The iterative two-stage population approach to
IVGTT minimal modeling: improved precision with reduced
sampling
Paolo
Vicini1 and
Claudio
Cobelli2
1 Department of Bioengineering, University of Washington,
Seattle, Washington 98195; and 2 Department of Electronics and
Informatics, University of Padova, Padua, Italy 35123
 |
ABSTRACT |
The minimal model
method is widely used to estimate glucose effectiveness
(SG) and insulin sensitivity (SI) from
intravenous glucose tolerance test (IVGTT) data. In the standard
protocol (sIVGTT, 0.33 g/kg glucose bolus given at time 0),
which allows the simultaneous assessment of
-cell function, the
precision of the individualized estimates often degrades and
particularly so in the presence of reduced sampling schedules. Here, we
investigated the use of a population approach, the iterative two-stage
(ITS) approach, to analyze 16 sIVGTTs in healthy subjects and to obtain refined estimates of SG and SI in the
population and in the individual subjects. The ITS is based on
calculation of the population mean and standard deviation of the
parameters at each iteration and then use of them as prior information
for the individual analyses. Theoretically, the use of a prior in the
ITS should improve the precision of the individual estimates. The
customary approach (standard two stage, STS), where modeling is
performed separately for each individual subject, does not take the
population knowledge into account. We used both frequent (FSS, 30 samples) and (quasi-optimally) reduced (RSS, 14 samples) sampling
schedules. For the FSS, STS gave estimates (mean ± SD) for
SG = 2.66 ± 1.09 × 10
2 · min
1 and SI = 6.46 ± 6.99 10
4 · min
1 · µU
1 · ml,
with an average precision of 51 (range 5-176) and 33%
(3-91), respectively. RSS radically worsened the
precision of both SG and SI. However, RSS and
ITS gave SG = 2.59 ± 0.73 and
SI = 6.06 ± 7.28, with an average precision of
23 (12-42) and 27% (),
respectively. In conclusion, population minimal modeling of sIVGTT data
improves the precision of individual estimates of glucose effectiveness and insulin sensitivity, as the theory predicts, and, even with reduced
sampling, the improvement is substantial.
glucose effectiveness; insulin sensitivity; parameter estimation
 |
INTRODUCTION |
THE MINIMAL MODEL of
glucose disappearance (5) is widely used to
estimate metabolic indexes of glucose effectiveness (SG) and insulin sensitivity (SI) in humans in both normal and
physiopathological conditions. Since 1994, ~280 peer-reviewed
scientific reports from the intermediary metabolism community have used
the minimal model approach or have discussed related methodology,
demonstrating substantial interest in the method; 65 of these reports
have appeared in the past two years. Use of the minimal model for
SI determination is often preferred over the more invasive
and labor-intensive glucose clamp technique, especially in large
clinical trials and epidemiological studies (26). Minimal
model-based insulin sensitivity (SI) functions very well as
a surrogate measure of cardiovascular disease risk in normal subjects
(18) and in type 2 diabetics (17). Currently,
several studies are using it to search for diabetes-relevant genes in
the human genome (15, 16). Caumo et al. (7)
have criticized this method because of its likely undermodeling of
plasma glucose kinetics. However, its rising popularity among several
circles warrants some effort to improve not only the data collection
portion of the experiment but also the related procedures of data
analysis, because physiologically significant conclusions will rest on
both of these aspects.
Because of its simplicity, the standard intravenous glucose tolerance
test (IVGTT) protocol (sIVGTT), based on the injection of a glucose
bolus at time 0 and subsequent sampling for 3 or 4 h,
is by far the most popular experimental protocol. However, the
precision of SG and SI estimates is not always
satisfactory, especially with reduced sampling schedules. The
insulin-modified (mIVGTT) protocol (13, 37), which
involves an infusion of insulin between 20 and 25 min after the glucose
bolus, has allowed considerable improvement in the precision of the
individual estimates of SG and SI. However, the
modified protocol becomes substantially more complex than the standard
procedure and may lead to hypoglycemic episodes in normal volunteers;
also, it does not carry over easily to pediatric populations. In
addition, the presence of exogenous insulin makes the determination of
concomitant
-cell activity difficult, and Pacini et al.
(21) have recently suggested that, in studies comparing
groups, use of the sIVGTT is more desirable. Improvement of the
precision of the estimates in the sIVGTT is therefore paramount.
