Vol. 85, Issue 3, 935-945, September 1998
State-space models of insulin and glucose responses to diets
of varying nutrient content in men and women
David J.
Holtschlag1,
Mary C.
Gannon2,3, and
Frank Q.
Nuttall3
1 Metabolic Research Laboratory
and Section of Endocrinology, Metabolism, and Nutrition,
Minneapolis Veterans Affairs Medical Center, and Departments of
2 Food Science and Nutrition
and 3 Medicine, University of
Minnesota, Minneapolis, Minnesota 55417
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ABSTRACT |
Discrete-time
state-space models were developed to describe contemporaneous responses
of plasma insulin and glucose of normal human subjects. Male and female
subjects ingested three consecutive identical meals from isocaloric
diets classified as high-carbohydrate, high-fat, high-protein, or
standard. Distinctly different glucose and insulin responses were
measured in men and women. A seven-state system of linear equations,
three in insulin and four in glucose, was identified and estimated to
describe responses in men. A six-state system, three in insulin and
three in glucose, describes responses in women. Model simulations at
15-min intervals closely match measured concentrations over a 12-h
period. Effects of diet content and meal timing on insulin and glucose
concentrations were quantified. Dynamic insulin and glucose responses
to isocaloric meals of pure carbohydrate, fat, and protein diets were
projected on the basis of models developed from mixed diets. The
symmetry of the projections indicates that positive excursions in
glucose concentrations associated with carbohydrate intake may be
matched with negative excursions associated with fat and protein intake
to help manage postmeal glucose excursions.
diabetes; glycemic index; meal composition; high-carbohydrate
meals; high-protein meals; high-fat meals
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INTRODUCTION |
THE METABOLIC RESEARCH LABORATORY at the Minneapolis
Veterans Affairs Medical Center is engaged in generating research
data and developing techniques for predicting the plasma glucose
response to meals of mixed nutrient content. This knowledge provides a basis for regulating plasma glucose concentrations by managing nutrient
intake. In addition, a prediction of the natural glucose response to
mixed meals in nondiabetics provides a reference for controlling
glucose excursions in diabetic patients requiring insulin.
This study develops state-space models to describe the dynamics of
insulin and glucose excursions in normal human subjects ingesting diets
of varying carbohydrate, fat, and protein content. Separate models are
developed to describe the distinctly different responses in men and
women. Effects of the nutrient content and meal timing are quantified
for isocaloric meals. Estimates of glucose and insulin responses to
meals of pure carbohydrate, fat, and protein are projected on the basis
of meals with mixed content.
Data for this analysis were obtained from average insulin and glucose
profiles published previously (8). In that study, 12-h profiles of
plasma glucose and insulin concentrations were obtained in 26 healthy
(nondiabetic) subjects. The subjects, 14 men and 12 women, were in
their midtwenties and were within 10% of their ideal body weight (8).
Each subject ingested three consecutive identical meals separated by a
period of 4 h from one of four isocaloric diets:
1) a high-carbohydrate diet,
2) a high-fat diet,
3) a high-protein diet, and
4) a standard diet consisting of a
mixture of carbohydrate, fat, and protein in proportions similar to
those common in diets of Americans and Northern Europeans (Table
1). Resulting insulin and glucose responses differed
among diets and between male and female subjects (8).
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METHODS |
Six bivariate series of insulin and glucose concentrations were
digitized from average profiles of male and female subjects on
high-carbohydrate, -fat, and -protein diets (8). An additional bivariate series resulting from a standard diet for male and female subjects, which was overlaid on each of the insulin and glucose series
for high-carbohydrate, -fat, and -protein diets, was also digitized.
The bivariate series for the standard diets was used to confirm the
accuracy of the digitized records and to further quantify the effects
of the three major nutrient components on insulin and glucose
concentrations.
Offsets for insulin and glucose concentrations were determined to
remove effects of fasting-level concentrations, i.e., baselines, that
were unrelated to nutrient intake during the study. Offsets were
determined independently for men and women by diet on the basis of
prebreakfast measurements. In men, offsets for insulin ranged from 13.2 to 20.4 µU/ml; offsets for glucose ranged from 88.8 to 95.8 mg/dl. In
women, offsets for insulin ranged from 11.4 to 15.6 µU/ml; offsets
for glucose ranged from 80.6 to 95.1 mg/dl. Deviations from offsets are
referred to as excursions or concentrations.
