Compartmental modeling of postprandial dietary nitrogen
distribution in humans
H.
Fouillet1,
C.
Gaudichon1,
F.
Mariotti1,
S.
Mahé1,
P.
Lescoat2,
J. F.
Huneau1, and
D.
Tomé1
1 Nutrition humaine et physiologie intestinale, and
2 Laboratoire de Nutrition et Alimentation, Unité Institut
National de la Recherche Agronomique, Institut National Agronomique
Paris-Grignon, 75231 Paris Cédex 05, France
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ABSTRACT |
A linear 11-compartment model was developed to describe
and simulate the postprandial distribution of dietary nitrogen. The values of its 15 constant diffusion coefficients were estimated from
the experimental measurement of 15N nitrogen kinetics in
the intestine, blood, and urine after the oral administration of
15N-labeled milk protein in humans. Model structure
development, parameter estimation, and sensibility analysis were
achieved using SAAM II and SIMUSOLV softwares. The model was validated
at each stage of its development by testing successively its a priori and a posteriori identifiability. The model predicted that, 8 h
after a meal, the dietary nitrogen retained in the body comprised 28%
free amino acids and 72% protein, ~30% being recovered in the
splanchnic bed vs. 70% in the peripheral area. Twelve hours after the
meal, these values had decreased to 18 and 23% for the free amino acid
fraction and splanchnic nitrogen, respectively. Such a model
constitutes a useful, explanatory tool to describe the processes
involved in the metabolic utilization of dietary proteins.
mathematical model; parameter estimation; kinetics; protein
metabolism; optimization
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INTRODUCTION |
THE ASSIMILATION OF
DIETARY PROTEIN is associated with a cascade of transient and
dynamic metabolic processes involved in controlling the distribution of
amino acids and nitrogen throughout the body. During the postprandial
phase, nitrogen and amino acids of dietary origin are submitted to
sequential metabolic processes, including gastrointestinal digestion
and amino acid absorption, amino acid deamination, subsequent transfer
to ammonia and urea, or incorporation into organs. These complex
processes take place at various rates and lead to different states of
equilibrium, dependent on both nutritional and physiological status
(14, 46) and diet composition
(25). The interrelations between the various parameters
involved in this equilibrium cannot be described in a simple way. Under
these conditions, a compartmental model, which is a mathematical
representation of the structure and dynamic behavior of a system, will
be particularly well suited to describing the complexities of
postprandial dietary nitrogen distribution in humans.
Compartmental modeling has been widely used in a broad spectrum of
research areas to investigate the distribution of materials in living
systems (8, 11, 24,
30, 44). The development of dynamic models to
predict amino acid fluxes has seen marked improvement over the past 20 yr, in parallel with the increasing use of stable isotopes in human
nutrition. Particular attention has been focused on amino acid kinetics
in the body, and modeling theories have been applied extensively to the
kinetics of leucine, as reported by Cobelli et al. (13)
and Wolfe (44). Although some complex models have been
proposed [as illustrated by the 16-compartment model proposed by
Carraro et al. (7) in dogs], they usually concerned the
metabolism of one or a few amino acids. In contrast, only a small
number of studies have addressed the problem of nitrogen modeling in
humans, because of the broad, multiple exchange kinetics (e.g., through
transamination) of nitrogen that make its study both practically
and theoretically complex. In fact, nitrogen tracers have mainly been
used to assess whole body protein turnover, and they offer a good
routine method for clinical studies (41, 45).
However, contrary to endogenous nitrogen metabolism, the fate of
dietary nitrogen compounds has rarely been studied. In this context,
the labeling of nitrogen represents the most suitable method, because
dietary protein can more easily be labeled uniformly with nitrogen than
with carbon tracers. Uniformity of labeling is crucial when dietary
protein utilization, i.e., the balance between catabolism and entry
into the anabolic pathways, is under investigation. Indeed, the
metabolism of one amino acid is not representative of that of all the
amino acids in a dietary protein, as recently illustrated in the work by Stoll et al. (39), who reported a broad range of
posthepatic availability, depending on the essential dietary amino acid
involved. Moreover, it should be recalled that the use of
13C tracers enables an assessment of carbon skeleton
sparing, in contrast to the 15N methods used to study the
amino residue. The uniform and intrinsic 15N labeling of
dietary protein has been widely used as an excellent tracer of dietary
nitrogen and enables investigation of the transfer of dietary nitrogen
into different metabolic pools, such as plasma amino acids, body urea,
and ammonia (6, 18, 20).
The aim of the present study was thus to develop and validate a dynamic
and mechanistic compartmental model describing the postprandial
distribution of dietary nitrogen in humans after the ingestion of a
protein meal. For the purposes of this model, we employed previously
reported experimental data concerning [15N]nitrogen
kinetics determined in the intestine, blood, and urine after the
ingestion of 15N-labeled milk protein in humans
(18). We resorted to modeling so that we could both
simulate exogenous nitrogen distribution in different body nitrogen
pools (including those not experimentally monitored) and predict the
further evolution of the system (11).
Compartmental modeling seems particularly suited to describing such a
complex system, because it entails reducing a markedly complex
physiological system into a finite number of compartments and pathways,
thus restricting the number of variables and parameters of the model.
This simplification reduces mismatches between the complexity of the
system and the limited data available from in vivo studies, especially
in humans (10, 11). The use of a compartment
is suited to the simplification process, because a compartment
represents a theoretical amount of material acting kinetically in a
homogeneously distinct way (11, 24). For instance, it was necessary to combine material with similar
characteristics (e.g., plasma free amino acids were defined as a single
compartment) while at the same time endeavoring to maintain a high
degree of physiological relevance when choosing the structure and
parameters for the model. Furthermore, at this stage of investigation
and in view of the experimental data available, we chose to develop a
linear compartmental model in which the flux of material from one
compartment to another depends on the mass of material in the source
(8). Interpretation of the data from this single input-multiple output experimental study required a model both to
integrate known information about the system and to fit experimental data. Thus development of the model combined both structural modeling and parameter estimation (11).
