Abstract
The most common approach for estimating substrate rate of appearance (R_{a}) is use of the singlepool model first proposed by R. W. Steele, J. S. Wall, R. C. DeBodo, and N. Altszuler. (Am. J. Physiol. 187: 15–24, 1956). To overcome the model error during highly nonsteadystate conditions due to the assumption of a constant volume of distribution (V), two strategies have been proposed: 1) use of a variable tracer infusion rate to minimize tracertotracee ratio (TTR) variations (fixedvolume approach) or2) use of two tracers of the same substrate with one infused at a constant rate and the other at a variable rate (variablevolume approach or approach of T. Issekutz, R. Issekutz, and D. Elahi. Can. J. Physiol. Pharmacol. 52: 215–224, 1974). The goal of this study was to compare the results of these two strategies for the analysis of the kinetics of glycerol and glucose under the nonsteadystate condition created by a constant infusion of epinephrine (50 ng ⋅ kg^{−1} ⋅ min^{−1}) with the traditional approach of Steele et al., which uses a constant infusion and fixed volume. The results showed that for glucose and glycerol the estimates of R_{a}obtained with the constant and the variable tracer infusion rate and the equation of Steele et al. were comparable. The variable tracer infusion approach was less sensitive to the choice of V in estimating R_{a} for glycerol and glucose, although the advantage of changing the tracer infusion rate was greater for glucose than for glycerol. The model of Issekutz et al. showed instability when the ratio TTR_{1}/TTR_{2}approaches a constant value, and the model is more sensitive to measurement error than the constantvolume model for glucose and glycerol. We conclude that the onetracer constantinfusion technique is sufficient in most cases for glycerol, whereas the onetracer variableinfusion technique is preferable for glucose. Reasonable values for glucose R_{a} can be obtained with the constantinfusion technique if V = 145 ml/kg.
 stable isotopes
 epinephrine
 glycerol
 glucose
in vivo measurement of the rate of appearance (R_{a}) of substrates in nonsteady state is of importance in physiological investigation. Many studies have been performed to validate methods for estimating glucose R_{a} (13, 69, 11, 12,15). However, this is not the case for other substrates, such as glycerol. The most common approach used to calculate R_{a} is the singlepool model proposed by Steele et al. in 1956 (16). It consists of calculating substrate R_{a} by estimating the model parameters from measurements of concentration [C(t)] and tracertotracee ratio (TTR) obtained after a constant tracer infusion [i(t)]
The first approach consists of infusing the tracer at a rate that matches the changes in endogenous R_{a} (and therefore requires some a priori knowledge of the expected R_{a} variations). In this way, TTR is constant and R_{a} is calculated by the following steadystate equation: R_{a}(t) ≅ i(t)/TTR(t).
The second approach can be accomplished by infusing two tracers at different rates. The timevarying volume is the volume that allows the correct calculation of the R_{a} in plasma of the first tracer from the R_{a} of the second tracer plus the plasma concentration of the two tracers. It has recently been shown that the estimated volume is experiment dependent and, therefore, does not reflect true changes in V (2). Nevertheless, this approach should give the best estimate of R_{a} by using the singlepool model, since it considers the information obtained by the contemporary infusion of two tracers and has the advantage of not requiring a preliminary study to determine the pattern of tracer infusion that matches changes in endogenous R_{a}. On the other hand, it is more expensive and time consuming, since it requires the contemporary infusion of two tracers of the same substrate. From a mathematical point of view (2), 1) this method is very sensitive to measurement error and2) this approach gives the exact estimate of endogenous R_{a} only if one of the two tracers is infused at a rate that matches perfectly changes in endogenous R_{a} (i.e., TTR is kept constant). If this is the case, however, we do not need the second tracer: if TTR is constant, the equation of Steele et al. gives the exact estimate of R_{a}, since it is not dependent on the choice of V.
The third approach, the multiplepool model, requires a more complicated mathematical analysis and is not treated here.