Linear interpolation was used between digitized measurements to obtain
15-min samples within the ~12-h monitoring period for each diet. This
sampling interval is longer than the measurement interval during the
1st h after meals, equal to the sampling interval during the 2nd h, and
one-half of the sampling interval during the 3rd and 4th h, when
fluctuations in insulin and glucose concentrations generally
diminished. The 15-min sampling interval was chosen as the shortest,
uniform sampling interval that was consistent with the majority of
measurement intervals and the local variability in the process
dynamics.
Effects of diet content were considered independently, but all diets
were analyzed together in a single series for each gender. This ensured
that the individual effects associated with carbohydrate, fat, and
protein content were estimable and yet that there would be no carryover
effects between diets. To accomplish this, the end of each three-meal
(12-h) series of 48 samples was extended to 100 time steps by adding
one-half of the difference between the previous 15-min value and the
offsets. Extended series representing the four diets were concatenated
to form two bivariate series of 400 pairs of insulin and glucose
values: one for men and one for women. Offsets for each diet were then
removed from each set.
Auxiliary series of diet content and meal timing were formed with the
same length as the concatenated bivariate series. The diet content
series contained three columns corresponding to the intake of grams per
kilogram of body weight of carbohydrates, fats, and proteins,
respectively, in any particular meal (Table 1). All food intakes for a
particular meal were represented as occurring in a single 15-min
interval. The meal timing series was an indicator matrix that consisted
of three columns identifying the meal as breakfast, lunch, or
dinner by use of a "1" in the appropriate column. All sampling
intervals not corresponding to meals contained zeros.
Model conceptualization.
The interrelated response of insulin and glucose to dietary intake is
complex. These dynamics have been approximated by a variety of
techniques, including minimal model analysis (10), extended Kalman
filters, and fuzzy filter models (12). The following interrelations
were considered in conceptualizing the state-space models developed in
this study. Effects of age, activity level, and concentrations of
epinepherine, glucagon, glucocorticoids, or growth hormones on insulin
and glucose concentrations, however, could not be quantified because of
data limitations.
Glucose absorbed after digestion of carbohydrate-containing foods is
the major determinant of the postmeal rise in glucose concentrations
(5). For most carbohydrate-containing foods, the digestion rate exceeds
the glucose absorption rate. Quantitatively, the amount of galactose
present in the diet is not metabolically significant. The amount of
fructose absorbed is significant, but it results in little rise in
plasma glucose concentration. In comparison to carbohydrate, ingested
protein has little effect on the blood glucose concentration or on
digestion and absorption of monosaccharides (6). Ingested fat may
affect the rate of digestion of carbohydrate and the absorption of
monosaccharides, but the effect is delayed and becomes progressively
more prominent with the second and third meals throughout the day (8).
Ingestion of glucose- and fructose-containing foods, as well as
protein, stimulates insulin secretion. Ingested fat also may potentiate
the amounts of insulin stimulated by ingested carbohydrates. The effect
of these ingested nutrients on insulin secretion is direct, through
stimulation of insulin secretion (via glucose and absorbed amino
acids), and is indirect, through stimulation of incretin hormone
secretion (via ingested proteins, carbohydrates, and fats). Ingested
protein and fructose stimulate glucagon secretion. Glucagon secretion
is inhibited by glucose-containing foods. Glucagon stimulates glucose
release by the liver. Insulin inhibits glucose release by the liver.
The net effect on glucose production, therefore, depends on the ratio
of glucagon to insulin. Glucagon does not affect insulin-stimulated
uptake and storage of glucose in skeletal muscle.
The mathematical model conceptualized in this study is a simplified,
linear, time-invariant approximation to the complex dynamics between
insulin, glucose, and nutrient intake. Simplifications were necessary
because of uncertainties in physical processes and limitations of
available data. These limitations restrict application of the model to
prediction of short-term (4-h) effects within the range of dietary
conditions represented by the data. The 15-min sampling interval
further restricts the resolution of the temporal dynamics. Finally,
circadian variations in these dynamics, associated with variations in
activity or other factors, are approximated only by meal-timing
effects. Despite these limitations, the model is intended for
simulation of postmeal insulin and glucose excursions with sufficient
accuracy to identify primary effects of dietary content and meal timing
and to aid in the management of glucose concentrations.
Model formulation.
