## Abstract

Several approaches for estimation of fractional zinc absorption (FZA) by calculating the ratio of oral to intravenous stable isotopic tracer concentrations (at an appropriate time) in urine or plasma after their simultaneous administration have been proposed in the last decade. These simple-to-implement approaches, often referred to as the double isotopic tracer ratio (DITR) method, are more attractive than the classical “deconvolution” method and the more commonly used single-tracer methods based on fecal monitoring and indicator dilution, after oral or intravenous tracer administration, respectively. However, the domain of validity of DITR for measuring FZA has recently been questioned. In this paper, we provide a theoretical justification of the validity of four different “approximate” formulations of the DITR technique by demonstrating mathematically that their accuracy is a consequence of the particular properties of zinc kinetics.

- kinetics
- compartmental model
- deconvolution

the validity of the double isotopic tracer ratio (DITR) method for measuring fractional zinc absorption (FZA) has recently been questioned (8). This method, first proposed by Friel et al. (3) for measuring FZA, is a modification of the DITR method for determining the fractional absorption of calcium (2, 10, 11) from the ratio of oral to intravenous stable isotopic tracers in a 24-h pooled collection of urine after tracer administration. Because the kinetics of zinc absorption into plasma are slower than those of calcium, Friel et al. suggested that DITR should be applied to either a 72- or 96-h pooled urine collection. The same group further suggested that accurate estimates of FZA could more easily be obtained by applying DITR to any spot sample collected after the time at which the slopes of the log transforms of both the oral and intravenous stable isotopic tracers in plasma became equal. The time of this occurrence was ∼40 h after simultaneous tracer administration in their study.

The DITR method for determining FZA has recently been compared with the fecal monitoring method (8), a commonly used technique for estimation of FZA. The fecal monitoring method, employing a single oral tracer with measurement of the tracer in each fecal sample, can provide an accurate estimate of FZA if the kinetics of tracer absorption can be separated from the kinetics of secretion of absorbed oral tracer, a problem not yet completely solved (9). Friel et al. (3) found “good agreement” between the two methods, but Rautscher and Fairweather-Tait (8) concluded that DITR does “not reliably predict” FZA. On the basis of a number of experimental and simulation considerations, we believe that the fecal monitoring method is an inappropriate reference for assessing DITR (5, 9). In addition, we believe that the latter method does provide a valid estimate of FZA (9). However, we understand that a point against DITR as a validated measure of FZA is the lack of a theoretical analysis of its domain of validity.

In this paper, we consider six DITR techniques for estimation of FZA employing the simultaneous administration of two tracers, an oral and an intravenous tracer. In a theoretical environment, i.e., with error-free and continuous-time tracer data, two of them are exact and will thus be used and referred to as “reference methods” (seereference methods to calculate fza). The other four methods will be referred to as “approximate methods,” because they provide, even in a theoretical environment, approximate estimates of FZA (seefza by approximate methods). However, these approximate methods are clinically appealing, given their simplicity of implementation. In domain of validity of approximate methods, we investigate the accuracy of these approximate methods against the two reference methods in a theoretical context. In particular, we show mathematically that the specific characteristics of zinc kinetics ensure accuracy of all the approximate methods.

## REFERENCE METHODS TO CALCULATE FZA

We describe here two methods that, in a theoretical environment (i.e., error-free and continuous-time data), allow the measurement of FZA without error from data after the simultaneous administration of two tracers, an oral tracer and an intravenous tracer. Both of these methods assume that oral zinc behaves identically to the intravenous zinc once in the systemic circulation, that tracer kinetics are linear, and that all of the oral zinc absorbed from the gastrointestinal tract reaches the plasma.

### Compartmental Modeling

Let us assume that the zinc metabolism in humans is described by the compartmental model shown in Fig. 1(6). In a given individual, FZA can be obtained by making use of the rate constants of the kinetic model by Equation 1with obvious meaning of notation (1).

It is worth noting that the practical use of *Eq. 1
* would require the identification of the model of Fig. 1 from tracer data. In particular, the model should be fitted against frequently sampled plasma, urine, and fecal concentrations of orally and intravenously administered stable isotopic tracers of zinc, ^{67}Zn-tr and^{70}Zn-tr, highly enriched in ^{67}Zn and^{70}Zn, respectively. This was the procedure used in Ref.6. It is also important to note that, because tracer concentration data are unavoidably affected by error, the FZA value obtainable by *Eq. 1
* is always uncertain (even assuming an error-free model structure). This uncertainty reflects that of the parameters obtained from model fitting. However, because the aim of this paper is to evaluate the theoretical foundation of the DITR method, we will consider an error-free context, and we will not take into account precision issues.

