest. As will be
illustrated in RESULTS, this prior, albeit rough, allows a
solution of the "zero" SI problem, because a confidence
interval, including only nonnegative values, is obtained. However,
because one additionally experienced problem with NLS is that the
confidence interval of SI tends to be large, it is
worthwhile trying to incorporate more knowledge into the SI
prior, as we will explain.
A refined SI prior.
Ideally, to define the a priori probability density function of
SI, one would need a large database, e.g., containing
thousands of subjects, to extract it directly. Unfortunately, such a
database is not available to us. However, several insulin sensitivity
NLS studies in diabetic subjects are available in the literature that allow one to obtain an approximation of the actual distribution of the
true SI values. From these investigations, one knows that it is unlikely that SI will exceed certain values. For
instance, in Ref. 18, 30 type 2 diabetic subjects were
studied, and the average SI was 0.7 (104
min
1/µU ml
1) with a standard error of
0.1. Of note is that, in this data set, no "zero" SI
problems were present. Thus one expects that SI values
greater than a certain threshold are less and less probable. At the
same time, because SI can be very small, although we do not
know how small it can be in a certain subject before seeing his or her
data, it is wise not to incorporate in the prior any beliefs about its
variability at low levels. Thus we have assumed for SI an a
priori probability density function in which SI values <2 × 10
4
(min
1/µUml
1) are equally probable, while
SI values >2 × 10
4 are less and less
probable, according to a decreasing exponential law (with exponent
equal to 10
4 µU ml
1/min
1).
The chosen threshold value of 2 × 10
4 min
1/µU ml
1, even if
not directly obtained from a distribution of SI estimates in NIDDM subjects, is realistic and is supported by the literature, where it is unlikely to find insulin sensitivity values in NIDDM subjects >2 × 10
4 min
1/µU
ml
1.
The MCMC method.
The goal is to obtain
f|y(
|y), i.e.,
the joint probability density function of the unknown minimal model
parameters, given the glucose data. This task is analytically intractable, both because of the complex relationships between parameters and data and because of the shape of the probability distributions involved. A solution suggested in the literature is to
derive f
|y(
|y)
in sampled form by adopting a simulation strategy known as Markov chain
Monte Carlo (13). MCMC methods are based on two steps. In
the first step, a suitable Markov chain that converges (in
distribution) to a target distribution,
f
|y(
|y) in our
case, is generated. Then, a Monte Carlo integration step is performed
to numerically compute the integrals of interest, e.g., Eq. 5. There are many MCMC methods in the literature; see Ref.
13 for a review. They differ from each other in the way the Markov chain is generated. However, all of the different strategies proposed in the literature can be traced back to the
Metropolis-Hastings algorithm (15). In this work, we
generated the Markov chain by a symmetric transition kernel that
extracts a sample of the chain around the previous one (this scheme is
usually referred to as "random-walk Metropolis") by use of
independent uniform probability densities. The variance of such
densities was chosen case by case. As far as implementation issues of
the MCMC method are concerned, we note that convergence of the chain
was assessed by the Raftery criterion (21), which is also
known in the literature as binary-control. To compute the number of
iterations necessary to estimate a posterior quantile from a single run
of a Markov chain, a pilot analysis of output samples was used to fit a
two-state Markov chain model, and from it one can calculate the length
of the burn-in period and, then, the number of further iterations required to estimate quantiles of interest with the required accuracy. In the sequel, we have been required to estimate quantiles 0.025, 0.25, 0.5, 0.75, 0.975 with precision factors of 0.02, 0.05, 0.05, 0.05, and
0.02, respectively, with a probability of 0.95. From these quantiles,
numerically robust confidence intervals can be calculated.
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RESULTS |
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Data
Plasma glucose and insulin concentrations of the 10 subjects are shown in Fig. 2.