The goal of this study was to test the ability of the fixedvolume [i.e., traditional model of Steele et al. (12, 16)] and the variablevolume singlepool model [i.e., the model of Issekutz et al. (9)] to estimate R_{a}under nonsteadystate conditions. In particular, we were interested in studying the kinetics of glycerol when lipolysis is stimulated, since glycerol has not been widely investigated from a modeling point of view. Many studies on lipolysis have used the traditional model of Steele et al. (16) to estimate glycerol R_{a} in nonsteadystate conditions. The crucial point is the choice of V, since the appropriate value is not clear from the literature. Moreover, it is generally difficult to design a study that minimizes variations in glycerol enrichment, since its kinetics are not as tightly regulated as those of glucose, and the inter and intrasubject variability is very high. Therefore, we wanted to know the extent to which limiting variations in enrichment is important in the estimation of glycerol kinetics and whether the infusion of two tracers allowed us to obtain a better estimate of R_{a}. Using the same approach, we also studied glucose kinetics so we could compare the results of the two models for two different substrates.
METHODS
Subjects
Five healthy volunteers (2 women and 3 men) aged 27–33 yr were studied. A medical history of the subjects was obtained; they were given a routine physical examination and also routine blood and urine screening tests, as well as an electrocardiogram. The nature, purpose, and possible risk of the experimental procedures were explained to each subject before his/her written consent to participate was obtained. The experimental protocol was approved by the Institutional Review Board of the University of Texas Medical Branch.
The subjects reported to the General Clinical Research Center of the University of Texas Medical Branch at Galveston on the day before the study to eat a standard highcarbohydrate meal at about 6 PM, and then they went home, where they were asked to drink two cans of Ensure Plus (Ross Laboratories, Columbus, OH) at about 10 PM to increase their glycogen storage. On the next morning at about 7 AM they were admitted to the General Clinical Research Center for the performance of the experiment.
Experimental Protocol
Indwelling catheters were placed into the antecubital vein of one arm for tracer and epinephrine infusion and into a contralateral dorsal hand vein for arterialized venous sampling by the heated hand technique. A blood sample was drawn before the tracer infusion was started to determine background enrichment. The glucose and glycerol tracers were infused following the primed constantinfusion approach (starting 120 and 90 min before epinephrine infusion for glucose and glycerol, respectively) to reach isotopic steady state. Then, attime 0, epinephrine was infused (50 ng ⋅ kg^{−1} ⋅ min^{−1}) for 1 h. Deuterated tracers were infused at constant rate ([6,6^{2}H]glucose: 0.61 μmol ⋅ kg^{−1} ⋅ min^{−1}, priming dose 48.8 μmol/kg; [1,1,2,3,3^{2}H]glycerol: 0.21 μmol ⋅ kg^{−1} ⋅ min^{−1}, priming dose 3 μmol/kg). A carbonlabeled tracer of each substrate, [1^{13}C]glucose and [2^{13}C]glycerol, was infused during the basal period at constant rate ([1^{13}C]glucose: 0.22 μmol ⋅ kg^{−1} ⋅ min^{−1}, priming dose 17.6 μmol/kg; [2^{13}C]glycerol: 0.1 μmol ⋅ kg^{−1} ⋅ min^{−1}, priming dose 1.5 μmol/kg) and during epinephrine infusion at a variable rate (Fig. 1, Table1) necessary to maintain isotopic steady state. The changes in ^{13}C tracer infusion rates were based on the changes in glucose and glycerol R_{a} observed in pilot experiments of identical design.
Blood samples (7 ml) were collected before the tracer infusion was started at −10, −5, and 0 min (where 0 min indicates the moment just before the start of the epinephrine infusion) and at 2, 4, 6, 8, 10, 12, 15, 20, 25, 30, 35, 40, 50, and 60 min.
Sample Analysis
Plasma glucose concentration was measured using a glucose/lactate analyzer (model YSI 2300, Yellow Springs Instruments, Yellow Springs, OH). The enrichments of [6,6^{2}H] glucose and [1^{13}C]glucose were determined as TTR (4, 5, 14), as previously described (17). Briefly, isotopic enrichment was determined on the pentaacetate derivative by gas chromatographymass spectrometry (model 5985, HewlettPackard, Palo Alto, CA) with use of electronic impact ionization by selectively monitoring ions of rounded molecular weight (rmw) 242, 243, and 244 for [6,6^{2}H]glucose and 331 and 332 for [1^{13}C]glucose. Correction was made for the contribution of singly labeled molecules (rmw 243) to the apparent enrichment of rmw 244 (17).
