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.
 |
INTRODUCTION |
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" (see
REFERENCE 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 (see
FZA 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
|
(1)
|
with obvious meaning of notation (1).

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Fig. 1.
Compartmental model of zinc metabolism with
representative kinetic parameters espressed in days 1
(6). These parameters yield a true fractional zinc
absorption (FZA) of 0.279.
|
|
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, 67Zn-tr and
70Zn-tr, highly enriched in 67Zn and
70Zn, 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 in
APPENDIX A, 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., 67Zn-tr and 70Zn-tr
|
(2)
|
where, in general, 67zi(t)
denotes the concentration (e.g., measured in mg/ml) in pool i (1 is
plasma; see Fig. 1) of 67Zn-tr after an oral intake of a
unit mass at time 0, and 70zi(t)
denotes the concentration in pool i of 70Zn-tr after the
pulse intravenous administration of a unit mass of 70Zn-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 70Zn-tr and
67Zn-tr in plasma, 70z1 and
67z1, respectively; methods 2 and 4 employ the 70Zn-tr and
67Zn-tr concentration in urine,
70z8 and 67z8, 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
|
(3)
|
where
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
|
(4)
|
where
is a time greater than
t*.
Method 3
The third approximate method provides
from a single plasma sample as
|
(5)
|
where
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
|
(6)
|
where t2 and t1
represent the beginning and end of the urine sample collection
interval, with t2 > t1 > t*
(t2 can be only a few hours greater than
t1), and
67z8(t2,t1)
and
70z8(t2,t1)
denote, respectively, the 67Zn-tr and 70Zn-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 67Zn-tr and 70Zn-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
67Zn-tr and 70Zn-tr concentration in a
complete, cumulative urine sample over several days is required.
Simpler still is method 3, in which measurement of
67Zn-tr and 70Zn-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 67Zn-tr and
70Zn-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 70z1(t) and
67z1(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
70z1(t) is 5 [the state variables
corresponding to compartments 4, 6, 8, and 9 do
not play any role in 70z1(t)],
whereas that of the system that generates
67z1(t) is 6 (the input occurs in
compartment 4). Indeed, the (early) time course in plasma of
67Zn-tr reflects the first-pass kinetics of movement
through and absorption from the gastrointestinal tract. However,
67Zn-tr that is absorbed into plasma equilibrates with the
extraplasma zinc pools and routes of renal and gastrointestinal loss in
the same manner as 70Zn-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
70z1(t) and
67z1(t) can be approximated by a sum
of the same number k (k < 5) of decaying
exponentials with the same eigenvalues
1,...,
k
|
(7)
|
|
(8)
|
For 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
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
67Zn-tr completely vanish only at infinity, the equal
sign only holds asymptotically).

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Fig. 2.
A: model-predicted 70Zn tracer
(70Zn-tr) plasma concentration after intravenous unitary
pulse administration at t = 0. B:
model-predicted 67Zn-tr plasma concentration after oral
unitary pulse administration at t = 0. C:
same curves as above on a semilogarithmic plot with an extended time
scale. Kinetic parameters are those of Fig. 1. Concentrations are
fraction of dose in plasma.
|
|
Remark 2.
Relationships between the parameters of the exponentials and the rate
constants kij 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
(t) as the function that makes the
following equation hold
|
(9)
|
The fact that, for times t greater than t*,
70z1(t) and
67z1(t) tend to assume the same
shape means that in Eq. 9 the function
(t)
tends, as time increases, toward a constant value
. Therefore, for
any t > t*, one could write
|
(10)
|
A grasp of Eq. 10 is offered by Fig. 2C,
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
|
(11)
|
where
is a generic time
greater than t*. By integrating
67z1(t) and
70z1(t) from
to infinity and exploiting
Eq. 10, we can write
|
(12)
|
Now, 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
|
(13)
|
is approximately constant and equal to
. Given that FZA is the
ratio of two real numbers, the fact that the ratio

67z1(
)d
/
70z1(
)d
is approximately
constant for any
greater
than t* allows us to state that the ratio
70z1(
)d
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
|
(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 qi denotes the mass of a zinc tracer in compartment i,
one has
|
(15)
|
Because no tracer is present in the system at time 0, q8(0) = 0. From Eq. 15 it can
be easily shown that
|
(16)
|
|
(17)
|
where
is a positive scalar. By dividing Eq. 16 by
Eq. 17, one obtains
|
(18)
|
Therefore, 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
|
(19)
|
Figure 3A 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.