In the standard approach to minimal modeling, each subject is analyzed
individually, and the additional information that all subjects belong
to a homogeneous population is never exploited, even when available.
With a single exception (10), minimal model analysis of
large groups has so far neglected the fact that each individual belongs
to a population of subjects who share certain quantitative traits.
Nevertheless, the fact that all subjects in similarly performed studies
have similar characteristics, like body weight, age, medications, etc.,
carries a certain amount of information. Thus the techniques of
population analysis could bring a significant contribution to the
minimal model method. Population analysis is the methodology used to
quantify between-subject (also called intersubject in the literature)
variability relative to a given population model. Its use is widespread
in pharmacokinetic/pharmacodynamic studies, because it helps solve
problems crucial in clinical studies such as sparse sampling and
protocol deviation. Similar issues will become increasingly important
in physiology as metabolic studies like the IVGTT move out of the
investigative stage (small number of subjects) into the clinical arena
(large number of subjects). The statistics literature has especially
studied population analysis, developing the related methodologies of
analysis of repeated measurement data, nonlinear mixed-effects
modeling, and analysis of longitudinal data (9, 28).
In this work, we will describe the results of population analysis
applied to glucose minimal modeling, and we will use the population
results to provide Bayesian priors for the individual estimates of the
metabolic indexes. The database consists of 16 sIVGTTs performed in
normal subjects; we analyze these data with both a frequent (FSS) and a
reduced sampling schedule (RSS). The use of an RSS is of interest in
experimental design, because it potentially reduces experiment costs,
laboratory worker biohazard, and patient discomfort. We will show that
the use of a population analysis iterative algorithm, the iterative
two-stage (ITS) approach, allows significant improvement in the
precision of individual parameter estimates for each subject, even with RSS.
 |
MATERIALS AND METHODS |
Data.
The data consist of a series of sIVGTTs performed in a population of 16 young adults. Experiments were performed at the Washington University
School of Medicine, St. Louis, MO, and the Department of Metabolic
Diseases, University of Padova, Padua, Italy. We published these data
previously in Vicini et al. (35), to which we refer the
reader for experiment details. Briefly, a glucose bolus (between 300 and 330 mg/kg) was administered at time 0. Samples were
drawn with an FSS of 30 samples, from which an optimal RSS of only 14 samples was extracted (0, 2, 3, 4,
5, 8, 10, 12, 14, 16, 18, 20, 24, 28, 32, 40, 45, 50, 60, 70, 80, 90, 100, 110, 120, 140, 160, 180, 210, 240 min; the RSS in italics). The
RSS is based on optimal glucose sampling schedule studies (8,
25). Because different laboratories gathered the data, the
sampling schedule differed slightly among subjects; the ones we report are only indicative.
Minimal model of glucose disappearance.
We describe the minimal model of glucose disappearance (5)
by
|
(1)
|
where D is the glucose dose (mg/kg body wt), Q(t) (mg/kg)
is glucose mass in plasma, G(t) (mg/dl) is glucose concentration, I(t)
(µU/ml) is insulin concentration, Gb (= Qb/V)
and Ib are their basal values, and X(t) is insulin action
(min
1). Insulin concentration acts as a known input
(without error) in the second equation, and we estimate model
parameters by fitting the model response, G(t), to glucose
concentration data. The model has four uniquely identifiable
parameters: SG (min
1), SI
(min
1 · µU
1 · ml),
p2 (min
1), the insulin action parameter, and
V (dl kg
1), the glucose distribution volume.
SG and SI are the minimal model indexes of
glucose effectiveness and insulin sensitivity, respectively, and
reflect the effect of glucose and insulin on both glucose disposal and
production. In particular, SG measures the ability of
glucose per se, at basal insulin, to stimulate glucose disposal and to
inhibit glucose production. Similarly, SI measures the
ability of insulin to enhance the glucose per se stimulation of glucose
disposal and the glucose per se inhibition of glucose production. The
minimal model parameter vector is thus p = [SG, SI, p2, V].
Standard two-stage approach.
