## 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

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/C_{ref}, where C and C_{ref} are the time-varying venous concentrations of the metabolite and reference tracers, respectively, C and C_{ref}being normalized to their respective arterial levels. In 1981, *Eq.1
* was used for the unidirectional influx by Bustamante, Mann, and Yudilevich (2)
Equation 1where Q is the blood flow rate, C_{a} is the arterial concentration of the unlabeled metabolite, and U_{max} 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 U_{max}. This was accomplished by using a mathematical model of solute movement and an algebraic derivation of *Eq. 1.* The U_{max} 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 (U_{max}), it is useful to define a new variable,*v*′, such that
Equation 2The 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 U_{max} 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 ^{51}Cr-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 CO_{2} and O_{2} (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
.

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 FC_{IF}, where C_{IF} 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 FC_{IF}, a mass balance in the interstitial compartment gives
Equation 3where C_{a} and C_{v} are the arterial and venous tracer concentrations, V_{IF} 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
Equation 4Substituting in *Eq. 3
* gives
Equation 5where
Equation 6
*Equation 5
* is a linear, first-order differential equation with constant coefficients. When arterial concentration increases from zero to C_{a} at time *t* = 0, *Eq. 5
* may be integrated to give
Equation 7where
Equation 8and C_{v} can be calculated from
Equation 9With appropriate values for the parameters PS, V_{IF}, Q, C_{a}, 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 C_{a}.

*Equations 3-9
* also apply to solutes limited to the extracellular space (such as mannitol) by setting F = 0. In this case, we obtain
Equation 10and
Equation 11
*Equation 11
* was published by Vargas et al. (24) in 1980. Note that C_{IF}, C_{v}, C_{a}, and φ in*Eqs. 10
* and *
11
* apply to the extracellular reference tracer.

The value of U at any time was obtained by dividing C_{v}obtained from *Eq. 9
* (simulating lactate venous concentration) by C_{v} 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: PS_{Lac} = 11.1 ml ⋅ min^{−1} ⋅ 100 g^{−1}, Q = 26 ml ⋅ min^{−1} ⋅ 100 g^{−1}, C_{a} = 1.6 mM, V_{IF} = 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 PS_{Lac}, Q, V_{IF}, 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}), ^{51}Cr-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 C_{IF} 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, C_{IF} would be very close to zero for lactate and the reference tracer. When C_{IF} = 0, then C_{v} = C_{a}
*e*
^{−PS/Q} (20). Therefore, for lactate and a reference tracer, the ratio of the venous concentrations at this moment may be approximated by
Equation 12where the concentrations are normalized to the arterial levels. Taking natural logarithms of both sides we obtain
Equation 13Rearranging and multiplying both sides by C_{a}gives
Equation 14Because 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 C_{a}. 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 (C_{a} in*Eq. 9
*) to give the ratio R_{Lac}. The corresponding ratio for mannitol, R_{Mann}, was obtained using *Eq.11
* and the mannitol tracer arterial concentration. R_{Mann} 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}, R_{Lac} remained less than R_{Mann}. When cell uptake capacity was set to zero (F = 0), then R_{Lac} rose above R_{Mann} because the faster diffusing lactate (larger PS) equilibrates with the interstitial fluid more rapidly than mannitol.

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.^{51}Cr-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^{51}Cr-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.

#### 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 FC_{IF} was 13.6. As expected, changing V_{IF} affected the rate of approach to steady state but had no effect on the maximum and minimum values of *v*′. The effect of changing PS_{Lac} was more complicated, because the early data are sensitive to PS differences (*Eq. 14
*). When PS_{Lac} 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 PS_{Lac} increased the error in *v*′. In summary, the failure of *v*′ to match FC_{IF} was not dependent on the choice of model parameters over a wide range of values.

#### Use of an analytic equation for v′.

The U_{max} 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 U_{max} method. Watt et al. (27) measured U_{max}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 U_{max} 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.

## 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^{51}Cr-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 ^{51}Cr-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 C_{a} = 1, and the expression for *v*′ (*Eq. 2
*) will not reduce to F or FC_{IF} when *Eqs. 9
* and *
11
* are substituted. It was concluded that published reports using the U_{max} 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 U_{max} 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 V_{IF} and the extravascular tracer PS. With V_{IF} 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.

## Acknowledgments

The author is grateful to Drs. W. N. Durán, D. Kim, and A. B. Ritter for helpful comments and suggestions.

## 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
Equation A1where *i* referred to the *i*th compartment, PS was PS for the whole tissue divided by 10, and V_{pl} was the plasma volume for the whole tissue divided by 10. Together with the equation for the interstitial compartment
Equation A2where 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 inmethods; 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 V_{pl} 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.

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).

## Footnotes

Reprints may be requested from Dr. Watson.

↵1 Jandel Scientific, PO Box 7005, San Rafael, CA 94912-7005.

- Copyright © 1998 the American Physiological Society