## Abstract

A compartmental model of zinc metabolism has been developed from stable isotope tracer studies of five healthy adults. Multiple isotope tracers were administered orally and intravenously, and the resulting enrichment was measured in plasma, erythrocytes, urine, and feces for as long as 3 wk. Data from total zinc measurements and model-independent calculations of various steady-state parameters were also modeled with the kinetic data. A structure comprised of 14 compartments and as many as 25 unknown kinetic parameters was developed to adequately model the data from each of the individual studies. The structural identifiability of the model was established using the GLOBI2 identifiability analysis software. Numerical identifiability of parameter estimates was evaluated using statistical data provided by SAAM. A majority of the model parameters was estimated with sufficient statistical certainty to be considered well determined. After the fitting of the model and data from the individual studies using SAAM/CONSAM, results were submitted to SAAM extended multiple studies analysis for aggregation into a single set of population parameters and statistics. The model was judged to be valid based on criteria described elsewhere.

- SAAM
- GLOBI2
- extended multiple studies analysis
- isotope tracer kinetics
- population studies
- trace element

although zinc is well established as a micronutrient of major practical importance in human nutrition, our understanding of how to prevent or to detect and treat human zinc deficiency continues to be hampered by our limited knowledge of zinc metabolism and homeostatic mechanisms under various dietary, physiological, and pathophysiological circumstances. A variety of mathematical models has been applied to the analysis of natural zinc and isotopic tracer data, contributing significantly to our knowledge of zinc metabolism and homeostasis. Compartmental modeling is an especially powerful analytic method that is particularly appropriate for investigating biological systems (10). Various compartmental models of human zinc metabolism have already been described (2, 24, 26, 45, 46, 59, 61, 63). The model described by Wastney et al. (59) is noteworthy because of its experimental basis, detail, and subsequent application.

All the existing models were judged to be inadequate for modeling our data, usually because the structures were too simple or too complex. For example, in using the model of Wastney et al. (59) to analyze some of our data, we found that*1*) the parameter values that could be well determined from our data were often significantly different (more than ±2 SD) than those of the Wastney model, *2*) our data were inadequate to determine many of the other parameters, e.g., those associated with slowly exchanging compartments and, given *point 1*, we did not feel that it was appropriate to assume the model's values for these parameters, and *3*) the absorption and excretion structures were too simple to provide a good fit for our data.

Furthermore, given our interest in homeostatic mechanisms involving intestinal transport processes and the rapidly exchanging (metabolically mobilizable) zinc in the body, we expected that a model developed with an emphasis on these areas and based on the abundant stable isotope data available from our multiple-tracer studies would be potentially more useful for our research. And, because the slowly exchanging body zinc was not of immediate interest to us, the potential inability to observe slow, long-term processes due to the sensitivity limitations of stable isotope methods would not be a problem. Therefore, we developed our own normal population compartmental model of human zinc metabolism using the SAAM31 and CONSAM computer programs (6, 9, 16). The details of model structure development and preliminary results have been described elsewhere (49). In the course of model development and parameter estimation we used several important methods and tools that have not been widely used in other compartmental modeling of mineral metabolism.

### Simultaneous Modeling of Kinetic Data From as Many as Three Tracers and Model-Independent Steady-State Data

On the theory that providing more data to the modeling process would best ensure the development of an appropriate model structure and the derivation of accurate and meaningful parameter estimates, we used as many as three isotope tracers in our studies and also incorporated various independently derived steady-state data into the parameter estimation process. All data were simultaneously fitted by, for the most part, a single set of parameter values.

### Structural Identifiability Analysis Using GLOBI2

Structural identifiability (also known as a priori or theoretical identifiability) refers to the property of a model that its parameters can be determined from a specific input-output experiment. That is, when the model is fitted to error-free data from the specified experiment, each of the parameters has a unique solution, i.e., is globally identifiable, or a finite number of solutions, i.e., is locally identifiable (10, 13, 29, 35). Until the structural identifiability of a model has been demonstrated, it cannot be known with certainty whether the parameter estimates resulting from its application to real data are the only possible solution (15). Formal proof of structural identifiability is a complicated mathematical undertaking for all but the simplest of models. Audoly et al. (1) recently described and made available a computer program called GLOBI2, which is capable of analyzing complex models to determine identifiability.

### Numerical Identifiability Evaluation of Parameter Estimates

Regardless of structural identifiability, parameter estimates resulting from the modeling of real data may not be meaningful because they have not been determined with sufficient certainty. This is generally the result of inadequate data quality and/or quantity or unjustified model complexity. Parameter estimation quality and its evaluation has been variously referred to as parameter estimability, identification, or numerical, a posteriori, or practical identifiability (10, 14, 36, 37, 42). SAAM/CONSAM provides several kinds of statistical data on parameter estimates that can be used to evaluate numerical identifiability.

