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2
*
Department of Radiology, University of California, San Francisco, CA 94143,
Western Human Nutrition Research Center, U.S. Department of Agriculture, Agricultural Research Service, University of California, Davis CA 95616 and
**
Department of Biological Sciences, University of Central Lancashire, Preston PR1 2HE, UK
2To whom correspondence should be addressed. E-mail: jking{at}whnrc.usda.gov
| ABSTRACT |
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KEY WORDS: zinc stable isotopic tracers zinc absorption compartmental modeling
| INTRODUCTION |
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We have formulated a detailed compartmental model (9)
of
zinc metabolism from dynamic zinc tracer and tracee data. The model is
based on six normal women in whom zinc tracer and tracee measurements
were performed in plasma, urine and feces for up to 12 d after
simultaneous administration of both oral and intravenous stable
isotopic tracers of zinc. The model is consistent with zinc metabolism
as currently understood, contains the minimal degree of mathematical
structure required to explain all of the data and is mathematically
unique, meaning that all 11 adjustable rate constants are determinable
from the data with good precision. Such large relatively complex models
of metabolic systems, based on rich databases of information, can be
used to evaluate the adequacy of simpler models that focus on specific
features (in this case FZA) of the larger metabolic system described by
the detailed compartmental model (10)
.
A recent report by us (11)
compared the results of the
simple techniques for estimation of FZA based on six individual
datasets with the FZA obtained from the compartmental model also
applied to each of the datasets. Because of the experimental and
biological variability in each of the datasets, the results reflected
this variability within the data as well as the logical adequacy of
each of the simpler techniques with respect to the compartmental model.
To overcome these problems, the current study was designed to simulate
time-dependent values for FZA for the simple techniques using the
compartmental model, and then compared these estimates with the known
FZA derived from the detailed model. This procedure eliminated any
confounding effects due to analytical variations and inadequate data
sampling, so that the underlying logic for the simple techniques of
estimating FZA can be evaluated, i.e., how well do the simpler
techniques of FZA estimation recover the value for FZA encoded into the
compartmental model assuming noiseless data with very fine temporal
resolution. In addition, the ideal time at which or over which the
simple technique should be applied to provide the best estimate of FZA
can be determined.
| METHODS |
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FM technique.
The FM method for estimating zinc absorption is based on the amount of
orally administered tracer cumulatively excreted during a 12-d fecal
collection period (2)
. The cumulative fecal excretion of
tracer expressed as a fraction of the orally administered dose,
fo, is measured from which FZA is estimated by
![]() | (1) |
This measure of FZA, FM, no correction (FM-N), is known to be an underestimate of FZA because fo contains endogenously secreted oral tracer, which has already been absorbed as well as the unabsorbed oral tracer.
To correct for the resecretion of absorbed oral tracer, which
contributes to fo, English et al.
(3)
presented a technique detailed by Krebs et al.
(1)
in which the cumulative excretion of tracer, expressed
as the percentage of administered dose, is plotted against time. The
rate of increase of fecal accumulation of tracer defined as the slope
between successive data points, rises rapidly at the beginning of fecal
collection due to the passage of unabsorbed tracer directly into the
feces and then decreases to an apparent slightly positive slope of
< 1%/d. It is assumed that this final positive slope is due
entirely to the resecretion of absorbed tracer back into the intestine
and subsequent excretion into the feces. To correct for this endogenous
excretion, a line is fitted by linear regression to the data
points comprising the slightly positive slope on the cumulative tracer
excretion plot and extrapolated back to the y-axis. The
percentage of the never absorbed oral tracer dose excreted in the
stools is then estimated as the intercept of this line on the
y-axis, y(0), under which conditions FZA is then
given by
![]() | (2) |
Values for FZA using Equation 2 are subsequently referred to as FM, English correction (FM-E).
Another correction for the resecretion of absorbed oral tracer
appearing in the feces has been developed by Rauscher and
Fairweather-Tait (4)
. It takes advantage of the fecal
accumulation of a second tracer administered intravenously at the same
time as the oral tracer. They define an equation for apparent
fractional absorption of zinc exactly as that formulated in Equation 1
and convert it to FZA using additional information provided by the
fraction of an intravenously administered tracer accumulated in the
feces, fiv. They suggest that:
![]() | (3) |
The product, (fo)(fiv), is assumed to correct for resecretion of absorbed oral tracer collected in the feces as part of fo. This measure of FINF>), is assumedZA is subsequently referred to as FM, Raucher correction (FM-R).