Glycerol concentrations were measured by enzymatic colorimetric assay (model RA500, Technicon, Tarrytown, NY). Isotopic enrichment of [1,1,2,3,3^{2}H]glycerol and [2^{13}C]glycerol was determined on the Tristrimethylsilyl derivative by gas chromatographymass spectrometry, as previously described (17). Ions of rmw 205, 206, 207, and 208 were monitored. [2^{13}C]glycerol was determined by monitoring the ratio 206/205. [1,1,2,3,3^{2}H]glycerol enrichment was determined by monitoring the ratio 208/205. Correction was made for the contribution of rmw 206 and 207 to the apparent enrichment of the ions of rmw 208 (17).
Calculations
Before modeling analysis, enrichment and concentration data were filtered by a splinefitting approach with use of a secondorder polynomial. TTR derivative was calculated analytically.
Approach 1: onetracer infusion.
Glucose and glycerol R_{a} were calculated using the equation of Steele et al. (12, 16) as modified for use with stable isotopes (13)
Approach 2: twotracer infusion.
R_{a} was also calculated according to the approach of Issekutz et al. (9), which involves estimating the timevarying V [V(t)] by using the information from the two tracers, [6,6^{2}H]glucose or [1,1,2,3,3^{2}H]glycerol (tracer 1), infused at constant rate (i_{1}), and [1^{13}C]glucose or [2^{13}C]glycerol (tracer 2), infused at a variable rate [i_{2}(t)] during epinephrine infusion. R_{a}was then estimated from TTR of tracer 1 (infused at rate i_{1}) or tracer 2 [infused at rate i_{2}(t)]
Statistical Analysis
Values are means ± SE. Variability of a measurex was expressed as variation from the expected value (x
_{m})
RESULTS
Concentrations and Enrichments
Plasma glucose (4.9 ± 0.17 mM basal) progressively increased during epinephrine infusion, reaching values 55% above baseline after 60 min (7.67 ± 0.31 mM at 50 min; Fig. 2,top). Plasma glycerol concentration (0.032 ± 0.003 mM basal) also rose progressively to five times the basal value (0.156 ± 0.016 mM at 25 min) and then started to decrease to reach a value equivalent to three times the basal measurement (0.091 ± 0.014 mM at 60 min; Fig. 2, bottom).
[1^{13}C]glucose (tracer 2) enrichment variations from the basal value were ∼5%, whereas [6,6^{2}H]glucose (tracer 1) enrichment decreased on average by 25% of the basal value (Fig. 2; T
Calculation of R_{a}
Approach 1: onetracer infusion.
Glucose (V = 145 ml/kg) R_{a} was estimated, with both tracers, to increase up to 2.5 times the basal value in the first 5 min (Fig. 3). However, R_{a} estimated fromtracer 1 varied up to 43% and up to 17% when tracer 2 was used (variability calculated according to Eq.7 , wherex _{m} is the R_{a} obtained with V = 145 ml/kg and when x is R_{a} obtained with V = 40 or 230 ml/kg; Fig. 3).
Glycerol (V = 230 ml/kg) tracers gave a similar estimate of the R_{a} pattern (Fig.4). R_{a}showed an increase up to five times the basal value, reaching a peak 20 min after the epinephrine infusion. Glycerol R_{a} showed a variability up to 20% when tracer 2 data were used and up to 40% when R_{a} was calculated fromtracer 1 (Fig. 4).
Approach 2: twotracer infusion.
The contemporary infusion of two tracers of glucose and glycerol allowed us to use the variablevolume singlepool model [approach of Issekutz et al. (9)] to estimate R_{a}. Figures5 and 6 show the glucose and glycerol R_{a}obtained from tracer 2 data with use of the equation of Steele et al. (16) compared with the R_{a} values obtained using the approach of Issekutz et al. (9) for each subject. On average, the R_{a} estimates were almost superimposable (data not shown). However, for a given subject, R_{a} calculated using the approach of Issekutz et al. was more variable than the estimates obtained using the traditional singlepool model, showing the presence of singularities when the ratio TTR_{1}/TTR_{2}approached a constant value (Figs. 5 and 6).