### Use of SAAM Extended Multiple Studies Analysis to Derive Population Parameter Estimates and Statistics

Extended multiple studies analysis (EMSA) is a global-iterative, two-stage algorithm for aggregating results from individual SAAM studies into a set of population parameter values (47,62). It has been implemented as a facility of SAAM31 and runs in batch mode only. As a global-iterative method it takes into account parameter variance-covariance information from the individual analyses and, therefore, is likely to produce more accurate population data than the standard two-stage method that simply calculates the mean and variance of the individual parameter estimates (18, 56).

To summarize, our objectives were *1*) to develop a compartmental model of human zinc metabolism consistent with current knowledge of zinc physiology that was capable of simultaneously fitting kinetic data from multiple isotope tracers, as well as various model-independent steady-state data, obtained from stable isotope studies of a heterogeneous group of normal subjects and *2*) to then derive population parameter estimates for this set of normal subjects. Furthermore, we strove to ensure the validity of the model and the quality of the parameter estimates by the evaluation of statistical information and the application of recently available software tools.

## METHODS

### Subjects

The subjects were five healthy, nonsmoking adults: four females and one male. All subjects gave informed consent. None of the women were pregnant or lactating during, or for the 6-mo period before, the study. None of the subjects were taking mineral supplements. Several basic physical characteristics of the subjects are shown in Table1. Body weight remained constant for the duration of the study. Subjects stayed in the Clinical Research Center for the initial 8–12 h of the study period, thereafter resuming normal activities and lifestyle and returning to the Research Center for any subsequent tracer administration and blood sampling.

### Diet

Fourteen-day diet records were obtained from each subject, and a constant daily diet typical of normal eating habits was planned for ingestion during the study. In particular, the constant daily diet was designed to provide an amount of dietary zinc similar to a subject's usual zinc intake as indicated by analysis of the diet records. Dietary zinc intakes are listed in Table 2. Subjects began eating the constant daily diet 7 days before the start of the study period.

### Tracer Administration

Zinc oxide preparations enriched in ^{67}Zn (89.55 atom %), ^{68}Zn (99.42 atom %) and ^{70}Zn (99.72 atom %) were obtained from the Oak Ridge National Laboratories (Oak Ridge, TN). Details of the preparation of the isotopically enriched zinc for oral and intravenous administration have been described elsewhere (48). Preparation of the intravenous tracer differs from that previously described in that the enriched zinc was equilibrated in 25% albumin for 4 h before administration.

The intravenous tracer was administered to all subjects at ∼0700 on the first day of the study. Approximately 0.4 mg of^{70}Zn-enriched zinc in albumin was infused into a forearm vein over a 2-min interval. At about the same time, the first three subjects were given 1.0 mg of ^{67}Zn orally in water. This oral tracer was administered to provide data on absorption of zinc in the postabsorptive state (10–12 h after the last meal). Shortly thereafter the breakfast meal was ingested. A second oral tracer (^{68}Zn) was given to the first three subjects with all the meals on the second day of the study. A total of 3 mg of^{68}Zn was administered with the three meals, the quantity of tracer given with each meal being in proportion to the natural zinc content of the meal. This tracer zinc, in water, was sipped at frequent intervals during the second half of the meal. The second oral tracer provided data on absorption of dietary zinc and is referred to as the prandial tracer.

Because of problems with the quality of ^{68}Zn isotope ratio measurements and blood sampling frequency limitations on the second day, it became apparent that we would be unable to obtain adequate data to model both the postabsorptive and the prandial absorption processes as well as we desired. It was decided to forgo the postabsorptive tracer modeling and modify the protocol to acquire improved data on tracer absorption with meals. As a result, *subjects 4* and*5* were given only a prandial tracer, 3 mg of^{67}Zn tracer, with all meals on the first day (as described for the ^{68}Zn). Although it was of lesser quality, we did model the ^{68}Zn data from the first three subjects because it was necessary for the population model.

### Sample Collection

Baseline blood, urine, and fecal samples were collected, and the hematocrit was measured before tracer administration. Frequent blood sampling began immediately after administration of the intravenous tracer. Sampling was done via an indwelling plastic catheter in a forearm vein in the arm opposite the infusion site. The initial sampling rate was every 2 min with the interval increasing to 8 h by the end of the first 24 h (a total of 24 samples in the first 24 h) and finally to daily sampling from 48 h to the completion of the study. Blood samples were separated into their plasma and erythrocyte components immediately after collection. Complete quantitative collection of urine and feces began at the time of tracer administration and continued until the end of the study. Although the duration of the study of *subject 1* was 9 days, the study periods were subsequently lengthened. Blood sampling was extended to ∼24 days, and urine and feces collections to 16 days, depending on practical considerations, e.g., continued subject cooperation. Samples were stored at −20°C until prepared for analysis.

### Sample Preparation and Analysis

Details of the sample preparation and analysis procedures have been described elsewhere (34, 40, 48). Zinc concentrations were determined using a Perkin Elmer model 2380 atomic absorption spectrophotometer fitted with a deuterium arc background correction lamp. Isotope enrichment was measured by fast atom bombardment-secondary ion mass spectrometry using a VG 7070EHF double-focusing mass spectrometer equipped with an Ion Tech atom gun and modified to perform enhanced secondary ion energy filtering and ion counting detection.