### Deconvolution

The theoretical rationale of this method is given inappendix
, where we also make clear the improper labeling of this method as “deconvolution.” In fact, there is no need for any deconvolution procedure to implement this method, which calculates FZA as the ratio of the areas under the curve (AUCs) of the oral and intravenous tracers, e.g., ^{67}Zn-tr and ^{70}Zn-tr
Equation 2where, in general, ^{67}z_{i}(*t*) denotes the concentration (e.g., measured in mg/ml) in pool i (1 is plasma; see Fig. 1) of ^{67}Zn-tr after an oral intake of a unit mass at time 0, and ^{70}z_{i}(*t*) denotes the concentration in pool i of ^{70}Zn-tr after the pulse intravenous administration of a unit mass of ^{70}Zn-tr at time 0.

Even if practical aspects are outside the theoretical focus of the present paper, it is worth noting that the implementation of *Eq.2
* would require frequent and prolonged sampling of intravenous and oral tracer concentration in plasma and some data extrapolation from the last measurement to time infinity. Moreover, because of data noise and interpolation/extrapolation errors, the FZA estimate provided by*Eq. 2
* would be uncertain.

#### Remark 1.

Throughout the paper, we will consider, for sake of simplicity, responses to unit oral and intravenous tracer doses. Obviously, considering nonunit and different intravenous and oral doses is possible by considering in the formulas the proper scale factors.

The implementation of either of the two methods presented in this section requires a large amount of data. This motivated the development and usage of simpler, albeit approximate, techniques for FZA estimation. These are described in fza by approximate methods. The same approximate methods will be assessed in domain of validity of approximate methods against the reference methods.

## FZA BY APPROXIMATE METHODS

In the previous section, we described two methods that, in a theoretical environment, allow the exact determination of FZA. There are at least four methods, numbered *1–4* below, that provide approximate measures of FZA. These approximate estimates will be denoted hereafter by the symbol
. *Methods 1 *and* 3*require knowledge of the concentration of ^{70}Zn-tr and^{67}Zn-tr in plasma, ^{70}z_{1} and^{67}z_{1}, respectively; *methods 2*and* 4* employ the ^{70}Zn-tr and^{67}Zn-tr concentration in urine,^{70}z_{8} and ^{67}z_{8}, respectively.

### Method 1

The first approximate method provides an estimate of FZA as a ratio of AUCs of functions fitted to plasma data between the beginning of the experiment and
Equation 3where
is a time greater than a certain threshold, *t**. For zinc, this threshold is typically 2–3 days.

### Method 2

The second approximate method provides an estimate of FZA by utilizing the concentration of each tracer in a total cumulative urine collection from time 0 to
Equation 4where
is a time greater than*t**.

### Method 3

The third approximate method provides
from a single plasma sample as
Equation 5where
is a time greater than the threshold time *t**.

### Method 4

The fourth approximate method requires the knowledge of the concentration of the two tracers in a single urine specimen, from which FZA is estimated as
Equation 6where *t*
_{2} and *t*
_{1}represent the beginning and end of the urine sample collection interval, with *t*
_{2} >*t*
_{1} > *t** (*t*
_{2} can be only a few hours greater than*t*
_{1}), and^{67}z_{8}(*t*
_{2},*t*
_{1}) and^{70}z_{8}(*t*
_{2},*t*
_{1}) denote, respectively, the ^{67}Zn-tr and ^{70}Zn-tr concentrations in urine.

When applied to real data, all of the approximate methods lead to FZA estimates affected by an error that reflects sparseness and noise of tracer data. Moreover, *method 1* would be particularly costly to implement, because it requires frequent sampling of^{67}Zn-tr and ^{70}Zn-tr concentrations in plasma over a large time interval (2–3 days). *Method 2* is more appealing for clinical purposes, because only one measurement of^{67}Zn-tr and ^{70}Zn-tr concentration in a complete, cumulative urine sample over several days is required. Simpler still is *method 3*, in which measurement of^{67}Zn-tr and ^{70}Zn-tr concentration in a single plasma sample obtained a few days after tracer administration is all that is required. Perhaps the simplest of all is *method 4*, which only requires a single measurement of ^{67}Zn-tr and^{70}Zn-tr concentration in a single spot urine specimen obtained a few days after tracer administration.