The typical method of estimating the mean (first-order moment) and the
variance (second-order moment) of the population distribution of
SG and SI consists of estimating SG
and SI separately in each subject via nonlinear regression
and then calculating the sample mean (µ) and covariance
(
2) of all the SG and SI
estimates. The results are grouped according to the population (e.g., a
population of normal controls and a population of diabetic patients
should be analyzed separately). This method is often called the
standard two-stage (STS) approach. The values of µ and
2 represent the mean and variance (or rather, the first-
and second-order moments, if we cannot assume normality) of the
population distribution. For the STS method, we used weighted nonlinear
least squares, as implemented in the kinetic analysis software SAAM II
(1), to determine each subject's parameters. We minimized
the weighted residuals sum of squares with respect to the vector of
model parameters for a subject j, pj
|
(2)
|
where Nj is the number of data points available for
the jth subject, ti,j and
Gi,jOBS are the ith time point and
data point, respectively, of the jth subject,
2i,j is the variance of the
measurement error of the ith data point, and
G(pj,ti,j) is the minimal
model prediction of glucose concentration for a given
pj.
We assumed measurement errors to be independent, Gaussian, zero mean,
and with a standard deviation given by a constant coefficient of
variation (CV) = 2% of the measured glucose concentration. We
calculated the variance (precision), Vj, of the resulting
estimate
j of pj
from the inverse of the appropriate Fisher Information Matrix
|
(3)
|
where the superscript T indicates vector or matrix transpose,
and [
G/
pj] is a 4 × N matrix that
contains the derivatives of the model output at given times with
respect to the parameters. We use this asymptotic approximation, which
is known to give a lower bound of precision (as shown by the
Cramèr-Rao theorem; see, e.g., Ref. 6,
p. 200), both because it is widely believed to be a good
index for "true" precision and for consistency with the usual
applications of the minimal model method. To mitigate the error of the
single compartment assumption, we did not use the early glucose data
(from 0 to
6 min) for model identification. We calculated the basal
concentrations of glucose and insulin (Gb and
Ib) as the end-test basal concentrations, i.e., the last available measurement (usually at 240 min). Figure
1 shows the individual fits, all
superimposed to the individual data, giving an idea of the degree of
variability in the individual measurements. The variability is
especially concentrated in the phase of acute response to the glucose
bolus (between 20 and 70 min from the start of the experiment).

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Fig. 1.
Plot of individual data points ( ) and individual
minimal model fits ( ) for the frequent sampling schedule (FSS) and
standard two-stage (STS) analysis.
|
|
We calculated the population mean for each parameter as the sample mean
of all the individual parameter estimates
(
j)
|
(4)
|
where N is the number of subjects, and we calculated the
population variance as the corresponding sample variance
|
(5)
|
From a methodological standpoint, this approach is undesirable
for a number of reasons. It neglects the fact that the precision of the
estimates of SG and SI can differ substantially
between individuals, and it therefore pools "good" individual
estimates together with "poor" individual estimates in Eqs. 4 and 5. The approach is also well known to overestimate,
to a varying degree dependent on the magnitudes of Vj, the
true population variance (9).
ITS approach.
The ITS is a parametric, iterative population analysis method based on
the concepts of population prior knowledge and maximum a posteriori
(MAP) probability empirical Bayes estimation. Steimer et al.
(34) proposed it as a computationally attractive
alternative to the nonlinear mixed-effects model approach
(9). The steps of the ITS follow.
Step 1: initialization. Estimate the STS mean and covariance
as in Eqs. 3 and 4. Define
µ(0) = µSTS,
2(0) =
STS2.
Step 2: k + 1, k
1. Perform parameter estimation on
each subject j again, this time minimizing the following extended MAP Bayesian objective function with respect to pj
|
(6)
|
where the distance of the current parameter estimate from the
population mean (the prior) is also penalized; we denote with pj,i the ith element of the
parameter vector for subject j and with Np the number of
elements in the vector pj. The estimate
j obtained by minimizing this objective
function is often called post hoc, or empirical Bayes, estimate. We can
again calculate Vj, the precision of the parameter estimate
j, from the Fisher Information
Matrix (see below); µi(k) is the value of the population
mean at the kth iteration of the method, and
i,i(k) is the ith diagonal element of the
population covariance matrix at the kth iteration.
Calculate, then, the updated population mean of the parameter vector
|
(7)
|
and the covariance
|
(8)
|
Step 3: k + 2, k
1. Check for
convergence of the population mean, the population variance, and the
individual parameter estimates; namely, determine whether or not the
current and the previous estimate differ by <1%. If so, stop; if not,
return to step 2.