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STRUCTURAL MODELING AND THEORETICAL IDENTIFIABILITY |
Because development of the model structure, parameter estimation,
and model validation was highly interwoven in an iterative fashion,
they are described here separately for more clarity. First of all, the
different stages of model structure development are described. The
selected model is then introduced, together with tests of its
theoretical identifiability.
Collection of experimental data.
Experimental data were collected as previously described
(18). Briefly, eight healthy fasting humans equipped with
an ileal tube and a catheter inserted in a forearm vein ingested a
protein meal made up of 30 g of 15N-labeled milk
protein. 15N enrichment was measured in intestine, blood,
and urine samples by isotopic ratio mass spectrometry (Optima, Fisons
Instruments, Manchester, UK). Ileal effluent samples were collected
over a period of 8 h, and 15N isotopic enrichments
were measured in the total nitrogen fraction. Urine was collected over
an 8-h period, and isotopic enrichments were determined in both urea
and ammonia. The cumulated exogenous nitrogen recovered in both ileal
effluents and urinary urea and ammonia was converted into
of
ingested nitrogen. Urinary data were interpolated, and ileal effluent
data were pooled in such a way as to obtain the same 1-h data step
size. Blood samples were collected over an 8-h period, and
15N isotopic enrichment was measured in both plasma free
amino acids and plasma urea. The amount of dietary nitrogen present in
plasma free amino acids was calculated by assuming that the plasma
amino acid concentration was 100 mg/l (1, 3,
5) and that the mean plasma volume represented 5% of the
body mass (17). Body urea was calculated using a formula
that took account of the total body water (TBW) value, which was
estimated with the equations established by Watson et al.
(42) that depended on age, sex, and anthropometric
characteristics. Plasma free amino acids and exogenous body urea were
expressed as
of ingested nitrogen. Mean experimental data are
reported in Table 1.
Structural modeling process.
The first objective when designing this model was to propose and
identify an adequate structure (8, 11).
Development of the model required the use of two modeling software
programs, SAAM II (36) and SIMUSOLV (15),
both with optimization capabilities. The first step in developing the
model structure was to decouple the system into different subsystems
accessible to measurement, and then to use the SAAM II forcing function
to select a separate model structure for each subsystem. SIMUSOLV was
then used to integrate the subsystems thus developed into a single
complete model that described the entire system. A major problem that
we encountered when using SAAM II was the need to assign a priori a
weight to each experimental datum. In this way, the estimated values of
parameters and their errors depend on the weights assigned. In
contrast, SIMUSOLV does not require advance knowledge of the error
structure, because it adds and optimizes a heteroscedasticity parameter
(
) representing the heterogeneous error of each experimental data
set. This is of particular importance when dealing simultaneously with
several sampled compartments with different scales and variances; thus
the choice of SIMUSOLV appeared to be appropriate in this situation, in
which no a priori information was available on the variance structure.
Nevertheless, the forcing function availability of SAAM II is of
particular value when subsystem structures are determined, thereby
justifying its use during the first stages of model development.
In both cases, by solving the linear ordinary differential equations
describing the model, simulation evaluates the responses of
compartments over the time period indicated. Levels of dietary nitrogen
in each compartment over time were computed by numerical integration by
use of the Rosenbrock integrator in SAAM II and Gear's algorithm in
SIMUSOLV for stiff systems. Model optimization enabled adjustment of
these simulations to the observed data by finding a set of adjustable
parameters that maximized or minimized a characteristic of the system,
the so-called objective function. The objective function minimized
during the iterative process of optimization in SAAM II is the extended
least squares (ELS), which is based on a modified function of the
weighted residual sum of squares (WRSS) (36). In SIMUSOLV,
the log of the likelihood function (LLF) is the optimization criterion
that is maximized during the iterative process of optimization
(15).
Development of model subsystems.
The model, developed with average data values, aimed to describe the
transfer of dietary nitrogen through the gastrointestinal (GI) tract,
the elimination of absorbed dietary nitrogen in the urine, and the
distribution of the retained dietary nitrogen in the body. Thus, as a
first step before construction of a single integrated model, it was
necessary to break down the overall problem of fitting all the sampled
pools simultaneously into three simpler ones (absorption, deamination,
and retention), each corresponding to a fitting activity and
constituting a specific subsystem (16, 37).
The system was dissociated so that it dealt independently with the GI
tract and deamination subsystems by use of the forcing function
machinery of SAAM II (16).
The GI tract subsystem was treated as the single entry point from which
other compartments of the system received transferred material, because
dietary nitrogen, once absorbed through the GI tract, is then
transferred to the blood and may either be retained in the body or
eliminated by deamination. To build the GI tract subsystem structure
separately, a forcing function, placed with SAAM II on the plasma amino
acid compartment, was assumed to be a substitute for the entry of
dietary nitrogen into the rest of the system (deamination and retention
subsystems). This function was created by linearly interpolating
between sequential pairs of plasma free amino acid data, so as to force
the contents of the corresponding compartment to be equal to a function
with the same characteristics as plasma amino acid kinetics.
Furthermore, certain statistical criteria were used to determine how
many GI tract compartments were required to fit the tracer data for
ileal effluents. The statistical tests used to determine the optimum structure compared the value of the objective function after the optimization process (12, 37). The first two
methods we used to discriminate between candidate models
embodied the principle of parsimony and consisted of testing
the goodness-of-fit of different models of increasing order and
retaining the simplest model structure that adequately fit the data vs.
higher-order models that did not significantly improve the fit
(11). The Akaike and the Schwarz criteria (AIC and SC,
respectively) take account of the goodness-of-fit and the number of
parameters; these can be used for linear compartmental models in the
case of independent and gaussian measurement errors (11,
27). The parsimonious model is that with the lowest AIC and SC values (11). Moreover, for nested models, i.e.,
when one structure is a subset of the other, with gaussian measurement errors, another alternative is to check with an F-test
whether the parameters added significantly to improve the fit
(24, 27). For the GI tract subset, a
three-compartment catenary structure was required as a minimum (Fig.