A formal proof of the reliability of these four approximate approaches for FZA estimation is not available in the literature. Such a proof will be developed in the following section by exploiting some specific characteristics of zinc kinetics, and, given its theoretical nature, it is obtained in a noise-free environment; i.e., we are concerned only with the accuracy of the four approximate methods and not with their uncertainty due to data noise.

## DOMAIN OF VALIDITY OF APPROXIMATE METHODS

### Method 1

Consider the kinetic model of Fig. 1. By using the average parameters of Ref. 6, one can obtain the simulation of ^{70}z_{1}(*t*) and^{67}z_{1}(*t*) in response to unitary intravenous and oral doses, respectively, displayed in Fig.2, *A* and *B*. It is worth noting that the order of the (linear) dynamic system originating^{70}z_{1}(*t*) is 5 [the state variables corresponding to *compartments 4, 6, 8, *and* 9* do not play any role in ^{70}z_{1}(*t*)], whereas that of the system that generates^{67}z_{1}(*t*) is 6 (the input occurs in*compartment 4*). Indeed, the (early) time course in plasma of^{67}Zn-tr reflects the first-pass kinetics of movement through and absorption from the gastrointestinal tract. However,^{67}Zn-tr that is absorbed into plasma equilibrates with the extraplasma zinc pools and routes of renal and gastrointestinal loss in the same manner as ^{70}Zn-tr administered intravenously. Consequently, at the time that the first-pass absorption process is complete or nearly complete (when all or nearly all of the unabsorbed oral tracer has passed out of the gut compartment), the plasma concentration time course of the oral tracer begins to assume the same shape as that of the intravenously administered tracer. In particular, because tracer kinetics can be assumed to be linear, for times*t* greater than a certain threshold *t**, both^{70}z_{1}(*t*) and^{67}z_{1}(*t*) can be approximated by a sum of the same number *k* (*k* < 5) of decaying exponentials with the same eigenvalues α_{1},…,α_{k}
Equation 7
Equation 8For instance, with the parameters of Fig. 1, for times greater than 2–3 days, the decay of the both the oral and intravenous tracers in plasma is well described by the last three modes of the system, with eigenvalues α_{j} equal to 0.854, 0.186, and 0.0017 day^{−1} (corresponding to half-lives of 0.81, 3.73, and 408 days, respectively). The larger the value of *t*, the more the approximations of *Eqs. 7
* and *
8
* improve (because the modes corresponding to the absorption process in^{67}Zn-tr completely vanish only at infinity, the equal sign only holds asymptotically).

#### Remark 2.

Relationships between the parameters of the exponentials and the rate constants *k*
_{ij} in Fig. 1 could be found by analytically solving the system of ordinary differential equations describing the kinetics, e.g., by software such as Maple or Matlab, which handle symbolic calculations.

Now, let us define ρ(*t*) as the function that makes the following equation hold
Equation 9The fact that, for times *t* greater than *t**,^{70}z_{1}(*t*) and^{67}z_{1}(*t*) tend to assume the same shape means that in *Eq. 9
* the function ρ(*t*) tends, as time increases, toward a constant value ρ. Therefore, for any *t* > *t**, one could write
Equation 10A grasp of *Eq. 10
* is offered by Fig. 2
*C*, where a scale factor between the oral and the intravenous tracer decay can be seen in their log transforms, which are, roughly speaking, “parallel.”

Now, let us consider *Eq. 2
*, from which true FZA can be obtained as
Equation 11where
is a generic time greater than *t**. By integrating^{67}z_{1}(*t*) and^{70}z_{1}(*t*) from
to infinity and exploiting*Eq. 10
*, we can write
Equation 12Now, consider the third term of *Eq. 11
*. Whereas the integrals from 0 to
increase with increasing
, those from
to +∞ decrease with increasing *
*. However, from *Eq.12,* it follows that, for
>*t**, the ratio
Equation 13is approximately constant and equal to ρ. Given that FZA is the ratio of two real numbers, the fact that the ratio ∫
^{67}z_{1}(τ)dτ/ ∫
^{70}z_{1}(τ)dτ is approximately constant for any
greater than *t** allows us to state that the ratio
^{70}z_{1}(τ)dτ of the other two functions involved in *Eq. 11
* is approximately constant as well for any
greater than *t**. The value of the constant ratio
must also coincide with its value when
tends toward infinity. Since
Equation 14(see *Eq. 2
*), it follows not only that for any
greater than *t***Eq. 3
* provides an approximate estimate of FZA but also that such an estimate approaches the true value of FZA when
is “very large.”