The iterative nature of the algorithm is apparent; it is also apparent
that the objective function in Eq. 5 takes explicitly into
account the available population information and that the computation
of the population covariance matrix in Eq. 8 includes available information on the precision of each individual estimate. The
STS, on the other hand, ignores subjects other than the current one; it
is essentially the initialization step of the ITS. The ITS has
had limited use in the pharmacokinetic literature to estimate population parameter means and variances from reduced data sets (11). The ITS has been implemented in a prototype
application built on SAAM II, with the maximum number of iterations set
to 50. We observed essentially no change in the individual and
population parameter values and precisions after 10-45 iterations
of the method.
We computed the precisions of the individual parameter estimates from
the inverse of the Fisher Information Matrix again. Unlike STS, which
uses only the kinetic glucose measurements, the ITS covariance formula
also uses the population information (20)
|
(9)
|
where
2 is the final estimate of the population
covariance from the ITS. Although it is apparent that Eq. 9
will always improve precision compared with Eq. 3 (because
2 is a positive definite matrix by construction), the
question arises of how to choose a proper value for
2.
The ITS provides an answer to that problem.
Statistical methods.
Because of both the nonnormality of the sample and the heterogeneity of
variances between the FSS and RSS groups (see Ref. 30,
p. 296), we compared the SI and SG
values obtained from different estimators and different sampling
schedules by means of the Wilcoxon signed-ranks test for matched pairs
(Ref. 30, p. 128). We set the significance
level at P = 0.05, and we used the nonparametric
Spearman rank correlation to measure agreement.
 |
RESULTS |
We report here the STS and the ITS results for both sampling
schedules (FSS and RSS). Table 1 (FSS by
use of STS and ITS) and Table 2 (RSS by use of STS and ITS) show
individual results for all of the subjects for all of the minimal model
parameters, whereas Fig. 2 displays a
visual summary of our findings on the percent precisions of parameter
estimates.

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Fig. 2.
Precision of glucose effectiveness (SG) and
insulin sensitivity (SI) estimates with different
sampling schedules [FSS or reduced sampling schedule (RSS)] and
population analysis approaches [STS and iterative two stage (ITS)].
Mean values are shown.
|
|
We first applied the STS method to both FSS and RSS data. We detected a
significant difference (P < 0.05) between
SI determined with STS analysis of the FSS compared with
the RSS data. To investigate the effect of the number of samples on
population analysis, we used linear regression to compare the
individual estimates across sampling schedules. From Fig.
3 we can infer that SI is
fairly stable (except when its value is very low, as in subject
11), whereas SG is slightly less stable. With the FSS
data, the average precision of the estimates, calculated as the mean of
all individual precisions, was 51, 33, 73, and 7%, respectively.
However, the average precision of the estimates worsened considerably
with the RSS data (79, 91, 227, and 11%, respectively). Therefore, the
RSS considerably hampers individual parameter reliability.

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Fig. 3.
Impact of the RSS on the SG and SI
estimates from the standard intravenous glucose tolerance test
(sIVGTT). Estimates are compared with the corresponding estimates
obtained under FSS. R2 values reported are
Spearman rank correlation results.
|
|
We then analyzed FSS and RSS data with the ITS method. Results for FSS
data were comparable to the STS results. The population spread was
slightly smaller than the STS estimate, as expected from theory, i.e.,
the STS overestimates the population covariance. Interestingly, we
found no statistically significant difference between STS-FSS and
ITS-RSS insulin sensitivity estimates, showing that the RSS is not
robust unless the data are analyzed with a population analysis method.
We can speculate that, in a data-rich situation, STS and ITS would
perform similarly; however, the average precision of the estimates
improved dramatically with ITS (24, 18, 35, and 4%,
respectively). This approach maximizes the information available
from both the rich data set of the FSS and the knowledge of the
population mean. The ITS, however, reveals its power in a (relatively)
data-poor situation like the RSS. The average precision of the
estimates in this case was very good: 23, 27, 37, and 4%, respectively. These numbers are similar to those obtained with ITS-FSS,
but the process employed fewer data points. The gain of the ITS with
respect to the STS, with both sampling schedules, is quite evident from
the summary of the estimated parameter precision in all four cases
shown in Fig. 2.