### Data Calculations and Analyses

#### Model-independent calculations of steady-state parameters.

Various characteristics of the tracee steady state of the subjects were measured or estimated using model-independent methods to provide additional data to be modeled with the tracer kinetic data and for evaluating modeling results. These methods are referred to as “model independent” because they do not rely on the assumption of a particular model structure (10). Total dietary zinc was determined from diet records.

The mass of the plasma zinc was estimated by two methods, and the average of the two results was used. First, the estimated enrichment at*time 0* was determined from the sum of exponentials decay function best fitting the initial intravenous tracer enrichment data from the plasma and the plasma zinc mass (in mg) calculated as equal to [tracer dose (in mg)/enrichment_{t=0}] − tracer dose (in mg). In the second method, the mean measured plasma zinc concentration was multiplied by the estimated plasma volume (49). The mass of circulating erythrocyte zinc was estimated in the same manner, i.e., the mean measured erythrocyte zinc concentration was multiplied by the estimated erythrocyte volume (49). The whole body zinc mass was estimated to be 28 mg/kg of fat-free mass (64). Fat-free mass was estimated to be 72% of whole body mass for the female subjects with normal body mass indexes (4) and was adjusted downward for the overweight female. It was estimated to be 85% of whole body mass for the male subject (4).

Other important model-independent steady-state parameters were calculated using previously described algebraic, graphical, and other methods. Fractional absorption (FAZ) of oral tracers was calculated using four methods: *1*) measurement of the excretion of oral tracer in the feces (22, 40), *2*) measurement of the ratio of intravenous and oral tracer enrichment in urine (27,40), *3*) measurement of the ratio of intravenous and oral tracer enrichment in plasma (similar to urine method), and*4*) deconvolution of the appearance of oral tracer in the plasma (10, 35, 55). We performed the deconvolution using SAAM as described in the SAAM User's Manual (6, 7).

Fecal excretion of endogenous zinc (EFZ) was determined from the measurement of intravenous tracer excreted in the feces over a period of, typically, 5 days beginning no sooner than 3 days after isotope administration (38, 40). The quantity of tracer excreted was then divided by the average tracer enrichment in the plasma or in the urine during the corresponding time period (the fecal measurement period minus the colon transit time from the model). The average intravenous tracer enrichment in the plasma and urine was determined by integration of the exponential decay function fitting the enrichment data. The size of the “pool” of mobilizable zinc that exchanges rapidly, i.e., within 3 days, with the plasma was estimated from the extrapolation to *time 0* of the linear regression line fitting the log-transformed intravenous tracer enrichment data (from plasma or urine) between 3 and 10 days after tracer administration (48). This zinc is referred to as the exchangeable zinc pool (EZP).

Multiple methods of calculating steady-state data were used to improve the certainty of the values used in the model and to provide data for other studies. Generally, whenever there were multiple determinations of a parameter, the average was used in the model and the uncertainty of the value was based on the variability in the determinations. The steady-state parameter data were entered into SAAM as general functions of adjustable kinetic and steady-state parameters and accompanied by “standard deviation” values, reflecting our confidence in the values. Data entered in this manner are weighted and fit in a similar manner to the kinetic data and are referred to as “statistical constraints” in the SAAM manual (6).

#### Model development and structural identifiability analysis.

All model development and fitting to data (parameter estimation) was accomplished using SAAM31 and CONSAM. The most recent version of SAAM31, WinSAAM (32, 62), was not yet available at the time most of the modeling of the individual subjects was performed. The plots in Fig. 2 were created with WinSAAM.

Development of the model structure is described elsewhere (49). The modeling of multiple isotope tracers was handled in SAAM/CONSAM by creating a separate, but identical, model for each tracer. These parallel models were then analyzed simultaneously. The corresponding kinetic parameters of the models were all defined to be equal, e.g., *k(2,1)* had the same value in each model. In effect, it was required that a single model and set of parameter values be used to model the kinetic data from all the tracers. The only exception to this was that, in the case of the first three subjects where two oral isotopes were administered, five parameters associated with the small intestine were allowed to vary separately for each oral tracer to accommodate differences in the postabsorptive and prandial absorption processes. In these cases, the values of the corresponding parameters of the intravenous tracer model were tied to the prandial parameter values because the prandial tracer model was used in all the subjects. Although we did account for the small perturbation of the plasma zinc by the bolus administration of the intravenous tracer, we did not model other non-steady-state conditions, e.g., the intestinal transport of exogenous zinc, because such processes were unobservable from the available measurement data.

At the time of model development, the necessary topological conditions for structural identifiability were evaluated manually. These conditions are *1*) the structure must be input connectable,*2*) the structure must be output connectable, and*3*) the number of unknown parameters must be equal to or less than the number of independent relationships between those parameters (10, 11, 35). A system is input connectable (also referred to as input reachable) if there is at least one path leading to each compartment from a tracer administration site and is output connectable (output reachable) if there is at least one path from each compartment to a measured compartment (10, 35). The connectability conditions were evaluated by visual examination of the model structure and the location of experimental inputs and outputs. Testing for the third condition was performed using a published algorithm (10,11). An informal identifiability analysis of the model was performed based on general concepts and published analyses of simpler models (10, 12).