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NLS Identification
Results of NLS for SG, p2,, G0, and SI, obtained with SAAM II (2), are reported in Table 1 together with the precision of the estimates. The SI estimates in all of the subjects are shown in Fig. 3, together with their 95% confidence intervals, obtained by adding and subtracting to the point estimate the quantity 1.96 SD (results of subjects 1-7 are those already reported in Table 2 of Ref. 1). As a consequence of the implicit Gaussian assumption, the confidence interval is symmetric and centered around the point estimate. As apparent from the picture, SI estimates are not equally satisfactory in all of the subjects; whereas in subjects 1, 2, 4, 5, 8, and 10 a narrow confidence interval lying on the nonnegative axis is obtained, in subjects 3, 6, 7, and 9, the low value of the SI estimate, combined with its poor precision, produce an unrealistic confidence interval. In fact, in each of these four subjects, the SI confidence interval tells us that the true SI can fall with nonzero probability in the negative portion of the x-axis.
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Note that, because the covariance matrix of the measurement error is assumed to be known from the data, NLS coincides with maximum likelihood (ML). If it were assumed to be dependent on the model, then NLS and ML are no longer equivalent on a theoretical basis. However, in this case, the two results are virtually superimposable (unpublished data).
Bayesian Identification
A nonnegativity prior on the four parameters.
The a posteriori probability density function of SI
estimated in each subject is reported in Fig.
4. As somewhat expected, in
subjects 1, 2, 4, 5, 8, and 10, where NLS was
successful, the posterior provided (in sampled form) by MCMC is
concentrated around the NLS estimate. However, the new method reveals
asymmetries in the distribution of the estimates, particularly in
subjects 1 and 4. This is not surprising, because
the symmetry of NLS confidence intervals reflects only the Gaussian
assumption implicitly made on the estimates. Moreover, in
subjects 3, 6, 7, and 9, MCMC analysis reveals
that the marginal posterior densities have a high peak located at low
and realistic SI values but that they have a long tail. The
confidence intervals, however, would be too pessimistic, because it is
known from the literature that such high SI values are
unrealistic, e.g., it is impossible that SI is, say, equal to 100 × 104 min
1/µU ml
1
(see subject 3 in Fig. 4). So, to obtain a more credible
confidence interval, it is necessary to adopt a more realistic
SI prior.
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A refined SI prior.
Results for SG, p2, G0,
and SI are reported in Table 2, together with the 95%
confidence interval. Results for SI are also displayed in
Fig. 5, where the estimated a posteriori
probability density function of SI in each subject is
reported (the axes are now different from those used in Fig. 4), and in
Fig. 3, where a summary of Fig. 5 results is depicted to make the
comparison with NLS easier. One can note, by comparing Figs. 5 and 4,
how the threshold influences the tail of the posterior distribution of
some subjects. One can also note that the posterior probability density
function is zero in an interval at the right of SI=0. This
allows us to exclude, on a probabilistic basis, that SI is greater than a chosen threshold. From the a posteriori density function
of SI, one can obtain the interval where the true value of
SI falls with 95% of confidence as the interval between
the quantiles 0.025 and 0.975. This interval is displayed in Fig. 3,
together with the posterior mean of SI. The novel features of the approach we propose emerges in subjects 3, 6, 7, and
9, where NLS results were largely unsatisfactory with
noncredible confidence intervals. One can see that the marginal a
posteriori probability density function of SI does not
collapse to zero (this also applies to Fig. 4, but it cannot be
visually appreciated due to the larger range of SI values
shown in the x-axis). In all of these subjects, the
posterior mean of SI is higher than the NLS estimate
(markedly in subjects 3 and 7), because the
minimum variance estimate takes into account the entire shape
of the posterior and, so, also the presence in some subjects of the
tails. More important, in each subject the confidence interval now lies
entirely in the positive axis and allows the investigator to infer that the probability that the true SI lies at the left of the
displayed bar is 2.5%. In summary, a nonzero SI value is
detected with high probability by Bayesian identification in these very
insulin-resistant individuals. This result is not always possible with
NLS, where, given the uncertainty of the SI estimate, one
can only conclude that SI is indistinguishable from zero
(Fig. 3, e.g., subject 7). A credible confidence interval is
obtained, which allows the minimal model user to determine with 97.5%
of confidence a (nonzero) lower-bound number for SI.