Last, we investigated the performance of the estimation methods applied
to the same sampling schedule. To give an idea of how individual
estimates from the STS map into individual estimates from the ITS, we
compared the individual STS and ITS parameter estimates via linear
regression; rank correlation of SG and SI was
always between 0.97 and 1.00.
 |
DISCUSSION |
Population analysis methods find their natural application in the
analysis of data-poor clinical studies, as when the number of samples
available for each individual is rather small (e.g., 3 or 4). The
natural arena is that of pharmacokinetics and pharmacodynamics, but
there have been some recent contributions in the metabolism arena
(24). We have found here that both rough (like the STS) and somewhat more sophisticated (like the ITS) methods give similar results for the population mean and variance. However, the
population-based approach obtained the same results for population mean
and variance with one-half of the total number of blood samples (an
attractive feature). Moreover, population analysis allows a significant
gain in the individual precisions of SG and SI
(Fig. 2), thus giving more confidence in the overall results. A
population method like the ITS allows the calculation of much more
precise individual estimates for all of the subjects, even for those
where the STS parameter precisions become unacceptable. For instance,
our results demonstrate that an RSS is feasible for the sIVGTT,
provided that a population-based method is used for data analysis; in
fact, we found a statistically significant difference between the FSS and RSS SI estimates with STS analysis. Note that the
overestimation error associated with the STS estimate of
between-individual variability increases as within-individual
variability (related to the precision of parameter estimates and also
referred to as intraindividual variability) increases (which is
particularly evident when comparing the estimate of
for
p2 in Table 2, STS vs.
ITS). This is the reason for the failure of STS when applied to
the RSS case. Therefore, in an experimental protocol with very
precisely estimated individual parameters, as in tracer studies
(8), or with the mIVGTT (36), the STS
would give quite reliable estimates of the population mean and
variance, because within-individual variability would have only a small
confounding effect. In contrast, in a situation where within-individual
variability is as large as or larger than between-individual
variability (i.e., the parameters are poorly estimated), the two
effects would confound each other. Thus the sIVGTT protocol with RSS
benefits considerably from population analysis.
The minimal model community widely uses measured values instead of
predicted values when calculating
2i,j in
Eq. 2. This practice, however, introduces measurement error in the weights. Inspection of weighted residuals from the individual fits indicates that model misspecification is small (data not shown),
thus suggesting that the difference would not be significant. Nevertheless, an extended least squares (ELS) or maximum likelihood criterion, where a term in the objective function with the logarithm of
the data variance balances the use of predictions in the weighting, would better account for the variance in the data (2,
3)
|
(10)
|
Equation 10 gives twice the negative logarithm of the
likelihood function of the data. If
2(pj,tij) did not depend on the
parameters and were constant across time (constant standard deviation),
then the two methods (weighted and extended least squares) would be
entirely equivalent.
Theory indicates that estimation of parameters with empirical Bayes
estimation will result in greater precision (as calculated from the
Fisher Information Matrix in Eqs. 3 and 9) than
with the STS approach. However, the purpose of this study was to
ascertain the magnitude and possible significance of such an
improvement a priori unknown. Indeed, with extremely large
variabilities of the metabolic indexes, we could expect negligible
improvement. Also, we should note the (historical) fact that, in the
clinic and even in large epidemiological studies (19), the
STS is invariably used for SG and SI
estimation, with both reduced and intensive sampling schedule
(18-30 blood samples). We hope that this work can begin to
introduce this large community to the concept that population-based
methods and/or empirical Bayes approaches improve individual/group
parameter quantification in the presence of large epidemiological data,
and that it is feasible to use priors (when available and appropriate)
to estimate individual patients' parameters (a practice sometimes
referred to in pharmacokinetics as "Bayesian forecasting").
The improved precision of parameter estimates in this subject sample is
due, at least in part, to the "shrinkage" of the individual estimates around the population mean (see Eq. 6). The
not-so-hidden assumption is that a population mean indeed exists and
that it is appropriate for all of the subjects involved in the study. Several levels of assumptions are embedded in kinetic modeling of
population data: 1) that the mathematical model of the
concentration time course is true; 2) that a certain
measurement-error statistical structure exists (which allows the
application of meaningful nonlinear regression); and 3)
that, in the case of our version of the ITS, the shape of the
population distribution of the parameters is multivariate normal.
Examination of the consequences of all of these assumptions does not
concern us here, since the present report is largely a "proof of
concept" of the usefulness of population analysis in a reduced-data
situation, and the ITS is only one possible algorithm for this purpose.
We are thus not concerned here with either model accuracy or methods
other than the ITS. However, some of the aforementioned assumptions,
such as normality or unimodality of the population distribution of the
parameters, can be relaxed, e.g., via mixture modeling, explicitly
including covariants in the model, or performing nonparametric
population analysis (see, e.g., Chapter 7 in Ref. 9).
