MODELING IN PHYSIOLOGY
Analysis of the paired-tracer method of determining cell uptake
Philip D.
Watson
Department of Physiology, University of South Carolina, Columbia,
South Carolina 29208
 |
ABSTRACT |
The
paired-tracer method has been used extensively for determining cell
uptake of numerous substances, although the method of calculating
uptake has no published theoretical support. We have investigated the
effect of capillary permeability of the tracers on v, the
uptake rate calculated directly from the ratio of tracer venous
concentrations. For a simple mathematical model of plasma-tissue
movement of lactate and an analytic expression for v, it has
been shown that values of v calculated in the first moments
after tracer injection depend almost entirely on the differences in
tracer permeability-surface area product (PS). The model indicates v would never give the correct value of cell uptake. It is also shown that PS differences alone can explain the published values for
lactate uptake obtained from v in skeletal muscle of the rat and dog.
lactic acid; mathematical model; capillary permeability
 |
INTRODUCTION |
A CLASSICAL PROBLEM in the study of living organisms
has been the separation of the kinetics of distribution from the
kinetics of metabolism. In whole organ studies, a popular approach has used multiple tracers to determine the transport parameters so that
metabolic factors can be identified (1). Direct methods can be used to
show increased apparent distribution volumes for solutes that penetrate
the cell (6), but to obtain cell uptake parameter values from tracer
data usually requires mathematical models and fast computers. In an
approach that avoided use of computers and models, Yudilevich and Mann
(29) described a simplified version of the multiple-tracer approach
that has been used to quantify cell uptake. Called the paired-tracer
method, this has proved very popular: at least 54 studies of the
movement of amino acids, glucose, and many other substances have been
published since 1979.
The paired-tracer method is similar to the double-indicator method (3,
5) usually used to measure capillary permeability. By using a
nonpermeating reference indicator with the substance of interest,
Chinard et al. (3) showed that many of the problems of substance
dilution and wash-in could be avoided. This use of a reference
indicator was extended to investigate blood-to-cell transport by
Yudilevich et al. (28) by replacing the intravascular marker with an
extracellular tracer as the reference for the metabolite under study.
After both tracers were injected intra-arterially as a short pulse, the
tracer concentrations in venous samples were determined. The uptake (U)
was calculated from U = 1
C/Cref, where C and
Cref are the time-varying venous concentrations of the
metabolite and reference tracers, respectively, C and Cref being normalized to their respective arterial levels. In 1981, Eq. 1 was used for the unidirectional influx by Bustamante, Mann, and
Yudilevich
(2)
|
(1)
|
where Q is the blood flow rate, Ca is the arterial
concentration of the unlabeled metabolite, and Umax is the
maximum value of U (29). With extrapolation from studies of the
blood-brain barrier, v was taken to be the unidirectional flux
into the cells; however, to our knowledge, there was no published
theoretical support for this interpretation.
Our interest was in lactate movement in skeletal muscle, and at least
three papers have used the paired-tracer method to measure lactate
uptake in that tissue (8, 26, 27). All these studies used radiolabeled
mannitol as the extracellular reference. The underlying principle of
the paired-tracer method is that mannitol, being unable to enter the
cells, accumulates more rapidly in the interstitial fluid (IF) than
lactate when both are injected into the arterial blood. The higher
interstitial concentration reduces the transcapillary gradient and the
transcapillary diffusive flux of mannitol more than those of lactate.
Hence, the venous concentration of mannitol rises above that of
lactate, and this difference is related to the rate of unidirectional
uptake of lactate by the cells.
Watt et al. (27) explicitly assumed that lactate and mannitol, being
"not restricted by the capillary wall," were indistinguishable in
their blood-interstitial space transport characteristics. However, because mannitol (mol wt 182) is twice the size of lactate (mol wt 89),
it would be expected that mannitol would diffuse more slowly from the
blood than lactate, i.e., that mannitol would have a lower
permeability-surface area product (PS) than lactate. This would cause
mannitol to have a higher venous concentration without any effect of
cell lactate uptake. It seemed likely that using mannitol as the
reference for lactate would always give cell uptake values that were
too high.