The dynamic response of insulin and glucose to nutrient intake was
formulated as a discrete-time, state-space model (4, 9). This model
contains two equations: a state equation and an observation equation.
The state equation describes the dynamics of the process at time steps
indexed by k + 1 on the basis of information available at and before time
k. The length of the state vector is
determined by the order of the difference equations (number of delays
of the response variables) needed to adequately describe the process
dynamics. The observation equation transforms the state vector by
reducing the dimension to the number of attributes measured on the
process at time k. The innovations
form, which includes estimation of the
K matrix below, of the state-space model was selected to help ensure that the model errors (innovation sequences) were uncorrelated. Uncorrelated innovation sequences simplify testing of the statistical significance of estimated model
parameters. Also, the innovations form is convenient for real-time
estimation of error-corrected predictions of insulin and glucose. The
innovations form of the state-space equation is expressed as
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where
x(k + 1) is a column vector sequence of length
n representing the state of the system
at time indexed by k + 1. The length
of the state vector is equal to n, the
sum of the orders ni and
ng needed to
simultaneously describe interacting insulin (ni) and
glucose (ng)
excursions. The index k identifies
discrete time steps separated by 15-min intervals. The initial state
(k = 0) was a zero vector.
A(
) is an
n
(row)-by-n (column) matrix of
coefficients and parameters relating the state at time
k to the state at time
k + 1. Coefficients refer to
structural elements of the A matrix,
whereas parameters refer to elements that are adjusted so that
simulated concentrations match measured concentrations.
B() is an
n-by-r
matrix of parameters relating the effect of diet and meal timing,
u(k),
to the state vector. The dimension r
is the number of columns in the forcing function (u) matrix.
u(k)
is an r row vector sequence containing
values of the forcing function. In this analysis,
r was equal to 6, corresponding to the
amounts of carbohydrate, protein, and fat (diet content) ingested at
time k and three separate column
indicators for breakfast, lunch, and dinner (meal timing).
K() is an
n-by-2 matrix of parameters relating
the innovations at time k,
e(k),
to the state at k + 1.
e(k)
is a two-row vector sequence of innovations (errors) between model
estimates and measured values of insulin and glucose at time
k.
y(k)
is a two-row vector sequence of measured excursions in insulin and
glucose.
C is a
2-by-n matrix of coefficients used to
extract simulated insulin and glucose concentrations from the state
vector.
Model identification and estimation.
Model identification determines the minimum model order (number of
delays of the response variables) needed to adequately describe the
process dynamics (2). In this analysis, alternate models between second
and fourth order were evaluated for male and female data sets. Given
the similarity of the insulin and glucose profiles, only models for
insulin and glucose that differed by one order or less were evaluated.
The primary criterion used to select model order was the minimization
of Akaike's final prediction error (FPE) (3). The FPE provides a
measure for comparing alternate models on the same data set by
adjusting the loss function, computed as the determinant of the
innovations matrix, to account for different numbers of parameters
among models. The adjustment compensates for a tendency to have an
automatic decrease in the loss function with an increase in the number
of parameters. In cases where the FPE contained more than one local
minimum for various model orders, the minimum model order was chosen
that was consistent with the data.
Once the model order was determined, estimates were computed for
parameters of the matrices A, B, and
K in the state-space equation.
Parameter estimates were computed by use of a
robustified1
quadratic prediction-error criterion minimized using an iterative Gauss-Newton algorithm (3). A stability test of the model was performed
to ensure that only models corresponding to stable predictors were
estimated. Finally, the statistical significance of each parameter was
evaluated on the basis of a Student's
t-test. Those parameters that were not
significantly different from zero were set to zero, and if the system
was stable under those initial parameter conditions, the system was
reestimated. If the FPE of the reestimated system was lower after
elimination of the parameter, the simpler model was retained. This
procedure was repeated until no additional parameters could be
eliminated.
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RESULTS |
Insulin and glucose dynamics in men.