With the availability of GLOBI2, the necessary conditions were tested and a full identifiability analysis was performed using the program. The model structure and experiment were created in GLOBI2 using point-and-click manipulation of component icons as described in the*GLOBI2 Users Manual* (28). GLOBI2 requires that a multiple input-multiple output experiment be specified as a set of single input-single output experiments. Our experimental protocol was represented by eight single input-single output experiments. We implemented two parameter constraints in the GLOBI2 model, making*k(11,10)* = *k(10,9)*because *compartment 10* is a delay compartment and specifying that *k(0,1)* is a known constant because it was set to a fixed value for the computer fitting. GLOBI2 requires REDUCE 3.6, an algebraic computations program, to perform some mathematical operations. We used the Codemist distribution of REDUCE 3.6 for Windows (Codemist Limited, Bath, UK).

#### Data-model fitting (parameter estimation) and assessment of numerical identifiability of results.

Data from each individual study were initially fit by manual adjustment of parameters and evaluation of results. The goodness of fit to kinetic data was evaluated by visual inspection of data-model plots and by monitoring sums of squares of residuals. Once adequate agreement of model and data was achieved by manual fitting, the resulting parameter values were used as the initial parameter values for SAAM/CONSAM iterative parameter adjustment. The standard deviation values attributed to kinetic data were determined by type of data and estimated analytic variability. Fractional standard deviation (FSD) values were used for enrichment data (assuming proportional variance) and absolute standard deviation values for cumulative tracer excretion data (assuming roughly constant variance). Some precision values were subsequently adjusted so that all data had appropriate statistical weights relative to the other data.

The following information provided by SAAM was used to evaluate goodness of fit and quality of parameter estimates: absolute and relative values of kinetic and steady-state parameter estimates, standard deviations of parameter estimates, parameter correlation matrix, absolute and relative sums of squares of residuals, and appearance of data-model plots. The standard deviations and correlation coefficients of parameter estimates are amenable to objective criteria for evaluating the numerical identifiability of the estimates. We considered an FSD >0.5 to indicate that a parameter has not been estimated with sufficient statistical certainty. Although higher FSD thresholds have been suggested (10, 51), an FSD <0.5 is significant because it indicates a 95% confidence that the parameter value is different from zero (62). High correlation between parameter estimates may indicate a condition analogous to multicollinearity in multiple linear regression (20, 53,54). In this situation, two parameters function in a similar manner in fitting model and data, indicating that both may not be needed to do the job, i.e., the model is too complex, or “overparametrized.” Correlation coefficient values ranging from 0.8 to 0.99 (absolute value) have been proposed as thresholds above which a problem with model complexity may be a concern (3, 31, 42, 53,62), although it has been noted that high parameter correlation may have other causes, e.g., peculiarities of the data or other factors unrelated to data or model (20, 54). On the basis of the literature, we chose to use a correlation coefficient threshold of 0.9.

In investigating the numerical identifiability of parameters, our focus was on evaluating the validity of the population model. We anticipated that the use of a single structure to model a diverse subject population would entail the possibility of poor parameter estimation in instances where data from a particular individual were inadequate to support some aspect of the model structure. Therefore, we were concerned to distinguish poor parameter results attributable to occasional inadequate data from those possibly caused by an inherent problem with the model.

In some cases residuals were tested for normality and plotted for examination. Normality was determined using the Kolmogorov-Smirnov test as implemented in GraphPad Prism 2.0 (GraphPad Software, San Diego, CA). Plots were examined for autocorrelation of residuals and to test assumptions regarding variance of the data.

#### EMSA and assessment of results.

The SAAM input files from the final computer fittings of the individual studies were combined to form an EMSA input file (47). Because the EMSA program requires input files that are identical as regards the unknown (adjustable) parameters, we initially considered removing the five intestinal parameters related to the postabsorptive oral tracer used in the first three subjects. This would have effectively excluded useful postabsorptive tracer data from the parameter estimation process. Therefore, we developed a method that used a complete set of adjustable parameters to model every subject, even though there were no data for the final two subjects with which to estimate the postabsorptive tracer parameters. Because there was no postabsorptive tracer data from all subjects, EMSA could not provide meaningful population data for the postabsorptive parameters. EMSA does not provide population data for steady-state parameters so these were manually calculated, and mass values were based on the mean of the plasma zinc masses from the individual studies. Population and individual parameter estimates were compared to detect any significant anomalies within the population.

## RESULTS

The steady-state data that were measured or estimated by “model-independent” methods are summarized in Table 2. Missing data were inadequate or unavailable. Discrepancies between zinc intake and excretion values are attributed to inaccurate calculations of dietary zinc from diet records or subject noncompliance with diet protocol. There is generally reasonable agreement between parameter results determined by different methods. Two notable exceptions are the plasma zinc estimates for *subject 1* and the fractional absorption results calculated from the^{68}Zn tracer data. Because the method used to determine the mass of rapidly exchanging zinc is predicted to overestimate the mass by 20–25% (Ref. 48 and unpublished data), these data were decreased by 20% for fitting.