We have developed an alternative formulation of the double tracer FM
technique based on different logic. Rewriting Equation 3
as
![]() | (4) |
It is seen that the final term of Equation 4
is given by the
fraction of oral tracer absorbed, FZA, multiplied by the fraction of
intravenously administered zinc tracer plasma tracer excreted in the
feces, fiv. Thus, Equation 4
becomes
![]() | (5) |
![]() | (6) |
Values for FZA determined using Equation 6 are subsequently referred to as FM, Shames correction (FM-S).
DA technique.
DA, used primarily as a method for estimating calcium absorption
(6
,12)
, can also be used to determine zinc absorption
(5)
. In this method, the first pass absorption of an oral
tracer dose can be determined if the tracer concentration responses in
plasma to both oral and intravenously administered tracers are
measured. Typically, the experiment is carried out by administering
different oral and intravenous tracers simultaneously, so that plasma
sampling has to be implemented only once. Under these conditions, the
function describing the tracer response in the plasma,
Rp,o(t), (% dose) to an orally administered
tracer is given by the convolution integral:
![]() | (7) |
where E(t) is the function describing the rate of first pass
entry of the orally administered tracer into the plasma compartment (%
dose h-1) and
Rp,iv(t) is the function describing the plasma
response to an intravenously administered tracer (% dose). Because FZA
is given by the integration of E(t) from 0 to infinity and the
integration of Eq 7
from 0 to infinity is
![]() | (8) |
![]() | (9) |
This value for FZA can be approximated by the ratio of the
integrals from zero to time t, assuming that t is great enough for the
first pass absorption process to be completed such that
![]() | (10) |
In practice, the numerator and denominator are plasma concentrations rather than total amounts of tracer in plasma, and consequently their ratio is equivalent to that in Equation 10 .
Because the amounts of oral and intravenous tracers in cumulative urine
are given by
0t Rp,o(
)d
and
0t Rp,iv(
)d
, respectively,
where
is a proportionality constant describing the fractional rate
of transfer of tracer from plasma to urine, the ratio of oral to
intravenous tracers in cumulative urine reduces to the same expression
as that seen in Equation 10
because
cancels out in both the
numerator and denominator.
DITR method.
The DITR method was first proposed for measuring FZA by Friel et al.
(7)
. They noticed that both the oral and intravenous
tracers in plasma and urine seemed to decline with the same slope after
a period of a day or 2 after simultaneous tracer administration and
assumed that the ratio of oral to intravenous tracer concentrations in
plasma or urine would be a good measure of FZA after correcting for
different tracer doses. These measures for FZA are given by the
following three equations for plasma, spot urine and cumulative 24-h
urine, respectively.
![]() | (11) |
![]() | (12) |
![]() | (13) |
where uo(t) and uiv(t) are concentrations of oral and intravenous tracer in a single spot urine specimen at any time t (d) > 2 and uo'(d) and uiv'(d) are concentrations of oral and intravenous tracer in a complete 24-h urine collection beginning after d 2. Estimates of FZA using Equations 1113 are subsequently referred to as DITR in plasma (DITRP), DITR in spot urine (DITRU) and DITR in 24-h urine (DITR24H), respectively.
ID techniques.
ID techniques for estimating FZA, FZAID,
described by Yergey (5)
and Jackson et al.
(8)
, are based on the accumulation of intravenously
injected zinc tracer in feces and urine or feces and plasma,
respectively, and provide a measure of the rate of endogenous fecal
zinc (EFZ) excretion, EFZ
(mg · d-1). When
combined with metabolic balance data on total dietary zinc and total
fecal zinc, these techniques can be used to estimate FZA. Assuming
intravenous administration of a unit dose of zinc tracer, one can show
that the ratio of EFZ to the rate of urinary excretion of zinc tracee,
U, (mg · d-1) is given
by
![]() | (14) |
where f* and u* are the accumulation of intravenously
administered zinc tracer in feces and urine, respectively, from time 0
to infinity. Assuming that this ratio reaches a limiting value at some
number of days t after tracer administration, Equation 14
can be
approximated by
![]() | (15) |
![]() | (16) |
where p(t) is the amount of zinc tracer in the plasma
compartment at any time t, P is the steady-state zinc tracee mass
in the plasma compartment (mg) and
is the fractional transfer of
zinc tracer and tracee from the plasma into the urine
(d-1), substitution into
Equation 15
yields
![]() | (17) |
where p'(t) is the zinc tracer-tracee ratio in plasma. Another
formulation of Equation 17
is given by
![]() | (18) |
where f'(t) is the zinc tracer-tracee ratio in feces and F
is the rate of fecal excretion of zinc tracee
(mg · d-1). Any of
these three equations for EFZ can be used to estimate FZA by the
steady-state balance equation
![]() | (19) |
where D and F are the dietary intake and fecal excretion, respectively, of zinc tracee (mg · d-1).