A coupled third-order insulin model and a fourth-order glucose model
were selected for the male series on the basis of the FPE criteria
(Table 2). Elimination of insignificant
parameters A(7,1)2
and A(7,2) and
B(1,1),
B(1,2), and
B(5,6) reduced the FPE of the male
model from 93.44 to 75.93. The estimated state equation for men
is
where * indicates that the parameter was set
to zero without significantly degrading model performance on the basis
of the FPE criteria; some parameters were not significantly different
from zero but could not be eliminated, because stable initial parameter
values could not be established; 
indicates an apparent
significance at the 0.05 probability level; and 
indicates an
apparent parameter significance at the 0.01 probability level. The
observation equation for men is
Finally, the covariance of innovations describes the
variance of the errors in insulin and glucose estimates on the diagonal terms and covariance between model error sequences on the equal, off-diagonal terms. For the model developed for men, the covariance of
the innovations is
The
innovation sequences were not significantly auto- or cross-correlated.
Model parameters provide some information on the process being
simulated. Elimination of elements
A(7,1) and
A(7,2) and the relatively small
magnitude (and significance) of element
A(7,3) indicate a relatively small
marginal effect (unaccounted for by other dynamics or food inputs) of
past insulin concentrations on future glucose concentrations. In
contrast, the significance of elements
A(3,4:7) indicates that past glucose
concentrations have a significant effect on future insulin
concentrations.
The B matrix indicates the relative
effects of nutrient components,
B(:,1:3), and meal timings,
B(:,4:6). In general, the signs of
these two sets of parameters are opposite for each element in the state
vector for the insulin components,
B(1:3,:), and the glucose components,
B(1:4,:). For example, parameters B(3,1:3) are all negative and
parameters B(3,4:6) are all positive, whereas parameters B(4,1:3) are all
positive and parameters B(4,4:6) are
all negative. Interrelations among the magnitudes of parameters in the
B matrix provide differentiation among
effects associated with nutrient components and meal timings.
A sampled and a continuous representation of simulated excursions of
insulin and glucose in men in response to standard, high-carbohydrate, high-fat, and high-protein diets are shown on Fig.
1. High-carbohydrate and standard diets
create similar abrupt increases in insulin and glucose concentrations
for all meals. In contrast, distinct effects of meals on insulin or
glucose excursions are less apparent from the high-fat diet.
High-protein diets are associated with minor glucose excursions;
however, distinct increases in insulin concentrations are apparent for
all meals.
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Fig. 1.
Insulin and glucose response of men to isocaloric diets high in
carbohydrate, fat, or protein. Solid line, modeled values;  ,
interpolated values; +, digitized values. Vertical bars, meal timing
[breakfast (B), lunch (L), and dinner (D)].
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In general, model-simulated values correspond closely to sampled
values. The simulated values differ from an explicit
application of the male state-space model in that no error correction
was involved [K() = 0].
Inclusion of error-correction components, e.g., computation of
one-step-ahead predicted values, would have resulted in a smaller
discrepancy between sampled and simulated values. However, error
correction would have made determination of the effects of diet content
and meal timing on insulin and glucose excursions more difficult.
Results of an analysis of simulated values and model errors during 4-h
periods after each meal are shown in Table
3. For insulin and glucose, sampled and
simulated values are highly correlated (R > 0.92) for the high-carbohydrate
and standard diets. Sampled and simulated insulin and glucose
concentrations have the lowest correlation for high-fat diets, although
the excursions (and the standard errors of the bias) in insulin and
glucose concentrations are also least for the high-fat diet. For the
high-protein diet, insulin concentrations are somewhat underestimated
and glucose concentrations are overestimated. These biases are likely
the result of the generally diminished correlation between insulin and
glucose excursions that is unique to the high-protein diet. Finally,
simulated glucose excursions overestimate sampled values for men after
breakfast. Some of the model error may be attributable to differences
in glycemic effects of foods in the various diets (7, 11, 13).
Insulin and glucose dynamics in women.
The identified female model is a coupled third-order system in insulin
and glucose (Table 2). Elimination of insignificant parameters
B(4,5) and
K(6,1) from the full model reduced the
FPE of the female model from 153.2 to 128.3. The estimated state
equation for women is
where *, , and
are as described in Insulin and glucose
dynamics in men. The observation equation for women is
Finally, the covariance of
the innovations sequence is
Again,
as in the male model, the innovation sequences were not significantly
temporally or intercorrelated.
In women, insulin
excursions are strongly influenced by second-order dynamics given the
large magnitude of the parameter associated with insulin at time
k 
1 [A(3,2) = 2.77] relative
to the parameter at time k
[A(3,3) = 0.16].