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DISCUSSION |
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In this article, we have attacked the identification of the minimal model of glucose kinetics in type 2 diabetic subjects in a Bayesian context. The motivation for this stems from the interpretative difficulties generated from what is usually referred to as the "zero SI" problem, i.e., in a significant number of cases NLS estimation returns an SI=0 when the minimal model program MINMOD (19) is used, or very small values with an unrealistic confidence interval including negative values with software packages like SAAM II (2) or ADAPT (10).
In this case study, the Bayesian approach allowed the derivation (albeit only numerically through MCMC) of the joint a posteriori probability distribution of the model parameters, and thus the determination of a nonnegative point estimate of SI, together with a plausible confidence interval. The differences of the point estimates between the Bayesian approach and NLS, at least in most subjects, are not large, but it is when the confidence intervals of the parameter estimates are considered that the Bayesian approach emerges as superior. In fact, because we think that the chosen prior as well as the confidence intervals employed are both reasonable, one can exclude on a probabilistic basis that SI in the critical subjects is higher than a given threshold. Moreover, from the confidence interval obtained by Bayes estimation, one can exclude with 97.5% of probability that the true SI lies at the left of the displayed variability range. In summary, the Bayesian approach is able to cope with those situations in which the confidence interval, obtained as two times the SD of the error given by the Fisher information matrix, is not realistic, because such a method is unable to cope with asymmetries in the distribution of the estimates, like those present in this study.
A critical point in Bayes estimation is the definition with a
probability density function of the information available on the
unknown model parameters before seeing the data. As far as SG, p2, and G0 are
concerned, we have limited ourselves to express the poorly informative
knowledge that they are nonnegative, i.e., a uniform distribution along
the positive axis was chosen. A more sophisticated prior was chosen for
SI, the very critical parameter of the model, based on the
observation that, for values of SI greater than a certain
threshold, the higher the candidate SI, the lower, in all
likelihood, its probability. We have translated these beliefs about
SI in statistical terms by describing its a priori
probability density function as a uniform distribution between zero and
a certain threshold (i.e., 2 × 104
min
1/µU ml
1, a value that appears as
prudential on the basis of published studies on insulin resistance in
type 2 diabetics) and an exponential decay for SI greater
than this threshold. It is important to note that, in the prior chosen
for SI, we do not include beliefs on how small it can be,
i.e., where the whole "zero SI" problem began. Choosing
also for SI a uniform distribution prior, as in Bayesian identification with only the nonnegativity prior, one would have obtained an SI a posteriori probability density function
lower than the one presented. This means that low SI values
are less probable and supports the robustness in each subject of the
0.025 quantile of Fig. 1, which allows us to exclude with 97.5% of
probability that the true SI lies at the left of the
displayed range. Another consequence of using a uniform prior for
SI along the positive axis is represented by a gratuitous
shift to the right of the 0.975 quantile. In fact, the new technique
permits us to show that, in some cases, the model defines likelihood
functions which assign a nonnegligible probability to high
SI values that are largely unrealistic. This kind of
information was impossible to obtain so clearly by the standard
Fisherian approaches. The rationale of choosing for SI a
prior more sophisticated than that employed for SG,
p2, and G0 was to determine with
95% confidence an interval as narrow as possible where SI
lies. The role played by this a priori knowledge is not hidden but
transparent in terms of its effect on the results. For example, a
strategy to obtain nonnegative confidence intervals by NLS could
consist in reparameterizing the model with the aim of estimating the
logarithm of the parameters. However, this would be obtained by
approximating the likelihood as a function of the logarithm of the
parameters, with a Gaussian distribution. This is questionable, because
it would not be possible to evaluate the quality of such approximation.