Davidian and Giltinan (9) comprehensively reviewed several
different proposed estimation methods for population analysis of
nonlinear models, all based on different degrees of approximation to
the maximum-likelihood objective function. The attractiveness of the
ITS derives from the need for a simple computational machinery based on
successive iterations of the single-subject identification algorithm
via a recursion formula. The ITS method is essentially an
expectation-maximization (EM) algorithm (27). A related
EM-type algorithm is the global two stage (GTS), which applies an
iterative algorithm directly to estimated parameters and their
within-individual covariances. Recently, Patron-Bizet et al.
(22) compared the STS and the GTS used with a
pharmacodynamic model and found that the GTS was superior. Steimer et
al. (34) and Racine-Poon and Smith (23) compared the GTS and ITS, finding that they perform quite similarly. Although the ITS is computationally more intensive than the GTS, we
chose to use the ITS here, because it allows a natural introduction of
the concept of empirical Bayes individual estimation, which is at the
root of the improved (asymptotic) precision. We have found the ITS
method quite robust with respect to different starting values, with the
main effect being the speed of convergence. After a few iterations, the
ITS algorithm gets reasonably near its limit point. The literature
suggests checking for convergence by calculation of the
maximum-likelihood objective function, or an approximation thereof, at
the point estimates (14).
Other available algorithms for the calculation of population parameters
belong to the family of mixed-effects modeling, where the mean and the
covariance of the population are fitted parameters in a
maximum-likelihood context, and the computationally demanding maximum-likelihood integral is calculated via various approximations to
the model function (29). Indeed, given that we have chosen a homogeneous population, mixed-effects modeling might be the best
choice. Some of these approximations are currently available in the
software NONMEM (28). Indeed, these algorithms are often preferred to the ITS, because they can be used even when individual estimates are not available in some subjects. They are based on the
minimization of a well-defined objective function and return the
precision of the estimates of population mean and covariance (and, in
principle, also of each individual estimate). In contrast, the ITS
algorithm does not readily allow calculation of the precision of
population mean and variance. Because we were primarily interested in
gauging the quantitative improvement of the individual posterior estimates, the ITS adequately met our purposes as a population analysis
approach. Interestingly, De Gaetano et al. (10) used the
first-order approximation to the population problem implemented in
NONMEM to estimate minimal model population parameters in a group of 20 normal subjects by use of sIVGTT data. In that article, the precision
of the population mean and covariance estimates improved with respect
to the STS method. The mean and covariance estimates, however, did not
change between NONMEM and STS (again, because the frequently sampled
sIVGTT is a data-rich experiment). However, the authors did not report
the values and precision of the individual parameter estimates
(possibly because they are not reported by the NONMEM software).
Allowing for precise estimates of the parameters at the individual
level in a reduced-data situation, and thus improving the a posteriori
identifiability of the model parameters in a population, is only one of
the possible applications of empirical Bayes estimation and population
modeling in glucose metabolism and in intermediary metabolism in
general. For example, there is ample pharmacokinetic/pharmacodynamic literature about the incorporation of demographic covariants into the
mathematical model (12). Thus it is conceivable that
population modeling of metabolic studies would, for instance, permit
understanding the relationships of important features like visceral
adiposity, weight, and age with metabolic indexes of insulin
sensitivity and glucose effectiveness.
 |
ACKNOWLEDGEMENTS |
We wish to thank our colleagues, Alan Schumitzky and Bradley M. Bell, for useful discussions and to gratefully acknowledge the
editorial help of Eileen Thorsos and the useful suggestions of the
anonymous referees.
 |
FOOTNOTES |
This work was partially supported by National Institutes of Health
Grants NCRR-12609 ("Resource Facility for Population Kinetics") and
GM-53930. Preliminary results from the material in this work were
presented at the Mathematical Modeling in Experimental Nutrition Conference held at the University of California, Davis, CA, on August
17-20, 1997, and at the American Diabetes Association 59th Scientific Sessions held in San Diego, CA, on June 18-22, 1999.
Address for reprint requests and other correspondence: P. Vicini, Dept. of Bioengineering, Box 352255, Univ. of Washington, Seattle, WA 98195-2255 (E-mail:
vicini{at}u.washington.edu).
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.
Received 23 August 1999; accepted in final form 13 September 2000.
 |
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