This characteristic may help explain the slower, smoother responses of
insulin and glucose in women than in men. Predicted insulin
concentrations (at k + 1) are
positively associated with glucose values at time
k, as indicated by
A(3,6) = 4.45. Predicted glucose
values are negatively associated with values of insulin at time
k, as indicated by
A(3,3) = 1.16, and positively
associated with values of glucose at time
k, as indicated by
A(6,6) = 4.04.
As in the case for men, the signs of the two sets of parameters in the
B matrix corresponding to the insulin
components, B(:,1:3), and the glucose
components, B(:,4:6), are generally opposite for each element in the state vector. For example, parameters B(3,1:3) are all negative and
parameters B(3,4:6) are all positive, whereas parameters B(6,1:3) are all
positive and parameters B(6,4:6) are
all negative. Here also, interrelations among the magnitudes of
parameters in the B matrix provide
differentiation among effects associated with nutritional components
and meal timings.
Sampled and simulated excursions of insulin and glucose in women
responding to standard, high-carbohydrate, high-fat, and high-protein
diets are shown in Fig. 2. Insulin and glucose
excursions have a lower, narrower peak for the standard diet than for
the corresponding high-carbohydrate diet. High-carbohydrate diets resulted in the greatest excursions in insulin and glucose; however, the peaks were distinctly broader and generally had slower responses than corresponding responses in men. Also, in contrast to men, insulin
and glucose responses in women to high-fat diets were distinct for each
meal, with a peak in the response after breakfast almost twice that
after lunch or dinner. Female responses were similar to the male
responses for the high-protein diet. Both sets showed distinctly
elevated insulin concentrations but only minor excursions in glucose.
Insulin and glucose excursions have a lower, narrower peak for the
standard diet than for the corresponding high-carbohydrate diet.
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Fig. 2.
Insulin and glucose response of women to isocaloric diets high in
carbohydrate, fat, or protein. See Fig. 1 legend for explanation of
symbols.
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Results of an analysis of simulated values and model errors during the
4-h periods after each meal are shown in Table 3. For insulin and
glucose, sampled and simulated values are highly correlated
(R > 0.96) for the high-carbohydrate
diet. Sampled and simulated insulin concentrations have the lowest
correlation (R = 0.786) for the
high-fat diet. No biases were detected with respect to diet contents or
meal timings in the female responses.
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DISCUSSION |
Insulin and glucose excursions were projected (extrapolated) to depict
the male and female responses to diets of pure carbohydrate, fat, and
protein by use of the linear state-space models (Figs. 3
and 4). The projections are based on isocaloric meals
of 38 cal/kg body wt of pure carbohydrate, fat, and proteins. By use of
data in Table 1, the calories per gram of carbohydrate, fat, and
protein were computed as 3.815, 8.546, and 4.563, respectively. Thus
nutrient inputs to the state-space model for projection of responses to
pure carbohydrate, fat, and protein diets were 9.961, 4.447, and 8.328 g/kg body wt, respectively.
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Fig. 3.
Projected insulin and glucose response of men to isocaloric diets of
pure carbohydrate, fat, or protein. Solid line, breakfast;  , lunch;
+, dinner. All responses converged to zero.
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Fig. 4.
Projected insulin and glucose response of women to isocaloric diets of
pure carbohydrate, fat, or protein. See Fig. 3 legend for explanation
of symbols. All responses converged to zero.
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In men, projected insulin excursions resulting from carbohydrate and
fat track projected glucose excursions. That is, positive (negative)
excursions in glucose are contemporaneously associated with positive
(negative) excursions in insulin. In women, projected insulin
concentrations decrease immediately after carbohydrate ingestion,
perhaps because the short-term increase in insulin production is
delayed or is not sufficient to offset increased requirements
associated with the increased production of glucose. In contrast, the
projected insulin response to fat ingestion in women apparently exceeds
the short-term requirements, resulting in a positive excursion in
insulin and predominantly negative excursion in glucose. Positive
insulin excursions are associated with protein ingestion in men and
women; corresponding glucose excursions are predominantly negative.
Differences among projected responses for breakfast, lunch, and dinner
were minor (Figs. 3 and 4).
A continuous representation of the discrete-time initial-condition
responses for insulin and glucose in men and women is shown in Fig.