Therefore, we tested the hypothesis that the capillary permeability of
the two tracers markedly influenced the values of v and
Umax. This was accomplished by using a mathematical model of solute movement and an algebraic derivation of Eq. 1. The
Umax method has been criticized in modeling studies
previously (21, 22), but the effects of PS differences were not
included.
 |
METHODS |
Because v is defined at a particular value of U
(Umax), it is useful to define a new variable,
v', such that
|
(2)
|
The first approach was to use a mathematical model of tracers moving
into a tissue with a predetermined unidirectional influx and to
calculate U from the venous concentrations predicted by the model and
v' from Eq. 2. v' was then compared with the
unidirectional influx set in the model. This approach assumed that the
model was a reasonable approximation to reality and that if the
Umax method could not give the correct influx values under
the ideal conditions of the model, it was very unlikely to be correct
in the real world. There are several extracellular solutes that could be used as reference tracers, and the effects of using solutes ranging
in size from sodium to 51Cr-EDTA were investigated.
Second, the expression for v' (Eq. 2) was derived
analytically from considerations of transcapillary transport for the
time when the capillaries were just filled with tracer. The resulting expression allowed the prediction of published lactate uptake data from
estimates of mannitol and lactate PS alone.
Model with known cell uptake.
Mathematical models have been used for studies of blood-tissue
transport since 1909 with work by Bohr and Krogh on CO2 and O2 (see Ref. 4 for review). The movement of nonmetabolized tracers was studied by Renkin (20) and Crone (4) among others, and more
complicated models have since been developed by Bassingthwaighte and
Goresky (1) and others to include the effects of cell uptake and
metabolism. The later models included a finite dimension in the
direction parallel to the capillaries (axially distributed), but this
important feature makes the equations so complex that numerical methods
are needed for analytic solutions, and numerical integration is usually
preferred (1). To use popular desk-top computers, we have extended the
work of Johnson and Wilson (14) and Vargas et al. (24), in which only
the capillaries were axially distributed, and the tracer concentrations
in the capillaries were assumed to equilibrate instantaneously with the
arterial inflow. This use of a steady-state equation for the capillary concentrations gives simple analytic solutions that can be handled by
investigators who are not expert in numerical methods. A minor disadvantage of using the steady-state capillary equation is that the
wash-in of tracer is omitted. Because the performance of the model
shortly after the injection of tracer is important, the steady-state
model predictions are compared with a dynamic distributed capillary
model in the APPENDIX.
A balance for the interstitial mass of each tracer was calculated. The
IF was assumed to be a well-mixed constant-volume compartment, with
lactate moving to and from the IF by diffusion across the capillary
wall and between the IF and the cell by Michaelis-Menten kinetics. To
analyze tracer data, it was assumed that the unlabeled lactate fluxes
and concentrations were in a steady state. This implies that tracer
influx into the cell is proportional to the tracer concentration in the
IF, so that tracer uptake into the cell is given by FCIF,
where CIF is interstitial tracer concentration, and F is
constant when interstitial concentration of unlabeled lactate is
constant. F may be called the cell uptake capacity and has units of
milliliters per minute. For simplicity, the model assumed that there
was no back flux of tracer or tracer metabolites from the cell. The
model was driven by a step change in arterial tracer concentration
rather than the short pulse used in most applications of the
paired-tracer method. This had the advantage of showing how uptake
varied with time.
The transcapillary flux of tracer lactate or mannitol into the IF from
plasma may be obtained from the product of blood or plasma flow and the
arteriovenous concentration difference. As influx into the cell is
given by FCIF, a mass balance in the interstitial compartment gives
|
(3)
|
where Ca and Cv are the arterial and venous
tracer concentrations, VIF is interstitial volume (ml), and
t is time. This equation assumes that all the radioactivity
entering the cell remains there, with no radioactivity returning to the
IF in any chemical form.