1, model B) and was built
using cumulative ileal effluent data modeled by a compartment with no output. The results reported in Table 2
show that AIC criteria, SC criteria, and F-tests all led to
selection of the order 3 model, which significantly improved the fit
compared with the order 2 model, in the absence of any significant
improvement between order 3 and order 4 models. The meal was considered
to enter the first compartment in the form of a bolus. The
physiological significance of the three compartments could be assumed
to represent dietary nitrogen in the stomach, in the lumen of the small
intestine, and at entry into the colon, respectively.

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Fig. 1.
Three different models tested for the gastrointestinal
(GI) tract subsystem: model A (A), model B (B), and
model C (C). A forcing function (FF) is substituted for
plasma free amino acid data to decouple the entire system. This is
accomplished by forcing the contents of compartment 2 to
equal a function created by linear interpolations between sequential
pairs of plasma free amino acid data. Bolus input is assumed to take
place in compartment 1, which represents the gastric
nitrogen content. The bullet labeled s1 is a sample
associated with cumulated ileal effluent data and occurs in
compartment 3. The 3-compartment catenary structure of
model B was finally selected for the GI tract subsystem.
Compartment 4 represents dietary nitrogen in the lumen of
the small intestine.
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Body urea, urinary urea, and urinary ammonia were grouped into the
second subdivision representing the deamination pool and were used to
build the structure of this subsystem. The subsystem chosen for
deamination was the minimum structure (3 compartments) necessary to
describe the elimination process and to fit simultaneously all of the
sampled compartments (Fig. 2, model
A). Urinary urea and ammonia data, calculated in terms of
cumulative excretion, were modeled using compartments with no outputs.
In the first instance, the approximation was made that plasma was the
only source of input in this deamination subsystem, which was therefore driven by the plasma forcing function. The criteria summarized in Table
3 showed no significant improvement in
fit when a compartment was added.

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Fig. 2.
Two different models tested for the deamination
subsystem: model A (A) and model B (B). Tested
models are driven by the plasma free amino acid forcing function (FF)
placed in compartment 2. Samples s1,
s2, and s3 represent body urea, cumulative
urinary urea, and ammonia, respectively, and are associated with
compartments 5, 6, and 7. Model A was
selected for the deamination subsystem.
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Integration of subsystems into a single whole model.
Once the GI tract and deamination subsystems had been determined, the
final structure that would take account of all data (ileal effluents,
plasma free amino acids, body urea, urinary urea, and ammonia) was
built, with particular attention paid to the retention subsystem, which
had to describe the retention of dietary nitrogen in the splanchnic and
peripheral areas. This final step was achieved by use of SIMUSOLV
software, which enabled the processing of variables with widely varying
scales and error structures, such as plasma free amino acid kinetics
and cumulated urinary urea data. For this final stage, a structure was
proposed on the basis of a priori knowledge of the retention subsystem and then modified until an adequate fit of the data was achieved (8, 34). Because dietary nitrogen is
transferred from the splanchnic to the peripheral areas via the plasma,
a catenary-type structure, with plasma free amino acids as the central
compartment, flanked by one splanchnic and one peripheral compartment
on each side, was thus the most appropriate starting point (Fig.
3, model A).

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Fig. 3.
Three different models tested for the whole system:
model A (A), model B (B), and model C (C). Bolus
input is assumed to take place in compartment 1. Samples
s1, s2, s3, s4, and
s5 represent cumulative ileal effluents, plasma free amino
acids, body urea, and cumulative urinary urea and ammonia,
respectively. Model C was finally selected to describe the
whole system. See text for further explanation.
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The connection between the GI tract and the retention subsystems was
achieved via the small intestine lumen and the splanchnic compartment,
because physiologically, the intestinal absorption of nitrogen leads to
its transfer toward the liver via the portal vein. Moreover, the
deamination subsystem was also connected to the splanchnic area,
because the main route of deamination, i.e., urea genesis, takes place
in the liver. This first model was not sufficiently consistent with the
data after extensive parameter changes had been explored, because a
simultaneous fit of plasma and urine data proved impossible. This major
inconsistency pointed to the need for more than three compartments in
the retention subsystem. The preliminary model structure was thus
modified so as to iteratively adjust the model structure and parameter
values in a physiologically reasonable way and to achieve an adequate match between observed and simulated data. We then tested models B and C of increasing order (Fig. 3, models
B and C), and we discriminated between these candidate
models by determining specific criteria, such as the F-test
and generalized likelihood ratio test. This ratio test compares the LLF
of two nested models and states that the quantity
2(LLF2
LLF1) follows a
2
distribution with r degrees of freedom, where the subscripts 1 and 2 denote the smaller and larger models, respectively, and r is the difference in the number of parameters between
models (37). After examining these statistical criteria
(Table 4), we finally selected
model C over the other two. No significant improvements have
so far been achieved by adding other compartments in the retention
area.
Model selected and theoretical (a priori) identifiability.
A linear, 11-compartment model was finally selected to fit the data
(Fig. 4). This model included all of the
sampled compartments [ileal effluents (E), plasma free amino acids
(AA), body urea (BU), urinary urea (UU), and urinary ammonia (UA)] to
cover all experimental data (34). A unidirectional chain
of three compartments was used to describe the GI tract:
compartment 1 corresponding to the stomach [gastric
nitrogen content (G)], compartment 2 to the intestinal
lumen (IL) nitrogen content, and compartment 3 to entry into
the cecum [ileal effluents (E)] from which fecal losses take place.