5. Figure 5 (top) shows the responses of
initially elevated insulin concentrations (100 µU/ml) and base-level
glucose concentrations (zero). Similarly, Fig. 5 (bottom)
shows the responses of initially elevated glucose concentrations (100 mg/dl) and base-level insulin concentrations. The simulated
initial-condition responses in men and women are conditionally stable,
because all concentrations tend to zero with increasing time. In
addition, the initial-condition responses to elevated insulin
concentrations and the male response to elevated glucose concentrations
monotonically tend to zero, indicating unconditional stability.
However, the slight rise in female glucose concentrations for the
elevated glucose simulation indicates that the stability of the glucose
response is dependent on the insulin concentration. Therefore, this
simulated response is not unconditionally stable. Further investigation
is needed to determine a stable parameterization.
Despite this limitation of the female model, two patterns are
consistent in the male and female models. First, elevated insulin concentrations lower glucose concentrations. This simulated effect is
consistent with insulin's role in inhibiting production of glucose and
in stimulating the removal of glucose from circulation. Second,
elevated glucose concentrations raise insulin concentrations. This
simulated effect is consistent with the stimulation of insulin produced by a rise in glucose and gut hormone-mediated release of
insulin secretion (7).
The model may be extended to facilitate real-time control of glucose
excursions in people with insulin-requiring diabetes. Extension will
necessitate a reformulation of the model to simultaneously predict
plasma glucose concentrations and insulin effectiveness as a function
of dietary inputs, insulin dosage, and motor activity levels. This
extension will necessitate additional clinical study involving people
with insulin-requiring diabetes, in which heart rate, perhaps, is
monitored as a measure of motor activity, with consideration of
possible confounding factors such as insulin absorption rates and
insulin injection sites.
A Kalman filter (1) implementation of the state-space equations may
facilitate real-time glycemic control by supplementing information on
model estimates with information on model uncertainties. Augmenting the
state vector with data on other hormone concentrations, particularly
glucagon, may help refine model simulations of insulin and glucose
dynamics. Results from future model studies may be applied to a larger
population if subjects from more than one age group are included in
clinical studies.
In summary, linear state-space models provide an effective mechanism
for simulating the short-term (4-h), contemporaneous response
of insulin and glucose to isocaloric meals for a wide range of nutrient
contents. These models describe the dynamic characteristics of insulin
and glucose responses with sufficient accuracy to identify primary
effects associated with specific dietary content and meal timing.
Separate models are needed to accurately describe the distinct dynamic
characteristics of responses in men and women. Some model error is
thought to be associated with variation in the glycemic effect of
selected dietary components and with circadian variations in subject
activity and hormone concentrations.
Responses of insulin and glucose to pure carbohydrate, fat, and protein
can be projected from data on meals of mixed nutrient composition. In
this study, projections for men and women indicate that glucose
concentrations rise after ingestion of pure carbohydrates and generally
fall after ingestion of pure fat and protein. The symmetry of the
projected responses with time indicates that state-space models may be
useful for designing the nutrient composition of meals to manage
plasma glucose excursions. In addition, the simulated glucose response
in nondiabetics may provide a reference concentration sequence for
control of glucose excursions in diabetic patients requiring insulin,
once a continuous, portable glucose-monitoring device is available.
Additional research is needed to extend the model applicability to a
wide range of caloric intakes and a wider variety of foods than were
available in this investigation.
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ACKNOWLEDGEMENTS |
We thank Claudia Durand for excellent secretarial assistance.
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FOOTNOTES |
This work was supported in part by Merit Review Research Funds from the
Department of Veterans Affairs and National Institute of Diabetes and
Digestive and Kidney Diseases Grant RO1-DK-43018.
1
The robust estimation technique limited the
influence of large individual prediction errors on parameter estimates.
Specifically, prediction errors that were >1.6 times the estimated
standard deviation of the innovations sequence,
e(k),
carried a linear, rather than a quadratic, weight.
2
Elements in matrices are identified by their
row and column indexes. Thus A(1,2)
refers to the element in matrix A in
row 1 (top row) and
column 2 (from the left). Furthermore, a colon is sometimes used to represent a sequence, so that
A(1:4,5:7) indicates a 4-by-3
submatrix of A consisting of
rows 1-4 and
columns 5-7. Finally, a colon
shown by itself indicates all rows or columns depending on its position
with respect to the comma separating row and column indexes.
Address for reprint requests: M. C. Gannon, Metabolic Research
Laboratory, 111G, VA Medical Center, Minneapolis, MN 55417.
Received 24 November 1997; accepted in final form 24 April 1998.
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Abstract
Introduction
