Although boluses of IV fluid are often administered to anesthetized or critically ill patients in an effort to increase arterial blood pressure, cardiac output, or urinary output, the clinical efficacy of fluid administration is often variable. Many investigators have studied the effects of fluid boluses on hemodynamics and plasma volume, but only recently have they applied principles similar to those used in pharmacokinetics to quantify the time course of plasma dilution after fluid infusions in volunteers ^{(1–5)}. Such human studies have provided valuable information about the influence of various fluids ^{(3)} and the effects of physiologic abnormalities such as hemorrhage on plasma dilution and kinetic variables ^{(4)}. Similar analyses have been performed in animals in an effort to define the influence of anesthetic drugs ^{(6)}, sepsis ^{(7)}, and hypoproteinemia ^{(8)}.

Most studies examining the kinetics of fluid have used relatively large volumes (e.g., 25 mL/kg) infused over 20–30 min. Such studies in conscious sheep ^{(8)} and healthy volunteers ^{(2,3)} have demonstrated that plasma dilution peaks at the end of the infusion at a maximal volume, which is a fraction of the infused volume, and then diminishes rapidly after the infusion is completed. Furthermore, kinetic variables remain fairly consistent when challenged by different infusion rates ^{(9)}. In contrast, isotonic and hypertonic fluids containing colloid result in a more sustained plasma dilution ^{(3,8)}.

We hypothesized that infusion of smaller volumes of 0.9% saline (NS) over a shorter duration (i.e., a bolus) would result in more pronounced plasma dilution, corresponding to preservation of a larger proportion of fluid within the vascular tree and would permit calculation of similar kinetic variables to those produced by larger infusions over a longer period of time. We also hypothesized that the peak plasma dilution during infusion of a hypertonic colloid solution, 7.5% saline/6.0% dextran 70 (HSD), would be similar after rapid or slow boluses consisting of the same volume of fluid and that kinetic variables would be similar between rapid and slow boluses.

## Methods

The experimental protocol of this study was reviewed and approved by the Institutional Animal Care and Use Committee of the University of Texas Medical Branch at Galveston and adhered to the National Institutes of Health Guide for Care and Use of Laboratory Animals.

Six adult female merino sheep weighing 36 ± 3 kg were studied in the fasting state. Surgical preparation was performed 24 h or more before any experiment and included splenectomy, insertion of a pulmonary arterial catheter (Swan-Ganz, Baxter, Irvine, CA), bilateral femoral arterial and venous catheters (Intracath, Becton Dickinson, Sandy, UT), and a urinary bladder catheter (Sherwood Medical, St Louis, MO).

All sheep were subjected to 4 protocols in random order on different days separated by at least 48 h. In Experiments 1 and 2, animals received 6.0 mL/kg of NS over 5 min and 24 mL/kg of NS over 20 min, respectively. Thus, the infusion rate, 1.2 mL · kg^{−1} · min^{−1}, was the same in both experiments. In Experiments 3 and 4, the animals were given 4.0 mL/kg of HSD over 2 min and 20 min, respectively. Thus, the infusion rate was 2.0 mL · kg^{−1} · min^{−1} in Experiment 3 and 0.2 mL · kg^{−1} · min^{−1} in Experiment 4.

The distribution and elimination of infused fluid was analyzed according to a three-volume kinetic model (Fig. 1). The application of this over-arching model depended on the infused fluid ^{(2–5)}. The isotonic fluid could expand the *V*_{1} (one-volume model) or both *V*_{1} and *V*_{2} (two-volume model). The hypertonic fluid could expand *V*_{1} and *V*_{2} or *V*_{1}, *V*_{2}, and *V*_{3} (three-volume model). In all cases, elimination of fluid from the system occurred according to a mechanism proportional by a variable *k*_{r} to the dilution of *V*_{1} and also by a baseline fluid loss *k*_{b}, which was set to 0.2 mL/min ^{(10,11)} to include correction for flushing of the catheters. Equilibration of fluid between *V*_{1} and *V*_{2} was proportional by a variable *k*_{t} to the difference in dilution between these body fluid spaces. Infusion of hypertonic fluid resulted in an osmotic fluid shift from a remote body fluid space, *V*_{3}, which was considered in the calculations (see Appendix). Fluid was also returned to *V*_{3} at a rate governed by the constant *k*_{13} (designated *k*_{23} if *V*_{3} was used).

The dilution of arterial plasma was used as the input variable in the calculations because it indicates the dilution of *V* (one-volume) or *V*_{1} (two-and three-volume models). Hence:MATH where [Hb]_{0} and [Hb]_{n} represent hemoglobin ([Hb]) at the beginning of the infusion (0) and at any time point (n), respectively. For this purpose, repeated measurements of [Hb] were performed at 5-min intervals or less on a HemaVet (CDC Technologies, Oxford, CT). A correction of the dilution was made for the loss of 2.0 mL of blood for each sample withdrawn throughout the experiments based on the assumption that the baseline blood volume was 6% of body weight (BW).