Compartment 4 corresponds to splanchnic free amino acids
(SA) exchanging bidirectionally with the intestine (absorption and
release into the intestinal lumen) and with two other compartments,
5 and 7. Compartment 5 represents
plasma free amino acids (AA). Compartment 7 corresponds to
the splanchnic protein (SP) pool, and reversible pathways between
compartments 4 and 7 reflect the
synthesis and degradation phenomena. Two irreversible losses occur from
compartment 4, one through the body urea (BU, compartment 9) from which UU is irreversibly lost
(compartment 10) and the other representing UA losses
(compartment 11). Finally, plasma exchanges occur
bidirectionally in a catenary structure with compartments 6 and 8, which represent peripheral free amino acids (PA) and
peripheral protein (PP), respectively.

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Fig. 4.
The model finally selected. Circles, compartments
representing kinetically distinct pools of dietary nitrogen; arrows
between compartments, transfer pathways; nos. by the arrows, fractional
transfer coefficients or transfer rate constants; bullets, those
compartments that were sampled. See text for further explanation.
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It was necessary for this first stage in model development, i.e.,
structural modeling, to be validated, so as to check whether all
unknown parameters could be uniquely or nonuniquely estimated, thus
ensuring the a priori identifiability of the model
(8, 16). This step is necessary to avoid the
choice of a model structure that would produce an infinite number of
solutions. This is particularly important when a physiological model is
developed, because different sets of parameter values can give rise to
different conclusions (16, 32). This question
is set in the context of an error-free compartmental model structure
with noise-free and continuous time measurements, i.e., ideal data. A
software package, GLOBI 2, has recently been developed for linear
compartmental models (2) and enables assessment of the
identifiability (unique or nonunique) or nonidentifiability of a model.
Application of GLOBI 2 showed that our selected model was uniquely
identifiable, i.e., all parameters had a unique solution
(12). It was then possible to turn to the problem of
numerical identification of the model, by estimating the numerical
values of unknown parameters from the noisy experimental data set
(12).
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PARAMETER ESTIMATION AND NUMERICAL IDENTIFIABILITY |
Once we had finally chosen a model structure, with transfer rate
constants that could theoretically be determined uniquely in an ideal
context of noise-free data, we used SIMUSOLV to perform the parameter
estimation from the real experimental data set leading to numerical
identification of the selected model.
Parameter estimation strategy.
The parameter estimation process sought parameter values that, in this
instance, would enable the model to generate the closest predictions
for five experimental data pools (ileal effluents, plasma free amino
acids, body urea, and urinary urea and ammonia) simultaneously.
The objective function used for the parameter estimation process in
SIMUSOLV is the likelihood function (LF), which represents the joint
probability of obtaining our experimental data for each sampled pool in
the context of a given set of fitted parameters and takes account of
the fact that an experimental error is always associated with
experimental measurements (36, 37). With the assumption that our measurement errors were normally distributed and
independent of each other, the LLF was used for the convenience of
mathematical manipulations. The optimization process, which is an
iterative method, attempts to locate a maximum point in the surface
defined by the LLF. At each step of this iteration, SIMUSOLV checks,
using a generalized reduced gradient method, whether further changes in
the values of the parameters can increase the value of LLF
(15). The estimation process is terminated when the LLF
has been maximized, i.e., the value of LLF cannot be increased by any
further variations in the parameters. Nevertheless, a trap occurred
when the starting values of some adjustable parameters were too far
away from the correct ones. Changes in parameter values may have an
insignificant effect on test criteria, and SIMUSOLV will cease to
change them. The result, of course, is an erroneous set of parameter values.
In this case, the problem was that we had little or no information
about the optimal values for parameters, so that it was difficult to
determine the adequate initial values necessary to start optimization.
To prevent this problem, we first explored a large domain of variations
in all parameters during the fitting process by testing different
initial values for parameter estimates. As this first step of intuitive
parameter estimation was very time-consuming, an additional sensitivity
analysis aimed to identify those parameters with the strongest
influence on model predictions and behavior (35,
37). This intuitive parameter estimation step and
sensitivity analysis enabled both a better understanding of model
behavior and circumscription of the domain of optimal values for
parameters. We thus focused on variations in parameters that had the
strongest influence on the system during the fitting process, by
optimizing them in the first instance and then by exploring widely the
effect of their conjoint variations on model fitting, before optimizing
all parameters simultaneously. Different values for initial parameter
estimates were tested to reduce the probability of falling into a local
optimum for the LLF value if the starting point was not in the
neighborhood of the global optimum. We then observed that the
goodness-of-fit in the final step of optimization could be improved by
allowing only a restricted group of less correlated parameters to
adjust simultaneously. This was due to a strong correlation between
certain parameters affecting the numerical ability of SIMUSOLV to find
the global optimum, leading to high standard deviations for the fitted
parameters and an overall poor fit, because the surface of the LLF
constructed by the fifteen parameters and eleven compartments of the
model was too complex to enable a successful search for optimum values (37). The correlation matrix provided from statistical
output showed that two parameters of one bidirectional pathway
(ki,j and
kj,i) were strongly correlated.
We therefore decided to keep the reciprocal kj,i
of each ki,j constant
during the fitting process and equal to a value determined during the
previous step of parameter estimation. Finally, the final parameter
estimates were obtained and verified as providing the best possible
fit, and not a local optimum.
Sensitivity analysis.
Sensitivity analysis of the model was performed by evaluating the
effect of a 1% change in parameter value on the prediction of a
variable response, i.e., by calculating a sensitivity coefficient for
each pair:
(model response)/
(model parameters). However, to
eliminate the bias caused by the magnitude in parameter values, the
sensitivity coefficients were log-normalized and calculated using the
direct decoupled method under SIMUSOLV (15).