Compartmental model simulations and reference value for FZA.
The compartmental model from which the simulated values for FZA were
generated according to the simpler techniques (Eqs. 13
, 6, 1013,
15, and 1719) is based on stable isotopic tracer studies from six
normal women (9)
. The structure of this model with average
values for the rate constants
(pools · d-1) used in
the simulations is shown in Figure 1A
, and the steady-state solution of zinc masses (mg) and
fluxes (mg · d-1),
based on the average plasma zinc mass of 2.02 mg, is shown in Figure 1B
. Using this model, simulations of tracer content in
plasma, urine and feces were performed for both intravenous and oral
administrations. The plasma responses for both tracers up to d 20 are
shown in Figure 2
; cumulative urine and fecal tracer responses also for 20 d are
shown in Figure 3A
for intravenous tracer and in Figure 3B
for oral
tracer. Because the compartmental model used for all simulations is
based on data obtained only to d 12 (9)
, extrapolation of
the model simulations is required between d 12 and 20. Although such
extrapolation does lead to an element of uncertainty in the simulated
results beyond d 12, this uncertainty is likely small due to the
limited dynamic range of the model responses between d 12 and 20. Most
importantly, the longer simulation of the FZA results beyond d 12 for
some of the simple techniques provides additional insight into the
logical adequacy of each of the simple techniques not completely
appreciated up to d 12.
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![]() | (20) |
All simulations were generated using the SAAM II computer program (SAAM Institute, Seattle, WA). Values for FZA using the simple methods were determined at multiple times over the course of the theoretical experiment to evaluate the optimal time interval for use of each method. Slope estimates obtained using the FM-E method were calculated daily beginning at d 5 using the simulated cumulative excretion values for the oral tracer for that day and the previous 2 d (total of three points for each regression line). Because of the suspicion that the FM and ID methods were sensitive to percent recovery of excreted tracer and gastrointestinal (GI) transit time, simulations of %FZA(t) were also performed assuming that fecal tracer recovery was only 95% complete and that the GI transit time was twice the average value used in the initial simulations.
| RESULTS |
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7 or 8 d after tracer administration. The FM-N (Eq. 1)
and FM-E (Eq. 2)
methods provided overestimates of FZA initially
and then crossed the expected FZA value (at 45 d for FM-N and at
d 9 for FM-E) and became underestimates thereafter. The degree of the
underestimate was less for FM-E than for FM-N. The worst of the
techniques was FM-R (Eq. 3)
, which always overestimated FZA, the
smallest overestimate (13%) occurred at d 6. Before and after d 6, the
overestimates of FZA by FM-R became progressively worse.
The results for the ID technique, FZAID (Eq. 19)
,
are shown in Figure 6
. This technique, like DA, provided an
underestimate of FZA that approximated FZAmod
asymptotically over time but was less accurate than the DA technique
for a given time, t. At d 5 after tracer administration,
FZAID returned a value just over 84% of the
expected value and not until d 11 was the estimate > 95% of the
expected FZA.
The simulated effects on FZA estimates using the FM and ID methods
resulting from incomplete fecal tracer recovery (95%) and prolonged GI
transit times (twice average) are shown in Figures 7
and
8
. A 5% loss of both oral and intravenous fecal tracer (Fig. 7A)
had a
huge effect on these FZA estimates. The theoretically most accurate of
the FM methods, that of FM-S, provided an overestimate of 15% and
no longer approached the expected value asymptotically. Values of FZA
from both FM-N and FM-E were increased for both short and long
fecal collection periods and paradoxically provided better estimates of
FZA for long fecal collections than would be the case assuming complete
recovery of fecal tracer. Values obtained using the FM-R method
were also higher for all fecal collection periods with the best value
again occurring at d 6 and providing a 25% overestimate of FZA. The
effects of a longer GI transit time (Fig. 7B)
revealed similar shapes
to those responses seen in Figure 5
but were considerably delayed in
time. The FM-S method was again the most accurate, although it did
not approach the expected FZA value until d 14 or 15. Both FM-N and
FM-E were, again, overestimates of FZA early and underestimates
later in the fecal collection period, with the crossover point being
significantly delayed compared with the shorter GI transit time. The
overestimate of FZA provided by FM-R was somewhat worse than that
seen in Figure 5
, the best value being approximately a 20%
overestimate occurring at d 10.