DISCUSSION
The goal of this study was to compare under nonsteadystate conditions R_{a} estimates obtained after the infusion of a single tracer [i.e., with the model of Steele et al. (12, 16)] with estimates obtained after infusion of two tracers [i.e., with the model of Issekutz et al. (9)]. In particular, we wanted to know how important in the estimation of glycerol kinetics it is to limit the variations in enrichment and whether the infusion of two tracers allows us to obtain a better estimate of R_{a}. We compared the results with those obtained using the same approaches to quantify the betterdefined glucose system. The nonsteadystate condition was created by a constant infusion of epinephrine (50 ng ⋅ kg^{−1} ⋅ min^{−1}). Epinephrine was chosen, since it creates rapid nonsteadystate changes in glucose and glycerol that peak and subside within 1 h (15). The high rate of epinephrine infusion was necessary to stimulate glucose production. For each substrate we infused two tracers:tracer 1 (a^{2}Hlabeled tracer) at constant rate and tracer 2 (a^{13}Clabeled tracer) at a rate that was designed to match the expected changes in endogenous R_{a}. These tracers have been shown to give the same estimates of R_{a}when infused simultaneously at the same rate (10, 17), and simultaneous use of the two tracers enabled us to eliminate intrasubject variability in the comparison of the constant and the variable infusion rate techniques.
Many studies have shown that the reliability of the traditional model of Steele et al. (16) for the estimation of glucose R_{a} in the nonsteady state is dependent on the choice of the effective V and, thus, is inversely related to the variation of the TTR (13, 69, 11, 12, 15). In other words, the higher the derivative of TTR, the higher the variability of the R_{a} estimate will be. The problem can be overcome1) by minimizing the variation in TTR (3, 6, 7) or 2) by infusing two tracers of the same substrate, one at a constant rate and the other at a variable rate, with use of the information of both tracers to estimate not only the R_{a} but also the variations of the apparent V, as proposed by Issekutz et al. (9). The first approach is simple but requires a priori knowledge of the pattern of the R_{a} response to the stimulus; the latter can often be estimated for glucose, but this is more difficult for substrates such as glycerol because of inter and intrasubject variability in response. The second approach requires the infusion of a second tracer to estimate the variations in V. This approach was designed to give a better estimate of R_{a} at the beginning of the stimulus, where the fixedvolume model may be adequate to describe the substrate kinetics. However, Caumo et al. (2) recently showed that, theoretically, the variablevolume model gives the exact estimate of R_{a} only if one of the two tracers is infused at a rate that exactly matches changes in endogenous R_{a}. If this is the case, however, there is no advantage in infusing a second tracer at constant rate, because if TTR is constant, we obtain an exact estimate of R_{a}, also with use of the traditional equation of Steele et al. (16). Moreover, the calculation of the variable volume is potentially dependent on the measurement error (2), and the model of Issekutz et al. (9) shows a singularity when the ratio TTR_{1}/TTR_{2}approaches a constant value (see ).
Approach 1: OneTracer Infusion
The model of Steele et al., (16) given its generality, can be used to study a large number of substrates. We studied the kinetics of glucose and glycerol in response to epinephrine infusion, since this hormone stimulates glucose production and lipolysis. The problems with this model for assessment of rapid changes in glucose R_{a} have been well documented. However, less information is available regarding assessment of glycerol R_{a} in the nonsteady state. Moreover, the optimal value of V for the calculation of glycerol R_{a} by use of the equation of Steele et al. in not known. We recently proposed (13) a value of 230 ml/kg, which corresponds to the volume of the extracellular fluid, but in theory the effective V for glycerol could vary between 40 (plasma volume) and 570 ml/kg (total body water). For glucose, reasonable values for V could range from 40 (plasma volume) to 230 ml/kg (extracellular fluid volume). We previously used 100 ml/kg during highintensity exercise (13), since in this case glucose turnover is increased and the apparent V is presumably smaller. In the present study we used 145 ml/kg, which corresponds to p = 0.63 of the total V (i.e., 230 ml/kg). We compared R_{a}estimates obtained using different values of V for both substrates (Figs. 3 and 4). The results for glucose showed that, as expected (3), the choice of V is less important if the TTR variations are minimized. Glucose R_{a} estimated fromtracer 2 with use of the fixedV singlepool model was comparable to that estimated fromtracer 1 (Fig. 3). However, the variability of R_{a} estimated fromtracer 1 with different values for V was higher than the variability obtained from tracer 2 data (43 vs. 17%; Fig. 3). During epinephrine infusion, gluconeogenesis is highly stimulated, and this can cause some problem in the estimation of TTR because of^{13}C label recycling. However, this does not seem to be a problem in this study, given that R_{a} estimates were comparable.