With the assumption that the intracapillary concentrations equilibrate
rapidly compared with the extravascular concentrations, a steady-state
equation can be used to describe capillary concentrations (14). Renkin
(20) showed that when diffusion is the only important transcapillary
transport process, tissue uptake is given by
|
(4)
|
Substituting in Eq. 3 gives
|
(5)
|
where
|
(6)
|
Equation 5 is a linear, first-order differential equation
with constant coefficients. When arterial concentration increases from
zero to Ca at time t = 0, Eq. 5 may be
integrated to
give
|
(7)
|
where
|
(8)
|
and Cv can be calculated from
|
(9)
|
With appropriate values for the parameters PS, VIF, Q,
Ca, and F, Eqs. 7 and 9 describe the
exponential rise of interstitial and venous concentrations of the
lactate tracer with time following a step in arterial tracer
concentration from zero to Ca.
Equations 3-9 also apply to solutes limited to the
extracellular space (such as mannitol) by setting F = 0. In this case,
we obtain
|
(10)
|
and
|
(11)
|
Equation 11 was published by Vargas et al. (24) in 1980. Note that CIF, Cv, Ca, and
in
Eqs. 10 and 11 apply to the extracellular reference
tracer.
The value of U at any time was obtained by dividing Cv
obtained from Eq. 9 (simulating lactate venous concentration)
by Cv obtained from Eq. 11 (simulating the
reference venous concentration). v' was calculated from Eq. 2.
To investigate how v' varied both with time and the capillary
permeability of the reference tracer, the model parameters were set to
values determined as follows. PS for lactate was determined in the
skeletal muscle of four cats in unpublished preliminary studies (P. D. Watson, M. I. Lindinger, M. T. Hamilton, and D. S. Ward) by use of the
methods previously reported (25). In these experiments, the following
values were set or found: PSLac = 11.1 ml · min
1 · 100 g
1, Q = 26 ml · min
1 · 100 g
1, Ca = 1.6 mM, VIF = 12.3 ml/100 g, net lactate flux = 0, and F = 17 ml · min
1 · 100 g
1. Under these conditions, the interstitial
concentration of unlabeled lactate was equal to arterial, and the
unidirectional influx that v' should give was 17 × 1.6 = 27.2 µmol · min
1 · 100 g
1. The sensitivity of the model output to the parameter
values was assessed by changing PSLac, Q, VIF,
and F separately by ±50%, and v' was calculated for the
period from 0 to 11 min. PS values for the extracellular reference
solutes were taken from Table 1 of Ref. 25. These were sodium chloride
(21 ml · min
1 · 100 g
1), 51Cr-EDTA (5.0), mannitol (7.5), and a
theoretical extracellular solute having a PS equal to that of lactate
(11.1).
Analytic derivation of Eq. 2 when CIF approaches zero.
In addition to the comparison with ideal data, the paired-tracer method
was analyzed further by deriving an analytic expression for v'.
Just after the start of the tracer infusion, when the capillaries have
just filled with tracer, CIF would be very close to zero
for lactate and the reference tracer. When CIF = 0, then Cv = Cae
PS/Q (20).
Therefore, for lactate and a reference tracer, the ratio of the venous
concentrations at this moment may be approximated by
|
(12)
|
where the concentrations are normalized to the arterial levels.
Taking natural logarithms of both sides we obtain
|
(13)
|
Rearranging and multiplying both sides by Ca
gives
|
(14)
|
Because the left-hand side of Eq. 13 is the definition of
v' (Eq. 2), Eq. 14 predicts that v'
will have an initial value determined entirely by the PS difference and
the value of Ca. It is emphasized that Eq. 14
applies only during the earliest moments of the infusions, just after
the capillaries have been filled with the tracers.
 |
RESULTS |
Figure 1 shows the model predictions
of tracer venous concentration ratios. For lactate, the venous
concentration calculated using Eq. 9 was divided by the
constant arterial tracer lactate concentration (Ca in
Eq. 9) to give the ratio RLac. The corresponding ratio for mannitol, RMann, was obtained using Eq. 11 and the mannitol tracer arterial concentration.
RMann starts above that for lactate because the mannitol
permeability is lower and less mannitol is extracted. When the cell
lactate uptake parameter was set at 17 ml · min
1 · 100 g
1, RLac remained less than
RMann. When cell uptake capacity was set to zero (F = 0),
then RLac rose above RMann because the faster diffusing lactate (larger PS) equilibrates with the interstitial fluid
more rapidly than mannitol.