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| DISCUSSION |
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The DA and DITR techniques (both using two tracers) were the most
robust of the simple methods because they were easy to perform and
provide relatively accurate estimates of FZA that were insensitive to
incomplete tracer collection and variability in colonic transit time.
The DITR estimates in urine were virtually the same as those in plasma
a few days after tracer administration, and little difference was found
between the DITR estimates in spot versus complete 24-h urine
collections (Fig. 4)
. Although the DITR values for FZA anytime after
the 1st d after tracer administration were overestimates, that provided
by DA was an underestimate of FZA, which approximates the correct value
asymptotically from below. In this analysis, based on zinc tracer
kinetics in six normal women, the DITR measures of FZA at times between
3 and 8 d after tracer administration were slightly more accurate
than that provided by DA, although eventually (e.g., at 20 d) the
DA estimate becomes more accurate. The DA and DITR estimates are both
ratio measurements and are insensitive to incomplete tracer collection
because both oral and intravenous stable isotopic tracers are measured
in the same sample of either plasma or urine. The colonic mean transit
time is also not a factor because measurement of fecal zinc tracer or
mass is not needed for the estimation.
All of the FM methods can lead to overestimates and underestimates of
FZA as shown in Figures 5
and 7
. Wastney and Henkin (13)
show, and we confirmed (Fig. 5)
, that use of FM-N (Eq. 1)
overestimates FZA for short fecal collections and underestimates it for
longer collections. The uncorrected technique provides an accurate
value for FZA only over a narrow band of days after tracer
administration. Because this ideal collection period depends on a
number of variables (rates of fractional absorption, fractional
secretion and GI transit time) the unknown values for which differ in
every subject, Wastney and Henkin (13)
argue that this
technique should not be used to estimate FZA. They point out that the
critical problem with this technique is the fecal excretion of
previously absorbed and resecreted oral tracer along with the
unabsorbed oral tracer.
An attempted solution to this problem of endogenous secretion of
absorbed oral tracer (FM-E; Eq. 2
) has been developed and carefully
described in a review by Krebs et al. (1)
based on work
presented earlier (3)
. The underlying assumption of this
technique is that the rate of excretion of resecreted absorbed tracer
is constant and can be determined from the slope of the fecal
accumulation of tracer plotted as a function of time after all of the
unabsorbed oral tracer has passed through the GI tract into the feces.
Once determined, this slope can be extrapolated back to zero time to,
thereby, account for all of the resecreted oral tracer that has been
absorbed. As shown in Figure 5
, this correction, just like that for
FM-N, also leads to an early overestimation and a later
underestimation of FZA, although not as great an underestimation as
that which occurs with no correction at all. The underestimation of FZA
by the FM-E method at later times results from the fact that the
rate of endogenous secretion of absorbed tracer is not constant but
falls with time probably in proportion to the fall in concentration of
the tracer in plasma, the latter peaking in the first 6 h after
oral administration (see Fig. 2
). Because the magnitude of the
correction slope is underestimated, Y(0) is overestimated and,
therefore, does not allow for complete correction for resecreted oral
tracer in the feces. The early overestimation of FZA using FM-E
results from the regression slope being applied to the cumulative fecal
tracer data before all of the unabsorbed oral tracer has exited the
colon producing a spuriously low value for Y(0).
Rauscher and Fairweather-Tait (4)
provide another
method for correcting for the secretion of absorbed oral tracer (FM-R;
Eq. 3
). This correction uses a second tracer administered intravenously
at the same time as oral tracer administration and assumes that the
fractions of absorbed oral and intravenously administered tracers
ending up in the feces would be the same. They estimate this amount of
absorbed but subsequently fecally excreted oral tracer as the product
of fo and fiv, where
fo and fiv are the
fractions of administered oral and intravenous tracers, respectively,
ending up in the feces. This amount of tracer is then added back to the
apparent absorption given by FM-N to correct for the underestimate
of FZA afforded by FM-N alone. As shown in Figure 5
, the
formulation offered by Rauscher and Fairweather-Tait
(4)
is inaccurate and overcorrects the underestimate of
FZA by FM-N, thereby producing an overestimate of FZA. This
overestimation by FM-R results from the substitution of
f0 for FZA in the correction term of Equation 3
.
