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Western Human Nutrition Research Center University of California, Davis Davis, CA 95616
Kinetic Analysis Associates, Inc Berkeley, CA
Department of Biological Sciences University of Central Lancashire Lancashire United Kingdom
Dear Sir:
Dainty et al make some good points in their letter about our article (1). They suggest that the comparison of the fractional zinc absorption (FZA) data of each subject obtained by using the compartmental model with the same data obtained by using simpler measures is not the best technique for determining the accuracy of the simpler techniques. They point out that random fluctuations in the data may be propagated in the simple estimates of FZA in unpredictable ways, possibly resulting in a spurious comparison. We agree and are currently studying a theoretical data set generated from the compartmental model with a precision much greater than what could be expected in an in vivo tracer study of zinc metabolism; we will use this data set to determine how accurately FZA values obtained with the simple measures compare with FZA values obtained with the compartmental model. With use of this strategy, only logical deficiencies associated with each of the simple techniques relative to the compartmental model would be elucidated.
Although we agree in part with the comments of Dainty et al, we disagree with their comments on the nature and usefulness of compartmental modeling. It is true that a compartmental model of a metabolic system is a kinetic hypothesis describing how that metabolic system functions dynamically and, therefore, it is open to criticism and further testing, as is any other hypothesis. Nevertheless, because a compartmental model (the parameters of which are estimable from the data) must be consistent with all of the data to which it is applied (ie, zinc tracer and tracee measurements in plasma, urine, and feces in our study), it is a more robust description of the physiology than is a simple measure or model of a limited portion of the entire data set. Thus, the compartmental model should serve as the gold standard against which other simpler measures of the data can be compared (the problem of "noisy" data mentioned above notwithstanding). In fact, if a simple measure of the data is a good estimate of a particular physiologic parameter or combination of parameters (eg, FZA) and is also estimable in the compartmental model, such a measure could be derived within the logical context of the compartmental model.
Dainty et al suggest that any model makes "gross simplifications of the way the body works." We agree. Nevertheless, however gross such simplifications may be in compartmental models, they are even more gross for simple measures of the data. Dainty et al mention the possibility of "false assumptions" in compartmental models, which is always a possibility, but they do not mention what these false assumptions might be in our model of zinc metabolism. They also criticize the "unjustified complexity" of our model. However, our model (2) is the simplest compartmental structure that fits all of our data. As is true with many mechanisms in nature, metabolic systems are complex and our "gross oversimplifications" (compartmental models) are often more complex than we would like. However, such complexity should not push us to retreat to gross oversimplifications and the use of simple unproven approaches for estimating various parameters (ie, FZA).
Dainty et al also refer to our "unsubstantiated claims of parameter precision." The precision of our parameter estimates from the SAAM II computer program (SAAM Institute, Seattle) uses relative data weighting of the highest quality, and the algorithms used are well documented (3). In brief, the fractional SDs for a data array are entered as input estimates of 0.1 (relative weights) into the SAAM II program. The program then adjusts this value up or down for each data set, depending on the quality of the least-squares fit of the model to the data. The uncertainty estimates for each parameter are then scaled accordingly. We apologize for a misprint in footnote 1 to Table 2 in our original article (2), which apparently has generated concern about the precision of our estimates. The uncertainty estimates for each parameter in that table are fractional SDs, not SDs, as indicated in the footnote. Thus, the average fractional SD of the rate constant k0,1, which describes the fractional movement of zinc tracer and tracee from the plasma to the urine per unit time, is 13%, not >60%.
In conclusion, we agree with Dainty et al that comparison of the adequacy of simple measures of FZA from a particular data set cannot be compared easily with the estimation of FZA from the compartmental model because of the uncertain way in which the random fluctuations in the data get propagated in the simple estimates of FZA. A more rigorous theoretical comparison of simple with compartmental modeling techniques is currently under way. Nevertheless, we believe that the comments made by Dainty et al about compartmental modeling are misguided. The more robust the hypothesis, ie, the more extensive and more different the data set explained by a model (in our case zinc tracer and tracee data in plasma, urine, and feces over 6 d after administration of oral and intravenous tracers), the more accurate a particular measure of a model (ie, FZA) is likely to be. Under such conditions, we believe it is appropriate to use a well-documented compartmental model as the gold standard for evaluating the accuracy of simple estimates of a parameter such as FZA.
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