#### Remark 3.

*Method 1* does not require additional hypotheses with respect to the reference deconvolution method from which it is derived. In fact, the proof reported above simply exploits the principle of indistinguishability of the oral and intravenous tracers and the linearity of their kinetics.

### Method 2

Let us consider (Fig. 1) the flux of zinc tracer from*compartment 1* (plasma) to *compartment 8* (urine). If q_{i} denotes the mass of a zinc tracer in compartment i, one has
Equation 15Because no tracer is present in the system at time 0, q_{8}(0) = 0. From *Eq. 15
* it can be easily shown that
Equation 16
Equation 17where γ is a positive scalar. By dividing *Eq. 16
* by*Eq. 17
*, one obtains
Equation 18Therefore, *method 2* provides the same estimate as*method 1*. Hence, as
tends toward infinity, the estimate provided by *method 2* also tends toward the correct value
Equation 19Figure 3
*A* allows the reader to grasp the goodness of the approximation provided by *Eqs. 3
* and *
4
* as
increases. Note that
by this method is always smaller than true FZA. We will now show theoretically why *methods 1 *and*2* give an underestimate.

Consider *Eq. 11
*. In light of *Eq. 15
*, ∫
^{67}z_{1}(τ)dτ and ∫
^{70}z_{1}(τ)dτ can be interpreted (if we ignore the common scale factor γ) as the relative amounts of oral and intravenous tracers, respectively, that traveled from plasma to urine from time zero up to infinity. At the same time, ∫
^{67}z_{1}(τ)dτ and ∫
^{70}z_{1}(τ)dτ can be regarded as the relative amounts of oral and intravenous tracer, respectively, that, after
, traveled from plasma to urine. Becasue the appearance of ^{67}Zn-tr in the urine is delayed with respect to that of ^{70}Zn-tr, we can write
Equation 20
*Method 1*, given by *Eq. 3
*, can be thought to be obtained from the reference method of *Eq. 11
*, coincident with *Eq. 2
*, by subtracting from the numerator ∫
^{67}z_{1}(τ)dτ and from the denominator ∫
^{70}z_{1}(τ)dτ the quantities ∫
^{67}z_{1}(τ)dτ and ∫
^{70}z_{1}(τ)dτ, respectively. Because in light of *Eq. 20
* the quantity subtracted from the numerator is greater, in relative terms, than that subtracted from the denominator, the ratio
Equation 21which gives a measure of FZA according to *methods 1* and*2*, is an underestimate of true FZA.

#### Remark 4.

*Method 2* is based on the same assumptions of the reference deconvolution method and on only one additional structure hypothesis, i.e., that the urine pool receives material only from the plasma compartment. In fact, it is this last hypothesis that leads to*Eqs. 16
* and *
17
* and thus to *Eq. 18
*; no other structural hypotheses are required. Therefore, *method 2* is, in theory, compatible with many kinetic structures other than that of Fig. 1.

### Methods 3 and 4

These two methods can be discussed together. In fact, for any pair of times (*t*
_{2}, *t*
_{1}), with*t*
_{2} > *t*
_{1} >*t** and *t** defined as the time after which both^{70}z_{1}(*t*) and^{67}z_{1}(*t*) exhibit a multiexponential decay with the same eigenvalues (parallel behavior on semilog plot; see Fig. 2, *B* and *C*), it can be shown (seeappendix
) that
Equation 22
*Methods 3 *and* 4* are thus equivalent. Let us now assess their accuracy in estimating FZA.

We first prove that *method 3* (and thus *method 4*as well) provides an overestimate of true FZA. From *Eq. 10
*, it follows that, for any *t*
_{1} >*t**
Equation 23From *Eq. 23
* and the rationale following *Eq.13
*, one has it that
Equation 24Now, by trivial calculations made on *Eq. 20
*, one can obtain that
Equation 25Hence, from *Eqs. 24
* and *
25
*, it follows that
Equation 26from which one can thus conclude that *method 3* (and thus *method 4* as well) provides an overestimate of the true FZA.