Sensitivity analysis was performed on each compartment and also on the
deamination and retention subsystems following the definition of new
variables, DEA and RET. For this purpose, DEA was calculated as being
the sum of the nitrogen content of BU, UU, and UA, and RET as the sum
of the nitrogen content of SP, SA, AA, PA, and PP. RET was also
subdivided into splanchnic (S = SA + SP) and peripheral (P = PA + PP) components, so as to evaluate the relative distribution of
retained nitrogen in those areas. Figure
5 shows the relative influence of the
fitted parameters on the subsystems. Whatever the compartment and
subsystem, k2,1 and k4,2
showed considerable initial influence, which then rapidly declined. In
the deamination subsystem, k9,4 had the
strongest positive influence on DEA (Fig. 5A). Moreover,
k5,4 and, to a lesser extent,
k7,4 and k3,2, had an
increasingly negative influence over time on the dietary nitrogen
content of the deamination subsystem. RET was most rapidly positively
sensitive to k5,4, k4,2,
and k7,4, in descending order, whereas it was
negatively influenced by variations in k9,4 and
k3,2, these trends increasing over time (Fig.
5B). As shown in Fig. 5C, the S content was most
positively sensitive to variations in k7,4 and
negatively to k5,4, and, to a lesser extent, to
variations in k9,4. Inversely, the P content was
most positively sensitive to variations in k5,4
and negatively to those in k7,4 and
k9,4 (Fig. 5D). To summarize,
k2,1 and k4,2 on the one
hand, and k5,4, k9,4, and
k7,4 on the other hand were identified as
important governing parameters. Consequently, we focused thereafter on
their variations within the fitting process. Similarly, we applied
physiological boundaries to the range of variations in the gastric
emptying rate k2,1 during parameter estimation,
because this exerts a primordial initial influence on the system.

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Fig. 5.
Relative sensitivity of fitted parameters
(k2,1, k3,2,
k4,2, k5,4,
k7,4, k6,5,
k8,6, k11,4,
k9,4, k10,9) to variables
representing deamination, DEA = BU + UU + UA (A); total
retention, RET = SA + SP + AA + PA + PP (B); splanchnic
retention, S = SA + SP (C); peripheral retention,
P = PA + PP (D).
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Results of parameter estimation.
The model was then quantified for each individual data set and for the
mean of data values using parameter estimation (13). The
model fitted all the data well for each subject, but a better fit was
obtained with the mean of the data. A typical fit (subject 8) is shown in Fig. 6 for each
sampled compartment, in terms of
of ingested dietary nitrogen. For
this typical subject and for the mean of subjects, optimization
criterion values and parameter estimates are given in Table
5. The distribution of parameter estimates did not differ significantly when obtained using the mean of
individually fitted parameters or when directly fitting the mean of
individual data (Wilcoxon matched-pairs signed-rank test with a
two-tailed P value of >0.99 was also considered
nonsignificant).

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Fig. 6.
A typical fit (subject 8). Observed vs.
predicted values for each sampled compartment. Each observed datum is
plotted by value ± 2 SD, with the weighting scheme determined by
, the coefficient of heteroscedasticity, during optimization.
A: plasma free amino acids (AA) and cumulative urinary
ammonia (UA). Lines, computer simulations; , AA
experimental data; , UA experimental data.
B: kinetics of body urea (BU) and cumulative urinary urea
(UU). Lines, computer simulations; , BU experimental
data; , UU experimental data. C: cumulative
ileal effluents (E). Lines, computer simulations; , E
experimental data.
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Table 5.
Values of estimated model parameters and respective CV values for
adjustable parameters obtained after optimization on a typical subject
and on the mean of individual data
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The model was optimized by allowing the values of
, the
heteroscedasticity parameter representing the heterogeneous error of
each sampled compartment, to be adjusted during the optimization process. In SIMUSOLV, this parameter may vary between 0 and 2, that is,
between the two extreme cases in which the absolute variability of each
datum from the same sampled compartment is constant for
= 0 (the standard deviation of each datum being independent of the value of
the data), and in which the relative variability is constant for
= 2 (the standard deviation being proportional to the value of
the data). The values obtained for
after optimization (Table
6) made it possible to gain information
about the error pattern of the experimental sets of data. For AA and BU
kinetics, we obtained the same classical error pattern, with a standard deviation proportional to the value of the data (
= 2). In
contrast, UA, E, and UU cumulated kinetics data exhibited different
error patterns, because their optimized values for
were 2, 1.1, and 0, respectively (Table 6). This last value (
= 0) made it
possible to force the optimization on the larger values of UU, i.e.,
data with lower relative variability. This seemed to be appropriate in
the case of a cumulated data set with a wide range of values, where the
latest and highest data are the most reliable. For E, SIMUSOLV found an
intermediate value for
(
= 1.1), probably because the range
in values for E was smaller. UA, the cumulated kinetics with the
smallest range of values, exhibited the same error pattern as AA and BU
kinetics (
= 2).
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Table 6.
LLF and heteroscedasticity parameter values for each sampled
compartment and P values of the corresponding runs tests
|
|
Numerical (a posteriori) identifiability of the model.
This second stage of model development, i.e., parameter estimation,
also required validation to ensure that parameters were estimated with
sufficient confidence to provide meaningful information about the
system under study. It was therefore necessary to check the a
posteriori, numerical, or practical identifiability of the model so as
to have confidence in its results and ensuing predictions. The
quantitative assessment of model quality from parameter estimation is
crucial, because the physiological conclusions drawn from model predictions depend intimately on estimated parameter values
(12). The parameter estimation process provides the model
fit to the data, the residuals (i.e., the difference between a datum
and its predicted value at each sampling time), and the precision of
estimated parameter values (12). All of this information was studied to evaluate the numerical identifiability of the model by
testing successively the goodness-of-fit, the randomness of residual
errors, and the reliability of parameter estimates.
