INTRODUCTION

TOP ABSTRACT INTRODUCTION METHODS RESULTS DISCUSSION REFERENCES APPENDIX

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 (1-3, 6-9, 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 single-pool 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 tracer-to-tracee ratio (TTR) obtained after a constant tracer infusion [i(t)]

(1)

where V is the volume of distribution. Since then, the model of Steele et al. has been widely adopted. During the 1980s, however, many studies revealed the inadequacy of this model for glucose during highly non-steady-state conditions (1, 3, 6-9, 12). The model error of the equation of Steele et al. is mainly associated with the choice of V, which is experiment dependent. V is usually calculated as an empirical fraction (p) of the total pool size to account for imprecision due to the fact that samples are obtained from the circulation, which equilibrates more rapidly than the other pools. The model error can theoretically be minimized by 1) limiting the variations in enrichment (3), expressed as TTR or specific activity for the radioactive case, 2) estimating the variations in the pool size by use of a second tracer [approach of Issekutz et al. (9)], or 3) using a multiple-pool model.

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 steady-state equation: R_a(t)  i(t)/TTR(t).

The second approach can be accomplished by infusing two tracers at different rates. The time-varying 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 single-pool 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 and 2) 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 multiple-pool model, requires a more complicated mathematical analysis and is not treated here.

The goal of this study was to test the ability of the fixed-volume [i.e., traditional model of Steele et al. (12, 16)] and the variable-volume single-pool model [i.e., the model of Issekutz et al. (9)] to estimate R_a under non-steady-state 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 non-steady-state 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

TOP ABSTRACT INTRODUCTION METHODS RESULTS DISCUSSION REFERENCES APPENDIX

Subjects

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 high-carbohydrate 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

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View larger version (16K): [in this window] [in a new window]   Fig. 1.   Glucose (top) and glycerol (bottom) tracer infusion rates. Glucose infusion started 120 min before epinephrine infusion; glycerol infusion started 90 min before epinephrine infusion.

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

Glycerol concentrations were measured by enzymatic colorimetric assay (model RA-500, Technicon, Tarrytown, NY). Isotopic enrichment of [1,1,2,3,3-²H]glycerol and [2-¹³C]glycerol was determined on the Tris-trimethylsilyl derivative by gas chromatography-mass spectrometry, as previously described (17). Ions of rmw 205, 206, 207, and 208 were monitored. [2-¹³C]glycerol was determined by monitoring the ratio 206/205. [1,1,2,3,3-²H]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

Approach 1: one-tracer 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)
where TTR(t) is the enrichment at time t calculated according to Rosenblatt et al. (14), i(t) is the tracer infusion rate (constant or time variable to match TTR), and C(t) is the endogenous concentration calculated from the measured concentration [C_m(t)] as

(2)

When stable isotope tracers are infused, the measured substrate concentration is the sum of the concentration of the endogenous substrate and the contribution of all the tracers injected, since their mass is not negligible

(3)

The volume of distribution V was assumed to be constant and equal to 145 ml/kg (P = 0.63 of total V) for glucose and 230 ml/kg (extracellular V) for glycerol. Bounds for the R_a estimate were calculated with the assumption that glucose apparent V varies between 40 ml/kg (plasma pool size) and 230 ml/kg (extracellular pool size), and glycerol apparent V was assumed to vary between 40 ml/kg (plasma pool size) and 570 ml/kg (total body water).

Approach 2: two-tracer infusion. R_a was also calculated according to the approach of Issekutz et al. (9), which involves estimating the time-varying V [V(t)] by using the information from the two tracers, [6,6-²H]glucose or [1,1,2,3,3-²H]glycerol (tracer 1), infused at constant rate (i₁), and [1-¹³C]glucose or [2-¹³C]glycerol (tracer 2), infused at a variable rate [i₂(t)] during epinephrine infusion. R_a was then estimated from TTR of tracer 1 (infused at rate i₁) or tracer 2 [infused at rate i₂(t)]

(4)

where

(5)

and

(6)

In the formula to calculate R_a, the concentration is that of the endogenous substrate [C(t)], which can be estimated from the measured concentration [C_m(t)] by correcting for the contribution of tracers 1 and 2, as described previously. However, by substituting Eq. 5 in Eq. 4, C(t) is cancelled out and is not necessary to calculate R_a (see Eq. A7).