In light of *Eqs. 14
* and *
19
*, *methods 1*and* 2* are asymptotically correct in an error-free environment. In contrast, as will be shown below, *methods 3*and* 4* provide only a biased approximation of the reference methods of reference methods to calculate fza and do not tend, as time increases to infinity, toward the true FZA. In formal terms, e.g., for *method 3*
Equation 27In particular, for any set of kinetic parameters, *methods 3*and* 4 *always overestimate the true FZA. It is thus important to assess under which conditions, i.e., kinetic properties, the overestimation made by these two methods is negligible.

If the amounts of ^{67}Zn-tr and ^{70}Zn-tr were administered simultaneously into the plasma (equal doses), the concentrations of ^{67}Zn-tr and ^{70}Zn-tr in the plasma would be equal at any time. Instead, the ^{67}Zn-tr is given orally and the plasma concentration of ^{67}Zn-tr is always lower than that of ^{70}Zn-tr. However, as discussed previously, at the time that the first-pass absorption process is complete or nearly complete, the plasma concentration time course of the oral tracer begins to assume the same shape as that of the intravenously administered tracer (see Fig. 2). This happens because the zinc system is generating, outside of the absorption process, eigenvalues that are slower than the slowest eigenvalue generated by the absorption process itself, so that the extraplasma distribution and the absorption processes are “completely separable.” In more formal terms, for any time
>*t**, one can think of the plasma kinetics of the orally administered tracer, ^{67}Zn-tr, as being the result of an “intravenous injection” of an apparent tracer dose D, delayed by an interval, *t*
_{d}, with respect to the intravenously injected tracer ^{70}Zn-tr
Equation 28If the administered doses of the intravenous and oral tracers are unitary, the apparent dose D coincides with the fraction of the orally administered tracer that is absorbed into the plasma. Therefore
Equation 29or
Equation 30In practice, *Eq. 30
* is not used (see *remark 5* below), and the following approximation is made
Equation 31which yields the estimate FZA shown in *method 3*
Equation 32The application of *Eq. 32
* instead of *Eq. 30
*will lead to an overestimation of FZA, because
will always be less than
−*t*
_{d}). The degree of FZA overestimation by*Eq. 32 *relative to* Eq. 30
* can be found by employing the following rationale.

For any
greater than t*, in light of *Eq. 23
*, *Eq. 32
*provides virtually the same value. To find what this value is, let us choose a very high value of
, say
> *t*
_{s}, where *t*
_{s} is the time after which only the slowest exponential, with eigenvalue hereafter indicated by α_{s}, is still active (e.g., with the parameters of Fig. 1, α_{s} = 0.0017 day^{−1}). By employing*Eq. 8
*, it can be seen that under these conditions
Equation 33Therefore
Equation 34For the reference parameters of Fig. 1, *t*
_{d}turns out to be ∼0.20 days, and α_{s} is ∼0.0017 day^{−1}. According to *Eq. 34
*, this leads to an overestimate of FZA by ∼0.03%. This overestimation perfectly matches that determined by numerical simulation, depicted in Fig.3
*B*.

#### Remark 5.

It is worth pointing out that, from Fig. 3, one cannot infer the average amount of error one could introduce in estimating FZA by the DITR method in practice. The purpose of Fig. 3 is simply to illustrate graphically the theoretical results on the approximate methods, e.g., asymptotic convergence to the reference value of *methods 1*and* 2*, and existence of a bias in *methods 3*and* 4*. Figure 3 was obtained by using the parameters shown in Fig. 1, but the same qualitative conclusion would have been obtained for any other set of zinc kinetic parameters (see also Ref.9 and *remark 6*). A possible way to quantitatively assess the average bias of the approximate methods would be to resort to stochastic simulation, but this is beyond the scope of this paper.