The first criteria to be satisfied during a numerical validation
process are goodness-of-fit and the randomness of the residual errors
obtained from the fitting process. Goodness-of-fit can be judged by
visual inspection of a plot of model predictions vs. experimental data
(as shown in Fig. 6), to ensure that datum points are randomly
scattered around the fitted curve (37). A more accurate
way of assessing goodness-of-fit is an analysis of residuals, providing
a check on the underlying assumption of the normality of the data error
distribution involved in optimization (37). If this
assumption is valid, standardized residuals, i.e., the difference
between a datum and its model prediction divided by the standard
deviation of the datum, should follow a normal distribution, with a
mean of 0 and a variance of 1. Thus, under this assumption, 95% of
standardized residuals should lie within the range of
1.96 to +1.96
(37). As shown in Fig. 7,
all standardized residuals of the sampled compartments were within or
close to the 95% interval range. The observation of nonrandomness in
residuals enables the detection of any systematic deviations between
experimental data and model predictions; it generally indicates that
the model is too simple to accurately fit the data and may require more compartments than those postulated (10-12).
Nonrandomness in the residuals can be tested formally using the runs
test, which counts the number of consecutive residuals with the same
sign and compares it with the number of runs expected if the residuals
were randomly scattered (27). As shown in Table 6, it
could be concluded for each sampled compartment that the residuals were
consistent with the hypothesis of randomness, because the P
value was higher than 0.75 (12).

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Fig. 7.
Weighted residuals vs. sampled time for each sampled
compartment after optimization on the mean of individual data: ileal
effluents (A), plasma free amino acids (B),
urinary urea (C), urinary ammonia (D), and body
urea (E).
|
|
The next criterion to be satisfied during a numerical validation
process is the reliability of parameter estimates. SIMUSOLV provides an
approximation for the covariance matrix of parameter estimates from the
inverse of the Fisher information matrix (15, 28). Thus, when the variances are known, the precision of
fitted parameters can be expressed in terms of percent fractional
standard deviation or coefficient of variation (CV), as follows
(8, 11, 12)
The smaller the CV value, the better the estimated value of the
parameter (16). Parameter values with a CV of <50% are usually judged to be adequately estimated (32). As shown
in Table 5, the highest CV for fitted parameters was <7%, so it was
thus possible to consider that the parameters had been estimated with
excellent precision. Furthermore, absolute values for correlation coefficients between fitted parameters ranged from
0.01 to 0.89 (Table 7). The fitted parameters were
consequently never strongly correlated, because the correlation
coefficients were always <0.9 (16).
 |
DISCUSSION |
The aim of this study was to develop and validate a compartmental
model describing the assimilation and metabolic distribution of dietary
nitrogen in the postprandial phase in humans. The model structure
included three subsystems: the GI tract, retention, and deamination.
The 11-compartment model that was selected to fit the experimental data
was validated at each stage of its development by testing successively
its a priori (theoretical) and a posteriori (numerical)
identifiability. We successively verified that the 15 parameters of the
model could theoretically be uniquely determined in an ideal context of
noise-free data, and then that all parameters could be estimated with
highly satisfactory precision from the experimental data set
(10, 28). An important outcome of the model
was a simulation of the kinetics of dietary nitrogen in different pools
in the body and a prediction of the further evolution of the system
(Figs. 8 and
9).

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Fig. 8.
Model predictions for the evolution of nonaccessible pool
sizes [gastric nitrogen content (G), intestinal lumen nitrogen content
(IL), splanchnic free amino acids (SA), peripheral free amino acids
(PA), splanchnic protein (SP), and peripheral protein (PP)] in % of
ingested nitrogen over time. Values are obtained by optimization on the
mean of individual data.
|
|

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Fig. 9.
Evolution of the distribution of retained dietary
nitrogen between free and bound amino acids in visceral or peripheral
areas. Values are expressed as % of ingested nitrogen and obtained by
optimization on the mean of individual data.
|
|
Our approach was to develop a multicompartmental model to describe the
digestion, absorption, and whole body metabolism of dietary nitrogen on
the basis of data obtained after a bolus administration in humans of
uniformly and intrinsically 15N-labeled milk protein. This
type of compartmental analysis requires the development of a complex
model of the system to investigate the distribution kinetics of dietary
nitrogen in different metabolic pools during the postprandial phase,
i.e., in the nonsteady state (8). Kinetic tracer studies
have been used extensively and analyzed using model-based compartmental
analysis to provide quantitative and predictive information concerning
the dynamics of numerous specific nutrient systems (30).
Among studies of protein metabolism, particular interest has been
focused on body amino acid kinetics, and many models have been
developed for leucine kinetics on the basis of infused tracer
experiments. The 10-compartment model developed by Cobelli et al.
(13) distinguishes intracellular and extracellular
ketoisocaproate pools and free or protein-bounded leucine, and the
16-compartment model designed by Carraro et al. (7) in
dogs distinguishes extracellular and intracellular free or
protein-bounded pools of leucine in different organs (liver, muscle,
and the like). As for the specific fate of exogenous nitrogen ingested
after a meal, the partioning of dietary nitrogen has been modeled in
preruminant calves (21) and growing pigs (33) on the basis of empirically derived components specifically related to
the studied organisms rather than on tracer studies. The metabolic fate
of 15N-labeled yeast protein in humans (45)
has been tentatively monitored after its ingestion and described using
a three-compartmental model that distinguished amino acids, proteins,
and excretion nitrogen pools. As far as we know, our work constitutes
the first attempt at modeling the distribution of dietary nitrogen from ingestion through its elimination or retention in the metabolic pools
of the body in a multitissue scheme (splanchnic and peripheral). Our
modeling approach is more closely related to that used to describe the
ingestion and whole body distribution of various micronutrients such as
zinc (16, 28, 32), selenium
(26, 34), or magnesium (38).