Because f0 is nearly always greater than FZA
(except perhaps in instances of FZA being very high in instances of
severe zinc deficiency), FM-R will always be an overestimate,
greater in relative terms at lower FZA and becoming a better estimate
at higher FZA when values for f0 and FZA become
more similar.
The correct formulation of FZA using the FM technique with intravenous
and oral zinc tracers was derived by us in Equation 6
and referred to
as FM-S. It provides the correct value for FZA if fecal collections
are carried out long enough for the unabsorbed oral tracer to
completely exit the GI tract as shown in Figure 5
. The correct value
for FZA is approached asymptotically from above at roughly 7 d
after tracer administration corresponding in this case to between 5 and
6 GI transit times.
The effects on FZA provided by the four FM methods due to incomplete
(95%) fecal tracer recovery and twice average GI transit time are
shown in Figure 7
. Incomplete fecal tracer recovery (due either to
measurement error or inadequate subject compliance) increased the
estimate of FZA in all four FM methods. The magnitude of the error due
to incomplete fecal tracer recovery is amplified by the FM
calculations. For example, a 5% loss of tracer results in a 15%
overestimate of FZA by FM-S. Ironically, a spurious increase in FZA
due to fecal tracer loss could convert what would have been a
significant underestimate of FZA by the FM-N method into a more
accurate result.
The problems associated with the FM techniques are further compounded
by the possibility of a delayed GI transit time as shown in Figure 7B
. All four of the FM responses shown in Figure 7B
are significantly delayed in time. The FM-S method
again asymptotically approaches the correct value for FZA, but not
until 1315 d after tracer administration. Terminating fecal
collections before d 1315 would result in larger FZA values for all
four of the methods compared with values based on shorter GI transit
times (Fig. 5)
. Such an effect, like that due to incomplete fecal
tracer recovery, leads to an apparent improvement in the accuracy of
the FM-N method. The GI transit time used in the simulations shown
in Figure 7B
of 2.66 d (twice the average of our six
subjects) is well within the 95% confidence limits determined by us
using compartmental analysis (9)
; it is still less than
half of the average value reported by Wastney et al. (14)
based on a similar compartmental analysis of 25 normal subjects. Using
their estimate of average GI transit time of > 5 d, fecal
collection would have to be done for > 20 d to exclude the
confounding effects of incomplete GI transit of unabsorbed tracer.
Thus, even if fecal tracer recovery was complete, a 13- to 15-d fecal
collection period may not be long enough to obtain an accurate estimate
of FZA if GI transit is long. Because of the high variability in GI
transit time in normal subjects, even the most accurate of the FM
techniques, FM-S, is not recommended for careful estimation of FZA.
Although logically correct, the estimate of FZA provided by the ID
technique (Eq. 19
using EFZ estimates from Eqs. 15
, 17 or 18) requires
a longer data collection period than the other correct methods to
provide the same degree of accuracy (Fig. 6)
. Moreover, this accuracy
is significantly degraded by incomplete zinc tracer and tracee recovery
in feces and prolonged GI transit time (Fig. 8)
.
The results of the theoretical analysis discussed above are generally
corroborated by our recent report (11)
in which each of
six detailed zinc tracer studies (9)
is analyzed according
to the simple techniques for FZA and then compared with the FZA
obtained from the compartmental model applied to that dataset.
Methodological problems with this study detailed in a Letter to the
Editor (15)
include the experimental and biological
variability in each dataset, the specific sampling frequency used and
the limited number of studies. By focusing in the current report on a
theoretical dataset with zero noise and very fine sampling frequency,
the logical adequacy of each of the simple techniques for FZA
estimation can be better compared with the FZA of the compartmental
model (from which the theoretical data were simulated) outside of the
any confounding effects resulting from noisy data with limited temporal
resolution.