Figure 1 shows the structure that resulted from our model development efforts. Each compartment is labeled with an arbitrary number and a physiological or kinetic description. For some compartments, the physiological/anatomic descriptions are clearly appropriate because they were measurement sites or the physiology/anatomy is unambiguous. Other tentative descriptions are based on existing knowledge of zinc physiology. For example, *compartments 2* and *3* are labeled “liver and other” because their size and rapid rate of exchange with the plasma suggest that liver zinc is a significant constituent of these compartments (17, 26, 57). It is also likely that other organ tissue zinc is represented here, e.g., pancreatic zinc (39, 58). Other compartments are distinguished only by the speed of their kinetics. On the basis of radioisotope tracer studies in humans and animals (26, 33, 39, 59), it would be reasonable to propose that the “other EZP” compartment includes zinc in viscera and soft tissues and the “slow exchanging” compartment consists primarily of zinc in bone and muscle.

*Compartment 14*, designated as the stomach, was added to produce a short delay needed to fit the initial appearance of oral tracer in the plasma. Because it was not a parameter of interest to us and was easy to fit manually with little consequence for the rest of the model, *k(8,14)* was set to a fixed value for the computer fitting. *Compartment 10*, representing the colon, is a nonmixing delay compartment. The resolution of the delay was determined manually and set to a fixed value in the range of 4–6. The delay time, *DT(10)*, was made an adjustable parameter so that it too could be estimated from the data. Being the output from a delay compartment,*k(11,10)* was set to a value of one as required by SAAM. Several of the subjects required an additional “loss” pathway, *k(0,1),* to fit the steady-state data. This parameter was also assigned a fixed value after manual determination. Although the model structure shown in Fig. 1 has 20 unknown parameters, there were a total of 25 parameters to be estimated for the first three subjects because separate postabsorptive and prandial values were estimated for the five small intestine parameters.

Our initial evaluation, confirmed by GLOBI2, indicated that the structure and experiment met all the topological necessary conditions for identifiability. We found that GLOBI2 would not complete an identifiability analysis of the structure shown in Fig. 1 but that it would analyze various slightly simplified variations created by removing compartments. The identifiability of the complete model was inferable from the analyses of the simplified structures. Results of the identifiability analyses indicated that all the parameters have a single solution except for *k(6,1), k(1,6), k(7,1),* and*k(1,7)*, which each have two possible solutions. Thus the model was demonstrated to be (locally) structurally identifiable, but, with the addition of a simple definitional constraint, it becomes globally identifiable. The above parameters have two solutions because*compartments 6 *and *7* are structurally indistinguishable and, therefore, can play interchangeable roles in fitting the data. If the roles are defined, e.g., in this model we have defined *compartment 6* as the more rapidly exchanging of the two, the number of solutions drops to one. As a result, all the parameters have unique solutions and the model is globally structurally identifiable.

All the kinetic and steady-state parameter estimations resulting from the computer fitting of the individual subjects and the multiple studies analysis are listed in Tables 3and 4. The fact that the steady-state parameter values shown in Table 4 are generally in agreement with the data in Table 2 was not a necessary outcome of using the latter data as statistical constraints being fitted along with the kinetic data. Our experience demonstrated that both the kinetic and steady-state parameter estimates were primarily determined by the more numerous kinetic data.

Regarding the numerical identifiability of the kinetic parameter estimates, Table 3 shows that there are six parameters that have at least one estimate with an FSD >0.5. Only two parameters,*k(5,4)* and *k(10,9)*, exhibit poor estimate precision in ≥50% of cases. It is noteworthy that among the four small intestine parameters, 58% of the estimates based on^{68}Zn tracer data had FSDs >0.5 but only 15% of the estimates from ^{67}Zn tracer data had high FSDs. An examination of the parameter correlation matrixes showed that there were nine instances of highly correlated (correlation coefficient >0.9) parameter pairs involving ten parameters. Three parameters exhibited highly correlated estimates in more than one instance:*k(1,9), k(9,8),* and *k(10,9).* Only one particular parameter pair, *k(9,8)-k(10,9),* was highly correlated more than once, i.e., in 50% of cases. Similar to the poor precision data, high correlation of intestinal parameters was more likely to occur among estimates from ^{68}Zn tracer data. Half of the instances of high correlation of intestinal parameters occurred in the results for a single subject, *subject 2*, whereas*subject 4* exhibited no high correlation of these parameters. The EMSA results showed two parameter pair correlations >0.9:*k(1,2)-k(3,2)* and *k(1,7)-k(9,2).* The first pair was highly correlated in only one of the individual studies, and the latter pair, which exhibited very low correlation in the individual studies, was clearly anomalous. Therefore, these correlations were not considered to be significant.

