As we have acknowledged, and as is the case with all models of biological systems and processes, the model we have developed is a simplistic representation. Beyond our limited knowledge of the process, the few available data with which to validate the model would not have supported the incorporation of more detailed knowledge of the absorption mechanisms, as demonstrated by addition of a passive absorption pathway. Furthermore, much additional complexity would require the use of more sophisticated analytical or numerical methods to solve the model equations and risks limiting the practical value of the model for routine application. Even at the basic level of our model, we encountered the need to use an approximation to avoid impractical complexity. Though rudimentary, the model may be judged to be valid for its intended predictive and explanatory purposes using various criteria, including soundness of the basis and derivation of the model, goodness of fit to the data, reasonableness and quality of parameter estimates, comparison to other models, and applicability and usefulness in current research (32
conveys essential information about the behavior of the model and its fit to the data. Although much of this information is presented here in other forms, the 3-dimensional representation best illustrates the distribution of the data in the predictor variable plane and the random nature of the residuals relative to all the variables. It is obvious from this graph that the model does not support the existence of a threshold for a phytate effect on absorption.
The goodness of fit as reflected in the R2 of 0.82 is very supportive of the model, indicating that 82% of the variance in TAZ is explained by the model. Analyses of the residuals also confirmed the quality of the fit and adequacy of the model while testing the assumptions underlying the regression analysis. These assumptions, which must be met for the regression to be valid, are that the errors, as reflected in the residuals, exhibit a Gaussian (normal) distribution, constant variance, and independence. Evaluating independence is necessary to detect nonrandom behavior in the residuals, indicating a systematic deviation of the model from the data. The residuals passed all the tests and visual examinations, although the presence of a datum that deviated noticeably from the model caused some results to be marginal. This point and its large residual are evident in –. Although this datum looks suspiciously like an outlier, we have examined the study in which it was measured and found no reason to reject it as an outlier at this time. Therefore, the point was retained for fitting the model and estimating the parameters. However, because of its disproportionate effect on the analyses of the residuals, the analyses without the discrepant point are thought to more accurately reflect the error characteristics on the whole.
The uncertainties around the parameter estimates are large, although AMAX may be considered to be adequately determined, because we can reasonably conclude that it is not 0 (P = 0.011). Though large uncertainty in the parameters is usually the result of a model having too many parameters (overparameterization) and collinearity of the parameters, it is not justified, or prudent, to simplify the model for the following reasons: 1) The model is supported by the other measures of validity; 2) the model is already of minimal complexity, given that there are 2 uncorrelated (r = 0.09) predictor variables and the relations are nonlinear and nonarbitrary; 3) each of the parameters has a well-defined relation to an essential element of the absorption process and cannot be eliminated without compromising the foundational validity of the model; and 4) the current data are limited in number and exhibit variability that may related to the multiple laboratories, analytical methods, and experimental protocols involved. It is expected that the quality of the estimates will improve notably with additional data.
Regarding the values of the parameter estimates, we have found virtually no existing data with which to compare them, although it is noteworthy that the relative magnitude of the equilibrium dissociation constants, i.e. KP
, is consistent with that reported by Wing et al. (6
) and with the deduction by O’Dell, from an experiment in which dietary EDTA alleviated the detrimental effect of phytate, that EDTA has a higher binding affinity for zinc than phytate and that mucosal receptors have a still higher affinity (Boyd O’Dell, University of Missouri, personal communication). Our previous application of a basic saturable response model (11
) to the data used by the Food and Nutrition Board (7
) estimated a value of 0.11 mmol/d for AMAX
, not significantly different (P
= 0.71) from the 0.13 mmol/d estimated here. It should be noted that, although we expect that this model may be appropriate for application to data from any human population, the parameter values derived here characterize absorption in healthy adults with assumed normal zinc status and, therefore, caution should be exercised in interpreting their predictive application to other populations until additional data have been modeled. As well as the predictive uses of the model that the parameter estimates permit, it is possible that the parameter values themselves will provide information about absorption. Because AMAX
is related to the number of transport receptors and KR
quantify the association/dissociation characteristics of the binding reactions, application of the model may contribute to our limited knowledge of receptor regulation and binding chemistry as it relates to bioavailability.
The opportunities for comparing the model’s predictions to other models are very limited. Not surprisingly, the TAZ
predictions from our model agree with those of the IZiNCG model (1
) with ±6% across the range of the data, but the models diverge at higher TDZ
values. The TDZ
at which divergence occurs varies inversely with the RPZ
. Above this, our model predicts that TAZ
increases at a lower rate with increasing TDZ
, which may be attributable to the additional high phytate data that we have used. Whereas more detailed comparison with modeling of the Food and Nutrition Board data (7
) was not possible because phytate data were not available, we did use the model to predict the phytate intake of the those subjects. The model predicted a mean TDP
of 0.45 mmol/d and a resulting RPZ
of 3.1. This is consistent with the fact that the Food and Nutrition Board data were selected from studies of very low phytate diets.
We have shown several variations of the model to accommodate the use of fractional absorption, FAZ
, or the phytate:-zinc molar ratio, RPZ
. Although they add flexibility for the model’s predictive applications, these forms of the model are not as well suited for fitting to data, because FAZ
are both ratios of the more fundamental variables, i.e. FAZ
. In the case of FAZ
, because there is a correlation between TDZ
= 0.73, P
= 0.0002), the relation between FAZ
is not as strong as that between TAZ
. This is manifest in our use of Equation 12
to fit the data. Although producing similar parameter estimates, the goodness if fit was notably inferior. The recommendation to relate zinc absorption to dietary component quantities rather than ratios has been made previously, although based on somewhat different considerations (4
). The use of Equation 14
, with the passive absorption parameter, to analyze these limited data is a good example of overparameterization, as evidenced by the very wide CI for all parameters. It is noteworthy that the goodness of fit was not improved with the added parameter, apparently indicating that these data do not exhibit evidence of passive absorption. Additional data will be required to discern the existence of passive absorption and demonstrate the usefulness of this version of the model.
In conclusion, we have developed a mathematical model of zinc absorption from a basic conception of the relevant intestinal biochemistry and fit it to selected existing data. Evaluation of the fit finds it to be good and in compliance with regression assumptions, thereby supporting the validity of the model. Evaluation of the parameter estimates and model predictions are also supportive of the model’s validity. We judge the model to be well founded, with immediate relevance and applicability to the study of zinc nutrition and metabolism and the estimation of dietary zinc requirements in varied populations. Furthermore, we anticipate the model’s evolution and improvement as new data are analyzed and further knowledge of the absorption process incorporated.