Furthermore, the pharmacokinetic literature was of value to our design
because of the extensive analogies between drug and nutrient kinetic
patterns. In particular, the "first-pass" pharmacokinetic model, in
which the liver (or, more generally, the hepatoportal system) is
represented by a kinetically distinct compartment, exhibits strong
similarity with our model structure (35).
Different approximations were made a priori for the calculation of
certain data used for model development. This was the case for data on
plasma free amino acids and urea. In fact, the size of the plasma free
amino acid pool varies after a meal, but different fasting and
postprandial values have been obtained during studies (1,
3, 5), depending on both the analytical
methods employed and the meal ingested. However, variations in pool
sizes are not so broad, as illustrated by the results of
Bergström et al. (3), leading to a plasma free amino
acid pool size ranging from 232 mg in the fasted state to a maximum
value of 348 mg 1 h after the ingestion of 50 g of bovine
serum albumin. These values ranged from 182 to 275 mg in the study by
Adibi and Mercer (1). Thus the amount of dietary nitrogen
present in plasma free amino acids was calculated by assuming that the
total amino acid level was constant and equal to a mean value of 300 mg
over time. We tested the differences for AA data when they were
calculated either with the assumption as constant of (300 mg) the total
amino acid level or by use of the variable pool size values obtained
from the data collected by Bergström et al.
(3). A repeated-measures ANOVA using a general linear
models procedure gave no statistical difference at each point and
regarding the global kinetics for these two methods. Moreover, the
exogenous nitrogen present in the urea body pool was calculated using a
formula that took account of the TBW value, which was estimated using
Watson's equations (Watson et al., Ref. 42). In the absence of more
straightforward experimental methods, these approximations were made to
obtain the data necessary to this first attempt at model development.
More accurate data would clearly be useful to further refine the
predictions of the model thus developed. Furthermore, the parsimonious
criteria used for structural modeling led to the choice of a model that
a posteriori neglected certain metabolic pathways during the period
considered. Both ileal effluents and compartments in the deamination
subsystem were modeled using compartments with no output and considered as sites of irreversible losses. The recycling of dietary nitrogen from
body urea in secondary metabolic pathways (23) was also neglected. It has been reported that ~20% of the urea produced is
delivered to the colon (23). When this figure is
considered, together with our observation of a maximum deamination
value of 24% of ingested nitrogen over the period considered
(18), it appears that neglecting this sparing phenomenon
gives rise to only a small error of 4-5% of ingested nitrogen.
Further development will probably involve the integration of these
aspects in a more complete model capable of predicting nitrogen
distribution over a longer period.
The model chosen monitors the fate of dietary nitrogen through the GI
tract. Gastric emptying delivers dietary nitrogen into the intestinal
lumen, where it is either absorbed or transferred to the ileal
effluents. The model simulates rapid emptying of the gastric content
with an emptying half-time of ~20 min. This prediction is consistent
with previous experimental results (29). Moreover, results
of the sensitivity analysis, which enabled identification of those
parameters with the greatest influence on the system, agreed with the
model structure and our knowledge of system behavior. The gastric
emptying rate (k2,1), and to a lesser extent the
intestinal absorption rate (k4,2), exert a
fundamental initial influence on the system. This is consistent with
the importance of digestive kinetics to protein metabolism that has
already been reported. The gastric emptying rate
(k2,1) is known generally as the principal kinetic parameter governing intestinal absorption, and its influence on
the system is greater than that of the intestinal absorption rate
(k4,2), because absorption capacities are seldom
saturated under normal physiological conditions (19,
43). Furthermore, even if the influence of
k2,1 and k4,2 on nitrogen
deamination (DEA) declines over time (Fig. 5A), sensitivity
to these parameters persists, suggesting that the flow rate of nitrogen
absorption and the early kinetics of amino acid delivery to the
splanchnic tissues partly determine the further entry of dietary amino
acids into the different catabolic pathways.
Once absorbed through the GI tract, dietary nitrogen is transferred to
the blood and then either retained in the body or eliminated by
deamination. The different compartments of the deamination subsystem
are all connected to the splanchnic free amino acid compartment of the
retention subsystem, because the oxidative degradation of dietary amino
acids takes place mainly in the splanchnic area. However, as far as the
elimination of NH3 is concerned, the direct relationship
between the splanchnic area and the deamination subsystem may appear
controversial. Indeed, although all of the tissues produce some
ammonia, it is usually assumed that the kidney, a peripheral organ, is
the main source of urinary ammonia. However, because the kidney is not
represented in our model, and a direct connection between either PA or
AA and UA provided a poor data fit, we preferred to retain the direct
elimination of dietary ammonia from the splanchnic area, the major site
of dietary nitrogen deamination. Sensitivity analysis showed that the
transfer rate of dietary nitrogen to body urea
(k9,4) had the major positive influence
persisting over time on nitrogen deamination (DEA) and thus a
considerable negative influence on nitrogen retention (RET). The
behavior of the deamination subsystem thus agrees with our knowledge of
the system, because the earlier deamination kinetics of splanchnic
dietary nitrogen in body urea (k9,4) further
determines the kinetics of its elimination from the urine.
Our model also enabled simulation of the distribution of absorbed,
nondeaminated dietary nitrogen between the different metabolic pools in
the retention subsystem. The retention subsystem proposed was built to
clarify nitrogen distribution between splanchnic and peripheral
tissues. Interestingly, the results indicated that the optimal
retention subsystem structure for both the splanchnic and peripheral
areas presented two compartments, leading to the conclusion that it was
necessary to distinguish between free and protein-bound amino acids.