The kinetic data and fit of the model for *subject 1* are shown in Fig. 2 to demonstrate the quantity and variety of data that were simultaneously fitted with, for the most part, a single set of model parameters. It is obvious that the model did not fit some of the prandial data well. This did not appear to be caused by an inappropriate model, because the pattern was not evident in the other studies. It was possible to substantially improve the fit to these data by decreasing absorption from the small intestine, but, assuming a steady state and no additional loss pathway unique to the prandial tracer, this adjustment entailed an increase in the dietary zinc intake to about twice the subject's known dietary zinc intake. Therefore, the problem with the prandial tracer data from the plasma, red blood cells, and urine was attributed to analytic error, and the final prandial parameter results were primarily determined by the other, more precise data.

The examination of residuals from a subset of the analyses showed that residuals were generally normally distributed and that assumptions regarding kinetic data variance were correct, i.e., that isotope enrichment data exhibited proportional variance and the variance of cumulative excreted tracer data tended to be constant. The results of the normality test may not be valid where there are a relatively small number of data points. There were several instances of significant autocorrelation of residuals that occurred when there was difficulty fitting multiple data sets with a single set of parameters. There was no pattern across individuals that would have indicated that bad fit was caused by an inadequate model.

## DISCUSSION

As a preface to the discussion we want to make explicit our understanding that the model we have developed is, like most other compartmental models of biological systems, a simplistic (lumped, linear, steady state) approximation of a very complex system and that we must be cautious in its interpretation and extrapolation. Nevertheless, these compartmental models are sophisticated means of integrating various data and information into a coherent, unified picture of the systems and processes being studied. As such they have synergistic potential for providing new understandings of these systems and processes in a variety of evaluative, explanatory, comparative, and predictive applications. Also, it is important to be mindful that a model must not be construed as a completed project. It should, instead, be “a working hypothesis and the best objective and subjective integration of the current state of knowledge” (8), i.e., it must evolve with newly acquired data and knowledge.

The population studied was a small and heterogeneous group of subjects. The heterogeneity of the preliminary group was necessary to ensure that the structure developed was adequate for modeling unique characteristics of varied individuals and, therefore, more likely to be suitable for representing the general population. Although the small number of subjects may limit the initial applicability of the model, it is, nonetheless, sufficient to produce a valid (preliminary) population model.

In assessing the validity of the model, it is significant that our model structure exhibits obvious similarities to the established radioisotope-based model of Wastney et al. (59), as well as to the models that preceded it (2, 26). This may be interpreted as confirming the validity of the common structural characteristics. At the same time, the structural differences are explained by differences in experimental methods. The ability of the radioisotope methods to measure tracer distribution in living systems and to detect tracer activity at very low levels is evident in the detail with which the body zinc is modeled. We, on the other hand, were able to provide some additional detail in modeling the absorption and excretion processes with the use of multiple isotope tracers and additional plasma, urine, and fecal sampling.

The general conception of plasma as a central transport compartment by which zinc exchanges bidirectionally with a large number of subsystems comprising the body zinc is widely, if not universally, accepted (17). For any of these subsystems, which may be anatomically, physiologically, chemically, and/or kinetically distinct pools, the default structure is a single compartment. Our data have required two-compartment structures for two of these subsystems, though. In both cases, the increased complexity is consistent with existing knowledge of zinc physiology.

Assuming linear kinetics, two compartments were required to fit the erythrocyte data, and a catenary structure was found to provide the best fit to the erythrocyte data and reasonable values for red blood cell zinc mass. This two-compartment representation of the red blood cells has been used in radioisotope models (2, 26, 59) and is supported by in vitro and in vivo studies showing that zinc uptake by erythrocytes exhibits two or three phase kinetics (19, 30,50). It has been suggested that the initial rapid phase(s) is uptake by cell membranes and exchangeable intracellular zinc, and the slower phase reflects uptake into intracellular zinc bound primarily to carbonic anhydrase and superoxide dismutase.

The two-compartment “liver and other” structure is also similar to the radioisotope models (2, 21, 26, 59), although two of the models (2,59) show additional slowly exchanging liver zinc not contained in these two compartments. The two-compartment structure is also consistent with animal studies that have demonstrated a two-phase uptake of zinc from plasma by liver cells (23,52). The first phase is thought to result from net accumulation in a labile pool that functions as a precursor to a slower exchange with the metalloprotein- bound zinc (17, 52). We have proposed that these two compartments also contain other very rapidly exchanging zinc, e.g., pancreatic zinc and, perhaps, some zinc in other viscera. One study of zinc kinetics in rats indicated that plasma zinc is taken up most rapidly by the liver, kidneys, spleen, pancreas, and intestine (39). A more recent study has proposed that the rapid uptake of radioisotopic zinc by the pancreas has been mistaken for uptake by the liver in previous studies using counting by probes placed over the liver (58). The existence of the pathway from *compartment 2* to the small intestine is also consistent with the compartment representing liver and pancreas, because pancreatic and possibly biliary secretions are believed to be the primary conduit for endogenous zinc losses to the gastrointestinal tract (2, 17, 39, 44, 57, 58).