#### Remark 6.

From *Eq. 34
*, it can be speculated that *method 3*(and thus *method 4* as well), albeit approximate, is quite robust. Even with values of *t*
_{d} and α_{s} quite different from those linked to the reference parameters of Fig. 1, the asymptotic error of *method 3* will likely be, in practice, small and less than that due to biological variability and the measurement errors associated with the determination of the plasma tracer concentrations. For instance, if the slowest eigenvalue α_{s} (linked to the exchange of zinc tracer with the slowly equilibrating pools) were 100 times greater than that obtained for the parameters of Fig. 1 (i.e., α_{s}= 100 × 0.0017 day^{−1}, corresponding to a half-life of 4.08 days) the overestimate of FZA would still be less than 3.5%. This explains why *method 3* (and *method 4*) can be safely applied to estimate FZA in practice, even if we have shown that, in theory, they do not tend toward the correct value of FZA as time increases.

#### Remark 7.

If the values of both *t*
_{d} and α_{s}were available, the formula (34) could be used to exactly correct by the proper scale factor the FZA estimate found by*method 3* when a very large value
, larger than*t*
_{s}, is used.

#### Remark 8.

*Equation 34
* can be also used, on a pseudoempirical basis, to predict the estimation error when *method 3* is used at times much earlier than *t*
_{s}. For instance, by interpolating with a straight line the log transform of two plasma samples, e.g., for intravenous response on *days 3 *and*4* (solid line in Fig. 2, *B *and* C*), one finds a value of 0.162 day^{−1}. If this value is plugged into*Eq. 34
* in place of the true α_{s}, the overestimate of FZA predicted would be ∼3.3%. This FZA overestimation closely matches that exactly determined by numerical simulation (see Fig. 3, *B *and* C*, in particular the *right*, and Ref. 9). Simulations show that similar results (not displayed) can be obtained with other values estimated from the log transform of couples of plasma samples collected at times much earlier than *t*
_{s}, i.e., when the next-to-last eigenvalue (here 0.186) is still the dominant one.

#### Remark 9.

If the apparent delay *t*
_{d} were individually available, one could collect two plasma samples, at times *t [*minus] *t*
_{d} and *t*, and use*Eq. 30
* as if it were a fifth approximate method. A population value could be also used for*t*
_{d}. However, drawing two samples instead of one would make this fifth approximate method less convenient than *methods 3 *and* 4*, especially in light of the fact that population values of α_{s} and*t*
_{d} (possibly obtained from a wider population than that studied in Ref. 6) could be employed to try to correct the (over)estimation provided by the single plasma sample*method 3*.

#### Remark 10.

Being a (biased) approximation of *method 1, method 3 *is based on the same assumptions (see *remark 3*). In light of*Eq. 34
*, we can state that its accuracy depends on the product α_{s}
*t*
_{d}. Therefore, the smaller the slowest eigenvalue of the system is and/or the faster the absorption process is, the higher the potential accuracy is of*method 3. Method 4* coincides with *method 3* if the additional assumption, that the urine pool receives material only from the plasma compartment, is verified (this is the same hypothesis behind *method 2*; see *remark 4*).

## CONCLUSIONS

Empirical approaches based on the plasma or urine measurements of the ratio of orally to intravenously administered zinc tracers, termed the DITR method, have been proposed in the literature to determine FZA. These approaches can have a significant clinical relevance in nutritional studies, because they allow the human and economic costs of such studies to be significantly reduced. However, the reliability of this technique has been questioned on empirical grounds, and a mathematical analysis of its domain of validity has been lacking.

In this paper, it has been shown in an error-free context that four simple-to-implement approximate approaches employing DITR provide estimates of FZA close to those obtainable by two reference methods. The basic assumptions behind these methods are the linearity of tracer kinetics, the physiological indistinguishability of the oral and intravenous tracers, the existence of a remote pool(s) slowly equilibrating with plasma (i.e., the slowest eigenvalue generated by the extraplasma zinc pools is smaller than the smallest of the eigenvalues that regulate the absorption process), and the existence of a single, unidirectional flux into the urine pool originating from plasma (for *methods 2 *and* 4* only). All of the approximate methods that we presented require the investigator to acquire data at or until the time at which the intravenous and oral tracer concentration responses in plasma begin to exhibit the same multiexponential clearance. *Methods 1 *and* 2*asymptotically tend toward the true fractional absorption value as time elapsed after tracer administration increases. In contrast,*methods 3 *and* 4* are biased, and their potential accuracy depends on the speed of the absorption process relative to the slowest mode generated by the extraplasma zinc pools. In practice, as discussed at the end of domain of validity of approximate methods, typical zinc kinetics ensure that the (overestimation) error associated with *methods 3* and *4* is small, irrespective of sex, age, and dietary intake (see also Ref.9 for simulations of large changes in the model parameter values of Fig. 1).