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Fig. 1.
Ratio of venous concentrations of tracer lactate (RLac) and
mannitol (RMann) to arterial concentration after a step
increase from 0 to 1 in arterial concentration at time 0. Initial values at time 0 are determined entirely by PS/Q. (See
METHODS for definitions of abbreviations.) Solid lines,
concentrations when cell uptake parameter (F) was set at 17 ml · min 1 · 100 g 1. Dashed line, lactate concentration ratio when no
cell uptake was present.
|
|
The lactate v' values for three commonly used extracellular
markers, and also for a theoretical extracellular marker having the
same PS as lactate, are shown in Fig. 2.
Each solid line indicates the variation in v' with time, and
the broken line at 27.2 µmol · min
1 · 100 g
1 indicates the actual unidirectional uptake rate used
in the model. The lowest curve shows the v' value expected if
sodium chloride were used as the extracellular reference. Sodium
chloride (mol wt 58.5) is smaller than lactate and diffuses more
rapidly from the circulation, having a PS of ~21 (25). For the first
40 s, v' is negative because the sodium chloride leaves the
capillaries faster than the lactate. v' rises as the solutes
equilibrate and eventually stabilizes at v' = 10.7. 51Cr-EDTA has been frequently used as an extracellular
space tracer (15) and is larger than lactate, with a PS of 5.0 in this
preparation (25). The top curve shows that the expected v' with
51Cr-EDTA starts at 9.8 µmol · min
1 · 100 g
1, falls, and then slowly rises to 10.7 µmol · min
1 · 100 g
1 at steady state. A reference solute having the same
PS as lactate would give an initial v' of zero, because both
solutes leave the plasma at the same rate (see Eq. 14). A
mannitol reference, initially leaving more slowly than lactate, would
create an initial v' value of 5.8 as predicted by Eq. 14, rising to the same steady-state value as the others. It
is clear that theory predicts that v' will vary markedly
depending on the size (PS) of the extracellular tracer employed, and
that no reference tracer will give the correct value for cell uptake.

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Fig. 2.
v' calculated from Eq. 2 by use of values
generated by Eqs. 9 and 11 for four reference tracers.
Different reference solutes give large differences in v' in the
1st min. The error between v' and true unidirectional flux
(dotted line) is large for all tracer combinations at all times.
|
|
Sensitivity of v' to model parameter values.
v' was only slightly altered by changing Q. When Q was changed
from +50 to
50% (39 to 13 ml · min
1 · 100 g
1), the greatest change in v' was 5.3%. When F
was increased 50%, v' increased by a maximum of only 24.4%,
increasing the error on the estimate of cell uptake rate. Decreasing F
by 50% reduced the error, but the smallest error in v' was
still 44%, i.e., v' = 7.7 µmol · min
1 · 100 g
1 when FCIF was 13.6. As expected, changing
VIF affected the rate of approach to steady state but had
no effect on the maximum and minimum values of v'. The effect
of changing PSLac was more complicated, because the early
data are sensitive to PS differences (Eq. 14). When
PSLac was increased 50%, at time zero v' was
increased to 18.6, closer to the correct value of 27.2 µmol · min
1 · 100 g
1 but still with considerable error. Decreasing
PSLac increased the error in v'. In summary, the
failure of v' to match FCIF was not dependent on
the choice of model parameters over a wide range of values.
Use of an analytic equation for v'.
The Umax method uses brief arterial pulses of tracer to
avoid back diffusion of tracer from the cell, so the accumulation of
solute in the IF is small. Although Eq. 14 only applies to
time zero, it would be expected that using Eq. 14 with
appropriate PS values should predict the published uptake data using
the Umax method. Watt et al. (27) measured Umax
in the rat hindquarters at elevated arterial lactate concentrations.