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Fig. 3.
A: time course of the approximate estimate of
FZA provided by methods 1 and 2 (solid line) and
method 3 (dashed-dotted line) obtained by numerical
simulation of the model of Fig. 1 (true FZA is 0.279). B:
time course of the percentage estimation error provided by method
3 for times greater than 1.5 day. C: zoom of
B.
|
|
Consider Eq. 11. In light of Eq. 15,

67z1(
)d
and

70z1(
)d
can be
interpreted (if we ignore the common scale factor
) 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,

67z1(
)d
and

70z1(
)d
can be regarded as the relative
amounts of oral and intravenous tracer, respectively, that,
after
, traveled from plasma to urine. Becasue the appearance of 67Zn-tr in the urine is
delayed with respect to that of 70Zn-tr, we can write
|
(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

67z1(
)d
and from
the denominator 
70z1(
)d
the quantities

67z1(
)d
and

70z1(
)d
, 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
|
(21)
|
which 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 (t2, t1), with
t2 > t1 > t* and t* defined as the time after which both
70z1(t) and
67z1(t) exhibit a multiexponential
decay with the same eigenvalues (parallel behavior on semilog plot; see
Fig. 2, B and C), it can be shown (see
APPENDIX B) that
|
(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 t1 > t*
|
(23)
|
From Eq. 23 and the rationale following Eq. 13, one has it that
|
(24)
|
Now, by trivial calculations made on Eq. 20, one can
obtain that
|
(25)
|
Hence, from Eqs. 24 and 25, it follows that
|
(26)
|
from 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
|
(27)
|
In 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 67Zn-tr and 70Zn-tr were
administered simultaneously into the plasma (equal doses), the
concentrations of 67Zn-tr and 70Zn-tr in the
plasma would be equal at any time. Instead, the 67Zn-tr is
given orally and the plasma concentration of 67Zn-tr is
always lower than that of 70Zn-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, 67Zn-tr, as being the result of an
"intravenous injection" of an apparent tracer dose D, delayed by an
interval, td, with respect to the intravenously
injected tracer 70Zn-tr
|
(28)
|
If 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
|
(29)
|
or
|
(30)
|
In practice, Eq. 30 is not used (see remark
5 below), and the following approximation is made
|
(31)
|
which yields the estimate FZA shown in method 3
|
(32)
|
The application of Eq. 32 instead of Eq. 30
will lead to an overestimation of FZA, because
will always be less than
td). 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
> ts,
where ts is the time after which only the
slowest exponential, with eigenvalue hereafter indicated by
s, is still active (e.g., with the parameters of Fig. 1,
s = 0.0017 day
1). By employing
Eq. 8, it can be seen that under these conditions
|
(33)
|
Therefore
|
(34)
|
For the reference parameters of Fig. 1, td
turns out to be ~0.20 days, and
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.
3B.
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 td and
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
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.,
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 td and
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
ts, 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 ts. 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
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 ts, i.e., when the
next-to-last eigenvalue (here 0.186) is still the dominant one.
Remark 9.
If the apparent delay td were individually
available, one could collect two plasma samples, at times t
[minus] td and t, and use
Eq. 30 as if it were a fifth approximate method. A
population value could be also used for
td. 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
s and
td (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
std. 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.
We thank the anonymous referees, whose constructive criticisms and
suggestions helped us to significantly improve the quality of this work.
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).
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