In summary, theoretical comparisons of the FM, DA, DITR and ID techniques for estimating FZA using a detailed compartmental model show that the FM technique, even when properly corrected for resecretion of absorbed oral tracer (FM-S), can lead to serious overestimates of FZA. The technique is highly sensitive to incomplete fecal tracer recovery (due either to inadequate subject compliance or analytical error) and prolonged GI transit time for zinc. Neither of these conditions is easily amenable to careful control, and the latter is highly variable among normal subjects. Because of these deficiencies, the FM technique cannot be recommended for accurate estimation of FZA. Nevertheless, when venous access is not possible or when only one zinc tracer is available, the FM-E technique may be the only technique applicable for estimation of FZA. Under such conditions, the FM-E technique may still be useful for relative nonquantitative FZA estimates in the same subject if careful attention is paid to complete fecal tracer recovery and maintenance of constant GI transit times between studies. Although logically correct, the ID technique based on a single intravenous injection of zinc tracer requires longer collection times than the DA and DITR methods for the same degree of accuracy. Moreover, the accuracy of the technique is seriously degraded by incomplete recoveries of fecal zinc tracer and mass and prolonged GI transit times. The DA and DITR techniques provide accurate estimates of FZA in both theory and practice. The DA technique requires multiple blood draws over a number of days to define the response of the orally and intravenously administered tracers in plasma with subject confinement for the first 78 h of the 1st d during which time high temporal resolution plasma zinc tracer measurements must be made. Alternatively, careful cumulative 24-h urine collections must be made over several days after oral and intravenous tracer administration. In contrast, the DITR technique requires only a single plasma sample, a 24-h urine or spot urine sample obtained 2 or more d after tracer administration. The spot urine sample requires the least amount of subject involvement of all the procedures. We, therefore, recommend the DITR technique, using a spot urine sample collected at least 2 d after tracer administration (or the average of several spot urine samples), as the method of choice for simple estimation of FZA when detailed compartmental modeling and the data acquisition it requires cannot be performed.
| FOOTNOTES |
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3 Abbreviations used: DA, deconvolution analysis;
DITR, double isotopic tracer ratio; DITRP, double isotopic
tracer ratio in plasma; DITRU, double isotopic tracer ratio
in spot urine; DITR24H, double isotopic tracer ratio in
24-h urine; EFZ, endogenous fecal zinc; FM, fecal monitoring; FM-E,
fecal monitoring, English correction; FM-N, fecal monitoring, no
correction; FM-R, fecal monitoring, Raucher correction; FM-S,
fecal monitoring, Shames correction; FZA, fractional zinc absorption;
GI, gastrointestinal; ID, indicator dilution. ![]()
Manuscript received December 13, 2000. Initial review completed February 5, 2001. Revision accepted March 23, 2001.
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J. Kim, H. Y. Paik, H. Joung, L. R. Woodhouse, S. Li, and J. C. King Effect of Dietary Phytate on Zinc Homeostasis in Young and Elderly Korean Women J. Am. Coll. Nutr., February 1, 2007; 26(1): 1 - 9. [Abstract] [Full Text] [PDF] |
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C. M Donangelo, C. L Vargas Zapata, L. R Woodhouse, D. M Shames, R. Mukherjea, and J. C King Zinc absorption and kinetics during pregnancy and lactation in Brazilian women Am. J. Clinical Nutrition, July 1, 2005; 82(1): 118 - 124. [Abstract] [Full Text] [PDF] |
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K M. Hambidge, J. W Huffer, V. Raboy, G. K Grunwald, J. L Westcott, L. Sian, L. V Miller, J. A Dorsch, and N. F Krebs Zinc absorption from low-phytate hybrids of maize and their wild-type isohybrids Am. J. Clinical Nutrition, June 1, 2004; 79(6): 1053 - 1059. [Abstract] [Full Text] [PDF] |
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M. Hambidge, L. Miller, N. F. Krebs, J. Westcott, and G. Grunwald Accuracy of Simple Techniques for Estimating Fractional Zinc Absorption in Humans J. Nutr., February 1, 2002; 132(2): 322 - 322. [Full Text] [PDF] |
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D. M. Shames, L. R. Woodhouse, N. M. Lowe, and J. C. King Reply to Hambridge et al. J. Nutr., February 1, 2002; 132(2): 323 - 323. [Full Text] [PDF] |
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G. Sparacino, D. M. Shames, P. Vicini, J. C. King, and C. Cobelli Double isotope tracer method for measuring fractional zinc absorption: theoretical analysis Am J Physiol Endocrinol Metab, March 1, 2002; 282(3): E679 - E687. [Abstract] [Full Text] [PDF] |
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