From the permeability values published by Haraldsson and Rippe (11), it
can be calculated that PS for lactate and mannitol would be 2.6 times
greater for the rat than for the cat (25). In the preliminary study,
cat PS values were 11.1 and 7.3 for lactate and mannitol, respectively. Hence, the PS difference in the rat would be 2.6 × (11.1
7.3) = 9.9 ml · min
1 · 100 g
1. The v' values in the rat experiments of Watt
et al. (27) predicted from this are compared with the published data in
Fig. 3. The upper dashed line is the
prediction of v' from the PS difference, and the data points
are the v' data of Watt et al. The dashed line falls below the
data but lies close enough to indicate that most of the apparent uptake
could have been due to PS differences. Also, with the assumption that
dog PS values are the same as those of the cat, then the
Umax data of Gladden et al. (see Ref. 9) can be predicted.
Figure 3 also shows that these dog gastrocnemius muscle data lie close
to the v' value expected from the PS values.

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Fig. 3.
Prediction of published v data for lactate using Eq. 14
and estimates of PS. Dashed lines, prediction; symbols, published data
[published data are from Gladden et al. (9) and Watt et al. (27),
with the assumption that dog PS values are the same as those of
cat]. Predictions are remarkably close, implying that v
data do not reflect lactate uptake, only PS differences.
|
|
 |
DISCUSSION |
The purpose of the study was to investigate the relationship between
the permeability of the reference tracer and the calculated uptake rate
in the paired-tracer method. For the case of lactate in skeletal
muscle, the numerical results in Fig. 2 clearly show that the apparent
uptake can vary from
15
µmol · min
1 · 100 g
1 with use of sodium to +10 with use of
51Cr-EDTA. Although both of these solutes may be considered
small and rapidly diffusing compared with plasma proteins, sodium has a
PS four times greater than that of 51Cr-EDTA, and this has
a large effect on the concentration ratios used to calculate U and
v'. The v' data in the first minute most closely
reflect the data that would be obtained with pulse injection rather
than a step, and this was when the variation in the tracer PS had the
most influence. Because none of the tracers approached the true value
of cell uptake and the different tracers could give widely different
cell uptakes, it was concluded that the simulation showed that the
paired-tracer method was seriously flawed. A very wide range of model
parameter values had little influence on the difference between
v' and cell uptake rate, suggesting that a more detailed model
of the transport pathways will not alter the conclusion that Eq. 1 does not give reliable information on cell uptake.
The analytic result, Eq. 14, indicates why the values of
v' are so PS sensitive. At time zero, v' is primarily
determined by the difference in the PS values of the tracer and tracee
and is not influenced at all by cell uptake. When this result was used to predict the results that the paired-tracer method would give in two
preparations in two species, the closeness of the prediction to
published data was surprisingly good (Fig. 3). In addition, note that
the ratio of Eq. 9 to Eq. 11 will not reduce to a
simple form like Eq. 2, even when Ca = 1, and the
expression for v' (Eq. 2) will not reduce to F or
FCIF when Eqs. 9 and 11 are substituted. It
was concluded that published reports using the Umax method with mannitol and lactate give data that largely reflect the difference between the PS values of these solutes and do not indicate the lactate
uptake.
These results apply to studies of many other tissues and metabolites
where the capillary wall is permeable to the reference tracer. These
tissues include the placenta (12), heart (18), salivary gland (17),
pancreas (16), skeletal muscle (13), liver (10), and intestine (23), to
cite only the more recent reports. The studies of the blood-brain
barrier uptake with small-molecule reference tracers, such as sodium or
mannitol (19), do not have the same problem, because the reference
remains intravascular.
It should be noted that it is the processing of the venous
concentration by Eq. 2 that creates the problems; the
underlying principle that IF accumulation influences venous
concentration is correct. This can be seen in Fig. 1, where cell uptake
generates a clear modification of the venous tracer lactate
concentration (plotted here as the ratio to the constant arterial
tracer concentration). There is an easily measured difference between
the tracer lactate concentration ratio without uptake (F = 0, dashed
line) and when cell uptake was present (lower solid line labeled
F = 17), after sufficient time has elapsed for the IF accumulation to
occur. The need for IF accumulation indicates that step infusion of
tracers should be more suited to measuring cell uptake than pulse
injections.