As already mentioned, switching from NLS to Bayes estimation is not
without a price, because the MCMC results are computationally more
demanding to obtain. However, this strategy offers a new tool that
could also become important in classification studies. In fact, it
would be interesting to reanalyze large databases of epidemiological
studies in which the standard NLS approach has produced SI
histograms with a likely artificial bimodal pattern (with one peak at
SI=0 and another peak at a positive SI value), like that qualitatively displayed in Fig.
6, which are difficult to interpret. The
value SI=0, which Fisherian techniques return as the joint
mode of a suitable objective function, can be a mathematical artifact,
because it can be far from the minimum variance estimate provided by
MCMC. Therefore, we believe that if the new strategy proposed in this
paper were applied in epidemiological studies like those mentioned
above, one would obtain a more reliable distribution of SI
estimates, e.g., changing the qualitative profile of Fig. 6A
into that shown in Fig. 6B. In this way, more reliable
information on NIDDM would be made available. Moreover, a reanalysis of
databases such as GENNID (20), FUSION (26),
and IRAS (16, 17), where the numbers of subjects are in
the range of thousands, could also allow a refinement of our MCMC
method, because a more accurate description of the a priori probability
density function of SI will be possible. It is clear, in
fact, that the refined prior used in Fig. 5 provides better results
than those obtained with the positivity prior and shown in Fig. 4.
However, the refined prior still has a certain degree of roughness. A
future evolution of the method could also envisage a population
approach with the aim of estimating, jointly with model parameters, the
common prior that has generated them.
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Finally, it is worth stressing that important questions concerning
possible problems introduced by the structure of the considered model
still remain open also in the light of our MCMC results. In fact, one
can clearly note that the a posteriori probability density functions of
SI of subjects 3 and 7 are quite
close to the prior. As a consequence, the point estimates are close to the mean of the prior, equal to 1.67 104
min
1/µU ml
1, suggesting that sometimes
the model extracts little information on SI from IVGTT data
(in fact the posterior distribution approaches the prior as the data
contain decreasing information about the parameters).
Moreover, the presence of long tails in the a posteriori probability density of SI of some subjects is a critical
issue that will have to be clarified in future studies. Our Bayesian analysis, including reasonable a priori information on SI,
improves the estimation process, reducing this kind of problem.
However, it is clear that no single parameter estimation technique,
however sophisticated, can be a panacea, able to compensate possible
structural defects of a model. So, further investigations are necessary
to clarify whether the cause of these problems is the fact that the model is structurally inadequate for the estimation of a low
SI. We are inclined to believe that, in many of such
critical situations, MCMC is really a more robust estimator than NLS or
ML and can significantly improve the estimation process. However, we
also believe that in the future it will also be worth exploring a
different description of the same process, like the two-compartment
model described in Refs. 7 and 8, which may overcome these
problems. Maybe, after including some suitable a priori information,
which is necessary because of a priori identifiability issues, the
problems mentioned here may be solved by resorting to a maximum a
posteriori estimator. In any case, it will be interesting to identify
the two-compartment model by resorting to the same MCMC approach
described here. In fact, even if more computationally demanding, this
technique is likely to provide a clearer picture of the advantages of
the two- vs. the one-compartment description.
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ACKNOWLEDGEMENTS |
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This work was in part supported by National Institutes of Health Grant RR-12609, by the MURST COFIN 2000 project, "Estimation of nonaccessibile parameters in physiological systems," by a University of Padova grant (STIM-PET) to G. Sparacino, and by a University of Pavia grant to P. Magni.
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FOOTNOTES |
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Address for reprint requests and other correspondence: C. Cobelli, Dipartimento di Elettronica e Informatica, Università degli Studi di Padova, Via Gradenigo, 6a, 35131 Padova, Italy (E-mail: cobelli{at}dei.unipd.it).
The costs of publication of this article were defrayed in part by the payment of page charges. The article must therefore be hereby marked "advertisement" in accordance with 18 U.S.C. Section 1734 solely to indicate this fact.
10.1152/ajpendo.00576.2000
Received 14 December 2000; accepted in final form 10 October 2001.
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