Statistical Analysis

(7)

	RESULTS

TOP ABSTRACT INTRODUCTION METHODS RESULTS DISCUSSION REFERENCES APPENDIX

Concentrations and Enrichments

View larger version (18K): [in this window] [in a new window]   Fig. 2.   Left, top: plasma glucose concentration (), [6,6-2H]glucose tracer-to-tracee ratio (TTR, ), and [1-13C]glucose (tracer 2) TTR (); bottom: plasma glucose concentration (), [1,1,2,3,3-2H]glycerol TTR (), and [2-13C]glycerol TTR (). Glucose and glycerol data were normalized to basal value. Values are means ± SE. Right: pattern of TTR derivatives calculated analytically by spline-fitting program from glucose (top) and glycerol (bottom) enrichments.

[1-¹³C]glucose (tracer 2) enrichment variations from the basal value were ~5%, whereas [6,6-²H]glucose (tracer 1) enrichment decreased on average by 25% of the basal value (Fig. 2; TTR^basal₁ = 0.062 ± 0.002, TTR^basal₂ = 0.025 ± 0.002). On average, [2-¹³C]glycerol (tracer 2) TTR first decreased and then increased, varying 15% around the basal value. [1,1,2,3,3-²H]glycerol (tracer 1) TTR decreased instead 75% below the basal value (TTR^basal₁ = 0.134 ± 0.026, TTR^basal₂ = 0.065 ± 0.012). The percent changes from basal of concentration and TTR are shown in Fig. 2.

Calculation of R_a

Approach 1: one-tracer 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 from tracer 1 varied up to 43% and up to 17% when tracer 2 was used (variability calculated according to Eq. 7, where x_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).

View larger version (14K): [in this window] [in a new window]   Fig. 3.   A: average glucose rate of appearance (Ra) estimated using a fixed pool volume [volume of distribution (V)] of 145 ml/kg with tracer 1 and tracer 2. B: average glucose Ra estimated from [6,6-2H]glucose TTR (tracer 1) with a fixed pool volume of 40, 145, and 230 ml/kg. C: average glucose Ra estimated from [1-13C]glucose TTR (tracer 2) with a fixed pool volume of 40, 145, and 230 ml/kg. Ra estimates were normalized to basal value. Values are means ± SE.

View larger version (15K): [in this window] [in a new window]   Fig. 4.   A: average glycerol Ra estimated using a fixed pool volume of 230 ml/kg with tracer 1 and tracer 2. B: average glucose Ra estimated from [1,1,2,3,3-2H]glycerol TTR (tracer 1) with a fixed pool volume of 40, 230, and 570 ml/kg. C: average glycerol Ra estimated from [2-13C]glycerol TTR (tracer 2) with a fixed pool volume of 40, 230, and 570 ml/kg. Ra estimates were normalized to basal value. Values are means ± SE.

Approach 2: two-tracer infusion. The contemporary infusion of two tracers of glucose and glycerol allowed us to use the variable-volume single-pool model [approach of Issekutz et al. (9)] to estimate R_a. Figures 5 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 single-pool model, showing the presence of singularities when the ratio TTR₁/TTR₂ approached a constant value (Figs. 5 and 6).

View larger version (15K): [in this window] [in a new window]   Fig. 5.   Glucose Ra estimated by [1-13C]glucose TTR (V = 145 ml/kg, solid line) and by approach of Issekutz et al. (Ref. 9; dashed line) for subjects 1 (A), 2 (B), 3 (C), 4 (D), and 5 (E). Ra estimates were normalized to basal value.

View larger version (16K): [in this window] [in a new window]   Fig. 6.   Glycerol Ra estimated from [2-13C]glycerol TTR (V = 230 ml/kg, solid line) and by approach of Issekutz et al. (Ref. 9; dashed line) for subjects 1 (A), 2 (B), 3 (C), 4 (D), and 5 (E). Ra estimates were normalized to basal value.

	DISCUSSION

TOP ABSTRACT INTRODUCTION METHODS RESULTS DISCUSSION REFERENCES APPENDIX

The goal of this study was to compare under non-steady-state 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 better-defined glucose system. The non-steady-state condition was created by a constant infusion of epinephrine (50 ng · kg¹ · min¹). Epinephrine was chosen, since it creates rapid non-steady-state 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 ²H-labeled tracer) at constant rate and tracer 2 (a ¹³C-labeled 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 (1-3, 6-9, 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 overcome 1) 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 fixed-volume model may be adequate to describe the substrate kinetics. However, Caumo et al. (2) recently showed that, theoretically, the variable-volume 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 TTR₁/TTR₂ approaches a constant value (see APPENDIX).

Approach 1: One-Tracer Infusion

Glycerol R_a, estimated from tracer 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 fixed-volume single-pool 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 variable-infusion 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: Two-Tracer Infusion

When all the data are considered together, it seems likely that under most circumstances the constant-infusion single-pool 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 two-tracer approach does not provide sufficient improvement to justify the extra effort and expense and, in some cases, may even be less accurate than the single-tracer approach.