Glycerol R_{a}, estimated fromtracer 2 with a fixed volume of 230 ml/kg, was similar to that obtained from tracer 1 infused at a constant rate (Fig. 4). Therefore, we conclude that the glycerol fixedvolume singlepool model is able to describe plasma glycerol data even when the enrichment drops by 75% in 30 min (Fig. 2). This is probably because of the high turnover rate of the glycerol pool compared with glucose. The high turnover rate of the glycerol pool results in an equilibration being achieved throughout the glycerol pool much more rapidly than is the case with glucose. This is also reflected by the fact that a much smaller priming dose is needed for glycerol than for glucose. Nonetheless, when glycerol TTR changes were limited by the variableinfusion technique, the variability in R_{a} estimates was lower (20 vs. 40%; Fig. 4). Thus, whereas it is possible to obtain a more precise estimate of glycerol R_{a} by limiting variations in glycerol enrichment, a constant tracer infusion approach does not affect the accuracy of the R_{a} estimate. This is an important result, since glycerol kinetics are not easily predictable, and therefore it is difficult to determine a priori the rate of tracer infusion needed to keep the enrichment constant.
Approach 2: TwoTracer Infusion
The second part of this study involved estimating glycerol and glucose R_{a} by use of the variablevolume singlepool model first proposed by Issekutz et al. (9). The approach of Issekutz et al. applied to the singlepool model was believed to give the best estimate of R_{a}, since it avoids the problem of choosing a constant value for V or trying to infuse the tracer at a rate that matches endogenous R_{a}. In this study the R_{a} estimated using this approach was generally the same as that obtained using the traditional equation of Steele et al. (16) (Fig. 5), but the model artifact predicted a sudden change. This can be explained by the fact that the approach of Issekutz et al. (9) is accurate when the two enrichments are varying independently but presents a singularity when the ratio TTR_{1}/TTR_{2}approaches a constant value: in this case, the denominator of V(t) approaches zero, which is reflected by a sudden change in R_{a}(see ). Moreover, the estimated V does not have a physiological meaning, and its value strictly depends on the protocols of infusion of the two tracers (2). The method of Issekutz et al. (9) is also sensitive to measurement error (in other words, the noisier the data, the more variable the R_{a} estimate), and it gives the exact estimate of R_{a} only if the TTR of one of the two tracers is kept perfectly constant and the other is varying (2). Considering these results, we conclude that the approach of Issekutz et al. (9) is not preferable to the traditional fixedvolume singlepool model, since it does not guarantee a more precise estimate of R_{a}, and it is more costly and labor intensive, since it requires the simultaneous infusion of two tracers.
When all the data are considered together, it seems likely that under most circumstances the constantinfusion singlepool model is adequate to quantify glycerol kinetics, whereas the variable tracer infusion is preferable in the case of glucose. However, even in the case of glucose, a pool size of 145 ml/kg enables a reasonable estimation of R_{a} in a rapidly changing situation. The twotracer approach does not provide sufficient improvement to justify the extra effort and expense and, in some cases, may even be less accurate than the singletracer approach.
Acknowledgments
The authors thank D. L. Chinkes, S. Klein, and J. I. Rosenblatt and the nurses and staff of the General Clinical Research Center at the University of Texas Medical Branch for their time and competent technical assistance.
Footnotes

Address for reprint requests and other correspondence: R. R. Wolfe, Shriners Burns Institute, 815 Market St., Galveston, TX 77551.

This work was supported by Shriners Hospital Grant 8490. The General Clinical Research Center at the University of Texas Medical Branch is supported by National Institutes of Health Grant M01–0073.

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 Copyright © 1999 the American Physiological Society
Appendix
With the model of Steele et al. (16), R_{a} can be estimated using the following equation