## 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 ^{15}N nitrogen kinetics in the intestine, blood, and urine after the oral administration of^{15}N-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

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^{13}C tracers enables an assessment of carbon skeleton sparing, in contrast to the ^{15}N methods used to study the amino residue. The uniform and intrinsic ^{15}N 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 [^{15}N]nitrogen kinetics determined in the intestine, blood, and urine after the ingestion of ^{15}N-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).

## 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 ^{15}N-labeled milk protein. ^{15}N 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 ^{15}N 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^{15}N 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 (γ) 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 2show 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.

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 Table3 showed no significant improvement in fit when a compartment was added.

#### 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*).

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(LLF_{2} − LLF_{1}) follows a χ^{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.

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).

## 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 (*k _{i,j}
* and

*k*) were strongly correlated. We therefore decided to keep the reciprocal

_{j,i}*k*of each

_{j,i}*k*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.

_{i,j}#### 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: δ(model response)/δ(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. Figure5 shows the relative influence of the fitted parameters on the subsystems. Whatever the compartment and subsystem, *k*
_{2,1} and *k*
_{4,2}showed considerable initial influence, which then rapidly declined. In the deamination subsystem, *k*
_{9,4} had the strongest positive influence on DEA (Fig. 5
*A*). Moreover,*k*
_{5,4} and, to a lesser extent,*k*
_{7,4} and *k*
_{3,2}, had an increasingly negative influence over time on the dietary nitrogen content of the deamination subsystem. RET was most rapidly positively sensitive to *k*
_{5,4}, *k*
_{4,2}, and *k*
_{7,4}, in descending order, whereas it was negatively influenced by variations in *k*
_{9,4} and*k*
_{3,2}, these trends increasing over time (Fig.5
*B*). As shown in Fig. 5
*C*, the S content was most positively sensitive to variations in *k*
_{7,4} and negatively to *k*
_{5,4,} and, to a lesser extent, to variations in *k*
_{9,4}. Inversely, the P content was most positively sensitive to variations in *k*
_{5,4}and negatively to those in *k*
_{7,4} and*k*
_{9,4} (Fig. 5
*D*). To summarize,*k*
_{2,1} and *k*
_{4,2} on the one hand, and *k*
_{5,4}, *k*
_{9,4}, and*k*
_{7,4} 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 *k*
_{2,1} during parameter estimation, because this exerts a primordial initial influence on the system.

#### 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 Table5. 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).

The model was optimized by allowing the values of γ, 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 γ = 0 (the standard deviation of each datum being independent of the value of the data), and in which the relative variability is constant for γ = 2 (the standard deviation being proportional to the value of the data). The values obtained for γ after optimization (Table6) 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 (γ = 2). In contrast, UA, E, and UU cumulated kinetics data exhibited different error patterns, because their optimized values for γ were 2, 1.1, and 0, respectively (Table 6). This last value (γ = 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 γ (γ = 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 (γ = 2).

#### 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).

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 and9).

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 ^{15}N-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 ^{15}N-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 (*k*
_{2,1}), and to a lesser extent the intestinal absorption rate (*k*
_{4,2}), 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 (*k*
_{2,1}) 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 (*k*
_{4,2}), because absorption capacities are seldom saturated under normal physiological conditions (19,43). Furthermore, even if the influence of*k*
_{2,1} and *k*
_{4,2} on nitrogen deamination (DEA) declines over time (Fig. 5
*A*), 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 NH_{3} 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 (*k*
_{9,4}) 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 (*k*
_{9,4}) 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 (*k*
_{9,4}) on deamination (see above). Moreover, the persistence over time and the positive influence of*k*
_{4,2} on nitrogen retention in the peripheral area (P, Fig. 5
*D*) 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 (*k*
_{5,4}) and transfer to the splanchnic protein (*k*
_{7,4}) constitute the last group with a major influence on nitrogen retention (RET) and deamination (DEA) in the system. *k*
_{5,4} and, to a lesser extent,*k*
_{7,4}, 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 (*k*
_{5,4}) seems to improve retention to a greater extent than transfer from SA to splanchnic protein (*k*
_{7,4}), 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 ^{13}C or ^{15}N 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.

## Acknowledgments

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.

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