The presence of multiple absorption pathways from the small intestine is consistent with the established view that zinc absorption occurs at multiple sites in the small intestine (41, 43). Although intestinal structures of eight compartments have been used in several radioisotope models (2, 26), the parameters of such a structure cannot be determined from available human data. A recent study of zinc kinetics in rats (33) was successful in estimating some of the parameters of a five-compartment representation of the intestine. Our two-compartment representation is probably the limit of complexity that can be supported by the data that are acquired in human studies, unless intestinal sampling is performed. Our model does not contain a pathway directly from the proximal gastrointestinal tract to the liver such as that which appears in the model of Wastney et al. (59) and was proposed to represent first-pass uptake by the liver of absorbed zinc from the portal circulation. The existence of this pathway, which was required to fit the initial appearance of radiotracer activity in the liver, was not discernible from our stable isotope data. Furthermore, the possible differences in the intestinal kinetics of endogenous and exogenous zinc (25, 33,41) are not apparent in the data from these studies.

Although the structure and parameters of Fig. 1 were necessary and sufficient to model the kinetic data from all the subjects and the steady-state data from three of the subjects, the steady-state data from *subjects 3* and *5* indicated that there were additional losses not accounted for by the model. Therefore, a pathway,*k(0,1)*, for losses from the system was added to the model. Because the pathway was not required to fit kinetic data and was not used by every subject, it was decided to estimate its transfer coefficient by manual fitting. Although we considered that this pathway might reflect endogenous losses via routes other than the intestine and kidneys, e.g., in semen or sweat, it is apparent from an examination of the data in Tables 2 and 4 that the total body zinc accounted for by the models was <40% of the mass predicted in these two subjects. We must, therefore, consider the likely possibility that *k(0,1)*also represents flow to other slowly exchanging zinc that was not detectable within the time frame of these studies. This is consistent with the kinetic characteristics of the very slowly exchanging pools (primarily bone and muscle) observed in the long-term radioisotope studies (59) wherein the turnover of zinc in bone was reported to be too slow to be determined even over several hundred days of measurements.

Foster et al. (26) and Lowe et al. (46) also observed that not all body zinc was apparent from the modeling of isotope data acquired over relatively brief study periods. Foster et al. (26) assigned the “missing” zinc to the compartment with the slowest turnover by calculating a fixed value for the compartment's outflow FTC. Similarly, Lowe et al. (46) created an additional compartment for the purpose of balancing tracee data and accounting for all estimated body zinc. The parameters associated with this compartment were calculated accordingly. Because our focus is on the more rapidly exchanging components of the model, we are not immediately concerned with the development of a whole body model and have not added a similar compartment to our model. Furthermore, any parameters assigned fixed values based on predicted body zinc could not be included in the population model derived by the EMSA, because it does not analyze fixed parameters. As a result, our population model is not to be considered a true whole body model of zinc metabolism.

It is not clear from our model and data why the individual tracer kinetics differed so that all the predicted body zinc was accounted for by the model in some individuals and only a small fraction in others. Given our model structure, this variation across individuals is chiefly dependent on the size of *compartment 7*, which, in turn, is determined primarily by fitting the later plasma data after accounting for losses from the system. None of these factors, i.e., model structure, later plasma data, or modeling of losses, provide a ready clue to sorting out the question. The fact that not all body zinc should be detectable from our tracer kinetic data suggests the possibility that the estimated body zinc values are inaccurately low in the instances in which there appears to be agreement between the estimate and the model. In addition, there is a statistically significant negative correlation between the fraction of estimated body zinc accounted for by the model and the body mass index of the subjects, and, if one outlying datum is removed, also between this fraction and the body weight of the subjects. One of the more plausible implications of this finding is that the proportion of body zinc in the very slowly exchanging pools increases with increasing body weight or BMI.

As for the numerical identifiability of the results, an examination of the data in Table 3 shows that the parameter estimates having high uncertainties (FSD >0.5) are confined to two areas of the model structure: the erythrocyte and small intestine subsystems. All the parameters for which estimates were highly correlated in more than one instance also belonged to the small intestine subsystem.

Foster et al. (26) encountered a similar problem determining red blood cell parameters. They reported that, although the second compartment was required to fit the erythrocyte data, the parameters between the two compartments could not be precisely estimated. In response, they elected to set one of the parameters to a fixed value to derive more precise results for the others. We felt it was potentially most informative to allow all the parameters to be estimated, even at a cost in precision. Furthermore, it was necessary to maintain parameter adjustability to obtain EMSA population data.

We believe that the lower precision of these parameter values is attributable to two aspects of our modeling methods. First, as in the radioisotope modeling of erythrocyte zinc, we fit the tracer data as the sum of tracer in both compartments. As a result, the erythrocyte data-model fit may be less sensitive to adjustment of the parameters between the compartments even though these parameters are necessary. Second, because we were fitting data from two or three tracers simultaneously with a single set of parameter values, any inconsistency between data sets would increase the uncertainty of the parameter estimates. Experimentation with fitting only one set of erythrocyte data (the intravenous tracer data) demonstrated that improved parameter estimates often resulted.