In conclusion, this paper demonstrates that the DITR method can provide a reliable measure of FZA. A single sample of urine or plasma after an appropriate time after oral and intravenous tracer administration can yield estimates of FZA that are only slightly different from the value obtained by the reference methods. In particular, from the theoretical analysis performed in this paper, it could be argued that the estimation of FZA by tracer ratio in a spot urine specimen with the use of *method 4* could be the method of choice in practice when data acquisition for compartmental modeling cannot be performed. In fact, in light of the analysis of domain of validity of approximate methods, it follows that the theoretical error generated by *method 4* will be small. Moreover, *method 4 *is simple to implement and requires limited subject compliance.

Finally, we would like to stress that the aim of this paper was to theoretically assess the accuracy of the DITR method in an error-free context. When any of the DITR methods is applied in practice, one must take into account that the FZA estimate will be unavoidably uncertain because of data noise and also because of sparseness of sampling for the methods of *Eqs. 1-3
*. A possible way to evaluate the precision of DITR methods in practice would be to resort to stochastic simulation, and this issue certainly deserves further investigation.

## Acknowledgments

We thank the anonymous referees, whose constructive criticisms and suggestions helped us to significantly improve the quality of this work.

## Appendix

### The Deconvolution Method

To illustrate the genesis of *Eq. 2
*, let u(*t*) denote the (unknown) rate of appearance in plasma of^{67}Zn-tr, orally administered at *t* = 0. The absorbed fraction of ^{67}Zn-tr is thus
Equation A1having assumed, for the sake of simplicity, the^{67}Zn-tr administered to be unitary in mass. *Eq.EA1
*, and thus the overall method, assumes that all of the oral zinc absorbed from the gastrointestinal tract travels from the gut directly to the plasma. Denoting by ^{67}z_{1}(*t*) the concentration in *pool 1* (plasma) of^{67}Zn-tr at time *t*, thanks to system linearity, one has
Equation A2where g(*t*) is the unit impulse response of the zinc-system; i.e., g(*t*) gives the zinc tracer concentration in plasma after a unitary intravenous pulse. The unit impulse response g(*t*) can be found by using the intravenous tracer, here^{70}Zn-tr. Letting^{70}z_{1}(*t*) = g(*t*) denote the concentration in *pool 1* of ^{70}Zn-tr after the intravenous administration of a unit bolus (for the sake of simplicity) at *t* = 0, from *Eq. EA2
* one has
Equation A3From *Eq. EA3
*, one could obtain an estimate of u(*t*) by deconvolution using^{67}z_{1}(*t*) and^{70}z_{1}(*t*) and then calculating FZA by integrating such an estimate from zero to infinity, as in *Eq.EA1
*. However, no deconvolution procedure is actually necessary to obtain FZA, because it is well known (e.g., Ref. 4), that the integration of *Eq. EA3
* yields
Equation A4from which it follows that FZA by *Eq. EA1
* is given by*Eq. 2.*

## Appendix

### Equivalence of Methods 3 and 4

From *Eq. 15
*, one obtains
Equation B1
Equation B2which yield immediately to
Equation B3From Eq. 10, if *t*
_{2} >*t*
_{1} > *t**, it follows that
Equation B4and thus
Equation B5However, the left-hand side of *Eq. EB5
* coincides with the ratio of the concentration of the two tracers in a single urine specimen when *t*
_{1} and *t*
_{2}represent the beginning and the end of the urine sample collection
Equation B6which, in light of *Eq. 10,* proves *Eq. 22
*.

## Footnotes

This work was supported in part by National Institutes of Health Grants RR-11095 and RR-12609.

Address for reprint requests and other correspondence: C. Cobelli, Dept. of Electronics and Informatics, University of Padova, Via Gradenigo 6/A, 35131 Padua, Italy (E-mail:cobelli{at}dei.unipd.it).

The costs of publication of this article were defrayed in part by the payment of page charges. The article must therefore be hereby marked “

*advertisement*” in accordance with 18 U.S.C. Section 1734 solely to indicate this fact.10.1152/ajpendo.00113.2001

- Copyright © 2002 the American Physiological Society