An alternative to the Umax approach is to use a formal
parameter identification method. This will usually require a vascular marker in addition to the extracellular reference and the metabolite tracer being investigated. The venous concentrations of the permeable tracers can then be normalized by the venous concentration of the
intravascular marker, as done routinely in PS measurements (25). When
the model described here is used, the extravascular reference ratio
would be fitted with Eq. 9 to obtain VIF and the extravascular tracer PS. With VIF known, Eq. 11
would be used to obtain F and PS for the metabolite. Because Eqs.
9 and 11 are algebraic, this fitting can be performed by
convenient packages, such as Sigmaplot,1 that automate the
Marquardt fitting algorithm. However, when we try to obtain cell uptake
parameters from experimental data, the simple model described here may
not be adequate. Further work is needed to compare the cell uptake
parameters obtained by this model with those found with more complete
models of the transport processes.
In summary, use of the paired-tracer method to obtain cell uptake in
tissues with capillaries permeable to the reference tracer will give
data that mainly reflect the PS differences between the two tracers.
Our view is that simple ratios will never give quantitative information
on cell uptake in complex systems like whole organs. That information
will always require explicit models of the transport processes with the
appropriate fitting of model to data. The difficulty of this procedure
will depend on the details of the model.
 |
APPENDIX |
The venous concentrations predicted by Eqs. 9 and 11 are not zero when time = 0. This is a consequence of
the assumption that the capillaries are always in a steady state (the
use of Eq. 4). To show that this somewhat unintuitive
behavior is not important, the venous concentrations given by Eqs.
9 and 11 were compared with those predicted by a
distributed capillary model, in which the capillary was represented by
10 well-mixed compartments in series, and convection alone moved solute
from one plasma compartment to the next. Diffusion moved solute between
a single well-mixed interstitial compartment and the capillary
compartments. The equation describing solute movement in each capillary
compartment was
|
(A1)
|
where i referred to the ith compartment, PS was PS
for the whole tissue divided by 10, and Vpl was the plasma
volume for the whole tissue divided by 10. Together with the equation
for the interstitial
compartment
|
(A2)
|
where the summation is over all 10 plasma compartments, there were
11 simultaneous ordinary differential equations, which were integrated
by the Euler method that used a time step 1/10 required
for stability. The parameter values used were those listed in
METHODS; i.e., Q = 26 ml · min
1 · 100 g
1, tissue PS for lactate was 11.1 ml · min
1 · 100 g
1, mannitol PS was 7.5 ml · min
1 · 100 g
1, tissue plasma volume Vpl was 2.0 ml/100
g, and the interstitial volume was 12.3 ml/100 g. Cell uptake was not
included for simplicity and because cell uptake has no influence on
venous concentrations in the first few seconds. The solid lines in Fig.
4 show the rapid rise in venous
concentrations to a value of ~0.7 as the wash-in is completed. This
whole period takes ~6 s.

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Fig. 4.
Comparison of lactate and mannitol venous concentrations predicted by a
dynamic distributed capillary model (solid lines) with calculated
concentrations given by Eqs. 9 and 11. Details are
given in APPENDIX. Mean transit time (vertical dotted line)
was calculated from Vpl/Q. Interstitial lactate
concentration prediction is also shown (slanted dotted line).
|
|
The broken lines are the venous concentrations calculated from Eqs.
9 and 11. They lie above the solid lines of the more
complete model mainly because there is no time delay in the
steady-state capillary model. If the two broken lines are moved to the
right by an amount equal to the mean transit time, then the broken
lines superimpose on the solid lines almost exactly. (It is not a
perfect fit because some solute does transfer in the wash-in period,
but this amount is very small.) Hence, the steady-state capillary model
is an excellent fit to the more complete model after the first second
or so. The plateau-like behavior of venous concentration is clearly
seen in experimental data (20, 25).
 |
ACKNOWLEDGEMENTS |
The author is grateful to Drs. W. N. Durán, D. Kim, and A. B. Ritter for helpful comments and suggestions.
 |
FOOTNOTES |
Reprints may be requested from Dr. Watson.
1
Jandel Scientific, PO Box 7005, San Rafael, CA
94912-7005.
Received 5 September 1995; accepted in final form 8 April 1998.
 |
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