The two-compartment small intestine structure was justified by the notable improvement in data-model fit that was often realized compared with that possible with a simpler structure. This structure provided the versatility required for the task of simultaneously fitting the appearance of oral tracers in the plasma and the appearance of all tracers in the feces with a single structure. The drawback of the greater complexity is, as we see here, the increased difficulty with adequately estimating parameters. Nonetheless, it is apparent that the parameters of this structure can be determined with acceptable certainty when there are sufficient data, as evidenced by the low incidence of poor precision among the estimates derived from^{67}Zn tracer data. Similarly, the correlation data appear to indicate that high correlation is related to peculiarities of subject data and not model structure.

Because the standard errors of the population parameter estimates reflect both the uncertainties of the individual estimates as well as the intersubject variability of the estimates, their interpretation as regards numerical identifiability is not straightforward. Furthermore, little formal work has been done regarding identifiability of population parameter data (5). Because the practical utility of the population model is dependent, in part, on the precision of the model parameters, it is encouraging that the fractional standard error values are generally <25%, especially given the heterogeneity and small size of the current population.

Table 5 presents a comparison of pertinent characteristics of our model with those of several other published models selected for the similarities in subject populations and experimental methods. The majority of transfer coefficient, transfer flux, and compartment mass values is not simply comparable between models because of differences in structures and compartment attributions. Therefore, we have not provided a comparison of these more fundamental characteristics.

On the basis of the following observations and findings, we conclude that the model structure and parameter results constitute a valid and useful linear compartmental representation of zinc metabolism in the population studied, with the only qualification being that it does not account for all the most slowly exchanging body zinc and, therefore, is not a true whole body model. First, the model (Fig. 1) is structurally identifiable in the context of our experimental protocol, as demonstrated with the assistance of the GLOBI2 software. Second, the goodness of fit of the model and data (from all studies) is generally very good as determined from small sums of squares of residuals, normality and general lack of autocorrelation of residuals, and examination of data-model plots. Third, the kinetic parameter estimates are generally well determined, i.e., numerically identifiable, as evidenced by our examination of the parameter uncertainties and correlations. Instances of poor parameter results are the consequence of inadequate data and/or the modeling techniques used and not shortcomings of the structure. Fourth, the steady-state parameters are consistent with the model-independent data in most cases. Fifth, the population parameter estimates calculated by EMSA are an accurate aggregation of the results from the individual studies. Sixth, overall, the model is judged to be valid in that it meets the following applicable criteria (10, 14). The model contains no logical, mathematical, or conceptual contradiction (consistency); it was derived using solution and simulation algorithms that are appropriate and lead to accurate solutions (algorithmic validity); it corresponds to the available experimental data (empirical validity); it is consistent with accepted physical theories and models (theoretical validity); and, to the extent that it has functioned thus far to provide scientific explanation and hypothesis testing, e.g., in discriminating between proposed structural subsystems, it is shown to have heuristic potential (heuristic validity). And, although implicit in several of the above criteria, it is important to note that the model is compatible with current knowledge of zinc physiology. It is expected that further heuristic potential as well as the pragmatic validity, i.e., the utility, of the model will be demonstrated by its application to current and future research.

### Perspectives

With the advent and refinement of zinc stable isotope techniques coinciding with notable recent growth in our appreciation of the importance of zinc in human nutrition and public health, there has been and will be a continuing escalation in the application of these techniques directed to advancing our understanding of human zinc metabolism and homeostasis. In turn, this has placed a premium on improved techniques for analyzing and interpreting experimental data. Compartmental modeling has a vital role not only in the analysis of more complex data sets but also in assisting in the validation of simple mathematical models that are applied much more frequently in human studies that have a specific and more circumscribed objective, e.g., studying the role of the intestinal tract in maintaining zinc homeostasis.

Future application will, logically, go hand in hand with further refinement of this model as our current, very limited database is enlarged. The sensitivity of this model will be fully challenged in its application not only to advancing our understanding of the physiology and pathophysiology of this metal but, especially, to the field of human nutrition. These results give reason for optimism that these techniques can make a vital contribution to better understanding of zinc homeostasis, dietary zinc requirements and the effects of inadequate nutrition, and/or stressing zinc homeostatic mechanisms on broader aspects of human zinc metabolism.

## Acknowledgments

The authors express appreciation to Drs. Claudio Cobelli and Luigi Andolfato for providing the GLOBI2 software and assistance with implementation. The authors also acknowledge the contributions of other members of the research team and trainees, including Jamie Westcott, Mary Jefferson, John Huffer, Donna Beshgatoor, and Dave Easley, and the support of Paul V. Fennessey, who provided the mass spectrometry facilities and contributed to establishing a research environment that made these studies possible.

## Footnotes

This research was supported by grants from the National Institutes of Health, General Clinical Research Centers, RR-00069 and RR-00051; the National Institutes of Diabetes, Digestive and Kidney Diseases, K08-DK-02240; Clinical Nutrition Research Unit, P30-DK-48520; Mental Retardation Research Center, HD-04024; and Metabolic Regulation of Fetal Growth, HD-20761.

Address for reprint requests and other correspondence: L. V. Miller, Campus Box C-232, Univ. of Colorado Health Sciences Center, 4200 E. Ninth Ave., Denver, CO 80262 (E-mail:leland.miller{at}uchsc.edu).

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