This is not surprising, because the kinetics of amino acids in each
compartment are known to differ (14, 40).
Indeed, it is classically assumed that intracellular free amino acids
are more direct precursors of protein synthesis than circulating amino
acids (4). The free amino acid compartment could be
considered as a buffer area, unlike tissue proteins, which present a
more limited ability to react to nutritional variations. Both the
splanchnic and peripheral free amino acid areas must be considered as
crucial, because they are likely to act as important regulators of the
transfer of dietary nitrogen to tissue proteins. Given the structure of
our model, the splanchnic free amino acid zone (SA) is particularly
well defined because it is flanked by three sampled compartments (AA,
BU, and UA). Sensitivity analysis indicated a central and regulatory
role of SA, which was consistent with current knowledge on nitrogen
metabolism in humans (14), because it regulates both the
kinetics of oxidative degradation in the deamination subsystem and the
delivery of dietary amino acids in the peripheral zone. This is first
perceptible from the influence of the transfer rate from SA to body
urea (k9,4) on deamination (see above).
Moreover, the persistence over time and the positive influence of
k4,2 on nitrogen retention in the peripheral area (P, Fig. 5D) is indicative of the pronounced influence
of dietary amino acid delivery kinetics in SA on their future disposal in the periphery. Furthermore, other disappearance rates from SA, i.e.,
the delivery of amino acids to the periphery
(k5,4) and transfer to the splanchnic protein
(k7,4) constitute the last group with a major
influence on nitrogen retention (RET) and deamination (DEA) in the
system. k5,4 and, to a lesser extent,
k7,4, have a negative influence on DEA and a
positive one on RET, both of which grow over time. Thus the transfer of
dietary nitrogen from SA to the periphery (k5,4)
seems to improve retention to a greater extent than transfer from SA to
splanchnic protein (k7,4), although temporary
storage in splanchnic protein may contribute to sparing some dietary
nitrogen from oxidative pathways initiated in SA. The model simulates
the replenishment of SA with a maximum value being reached at 50 min,
representing 45% of ingested nitrogen (Fig. 8). The dietary contents
of this compartment are almost completely emptied 12 h after the
meal, whereas dietary nitrogen is partly redistributed to splanchnic
protein and the peripheral zone. The maximum value in the size of the
PA compartment is achieved later (3 h 20 min after the meal) and
reaches 26% of ingested nitrogen.
An important outcome of the simulation was an evaluation of the
partitioning of dietary nitrogen between splanchnic and peripheral protein. Our model simulates the incorporation of dietary nitrogen into
splanchnic protein decreasing after 4 h 20 min, whereas it was
still increasing after 12 h in peripheral protein. Moreover, the
maximum values achieved over the simulation period were 22 and 43% of
ingested nitrogen in splanchnic and peripheral protein, respectively.
These findings are consistent with the differential size of the
peripheral and splanchnic protein pools (14) and with the
higher turnover already reported in splanchnic tissues compared with
peripheral ones, especially muscle (31). Figure 9 shows
the distribution of dietary nitrogen between free and bound amino acids
in visceral or peripheral areas. Indeed, the model predicted that the
dietary nitrogen retained in the body 8 h after the meal consisted
of 28% free amino acids and 72% protein, ~30% being recovered in
the splanchnic bed vs. 70% in the peripheral area. Twelve hours after
the meal, these values had decreased to 18 and 23% for the free amino
acid fraction and splanchnic nitrogen, respectively. Few data
concerning the fate of dietary nitrogen in the organs are available to
assess the extent to which the model is compatible with current
knowledge of the system. Our model showed that splanchnic utilization
(SA + SP + DEA) of dietary nitrogen reached 47% of dietary input
6 h after the meal, with 21% of ingested nitrogen incorporated
into protein. By comparison, splanchnic extraction of dietary leucine
in humans is reported to reach 30-40% (9,
25). However, these values are hardly comparable, because
leucine is not representative of all amino acids and undergoes less
catabolism in the splanchnic zone (22). According to the
results obtained in piglets, and taking account of an averaged value
for the removal of four amino acids (Leu, Lys, Phe, and Thr),
splanchnic utilization reached 53% of dietary input 6 h after the
meal, with 15% of ingested nitrogen incorporated into splanchnic
protein (39). This indicates that the predictions of the
model were close to those reported in the literature, although the data
obtained during 13C or 15N tracer studies are
not directly comparable. Nonetheless, those findings emphasized the
validity of the model (10).
In conclusion, we have developed a descriptive and predictive model of
postprandial dietary nitrogen distribution in humans, which enables the
simulation of exogenous nitrogen kinetics in the different metabolic
pools of the body and the prediction of system evolution. It will now
be used to compare the differential distribution of dietary nitrogen
under different conditions, i.e., type of the nutritional status, type
of meal, or type of dietary protein. This will then enable testing of
both the validity of the model against data independent of those used
for the fitting process and the capacity of the model to discriminate
among several nutritional conditions. This model should constitute a
useful explanatory tool to describe the processes involved in the
differential metabolic utilization of various protein meals.
 |
ACKNOWLEDGEMENTS |
Use of the Globi 2 software was made possible through the
generosity of Prof. Claudio Cobelli. We also acknowledge the
contribution of the modeling work group at the Institut National
Agronique Paris-Grignon to stimulating discussions during the course of this work.
 |
FOOTNOTES |
This work was supported by ARILAIT Recherches.
Address for reprint requests and other correspondence: Claire
Gaudichon, Unite INRA Nutrition humaine et physiologie intestinale, Institut National Agronomique Paris-Grignon, 16 rue Claude Bernard, 75231 Paris Cédex 05, France.
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. §1734 solely to indicate this fact.
Received 6 October 1999; accepted in final form 1 February 2000.
 |
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