Each dilution-time curve was modeled using Matlab version 4.2 (Math Works Inc, Notich, MA), whereby a nonlinear least-squares regression routine, based on a modified Gauss-Newton method, was repeated until none of the variables in the kinetic models (such as *V*_{1}, *V*_{2}, and *k*_{t}) changed by more than 0.001 (0.1%) between the penultimate and final iteration ^{(3,4)}. Because this mathematical procedure does not yield a precise result, the output consists of the best estimate of each variable and the associated degree of uncertainty, the latter given as se. The kinetic program determines by using an F test ^{(12)} whether it is statistically justified to fit a mono-exponential, bi-exponential, or tri-exponential decay curve to each dilution-time curve.

Plots of the differences between the measured dilution and dilution predicted by the model were generated to determine the closeness of fit of each individual dilution-time plot to the mean dilution-time curve. In these plots, a difference of zero indicates no difference between model-predicted dilution and actual dilution.

The approach used to assess predictive performance is similar to that used in an evaluation of infusion pumps ^{(10)}, but in this study, many observations were nearly at baseline; therefore, we chose to report absolute errors instead of relative errors. The median absolute performance error (MDAPE) and the median prediction error (MDPE) were calculated. These are based on the difference between the measured dilution in all 6 sheep during infusion and 90 min thereafter and the computer-generated expected overall dilution response from the same series of infusion experiments or from another series. The first error, MDAPE, denotes the accuracy of the prediction and the second, MDPE, its grade of bias. Thus, the MDAPE describes the magnitude of the error in model-prediction of actual dilution, and the MDPE describes whether measured dilution is systematically more or less than the model-predicted dilution. Calculations of both errors were made using the two-stage approach ^{(10)}.

Although the kinetics of each individual experiment were described by one single set of variables, the expected overall dilution response to infusion of a fluid was obtained as the mean of all six simulated dilution-time curves from the respective series of six experiments, each one based on the appropriate 1-, 2-, or 3-volume model as indicated by the F test.

In the same way, we compared dilution-time data in sheep with similar theoretical dilution-time data derived from previous studies by Svensén and Hahn ^{(3)} and Drobin and Hahn ^{(5)}. To determine whether or not the human and sheep variables were comparable, the infused fluid volumes from the human studies were adjusted to the mean BW of the sheep.

Data are presented as the mean ± (sem). The groups were compared by the paired *t*-test or the Mann-Whitney *U*-test, as appropriate. Correlations were studied by simple linear regression. *P* < 0.05 was considered statistically significant.

## Results

Although the volume of the 20-min bolus was four-fold larger than that of the 5-min bolus, the maximum plasma dilution at the end of the infusion was only slightly more than twice as large (22% versus 10%;Fig. 2, A and B). In four experiments in each protocol, the one-volume model was statistically preferable. The two-volume model was statistically preferable in two of the six experiments in each group (Table 1). The elimination rate constant (*k*_{r}) was higher in the former model, whereas the expandable body fluid spaces (*V*, *V*_{1}, and *V*_{2}) were slightly larger for the short infusions (Table 1). Despite these small differences, simulations based on all kinetic data from each of the two series of experiments showed very similar dilution-time profiles for an infusion of 1 L of NS given over various periods of time (Fig. 2, C and D).

The elimination of NS as calculated by the model agreed reasonably well with the measured urinary excretion. Hence, *k*_{r} obtained by all curve-fitting procedures shown in Table 1 averaged 187 ± 40 mL/min and *k*_{r} predicted from the urinary excretion was 215 ± 62 mL/min ^{(11)}. The two calculations were highly correlated (*r* = 0.79;*P* < 0.003).

Urinary output was larger for experiments in which the kinetics of 0.9% saline best fit the one-volume model. Sixty minutes after the experiment started, 51% of the infused fluid in these experiments had been excreted, whereas the corresponding amount was 36% in sheep that handled the infused fluid according to the two-volume model (Table 2[top]). The slope of the dilution-time curve, which is expressed by *k*_{r}/ *V* (or *k*_{r}/ *V*_{1}), was also less steep for the latter experiments (0.05 ± 0.02 min^{−1} versus 0.13 ± 0.02 min^{−1};*P* < 0.05).

Plasma dilution averaged 24% at the end of the 2-min infusions and 21% at the end of the 20-min infusions (Fig. 3, A and B). Although the plasma dilution returned nearly to baseline during the 180 min of the study in both protocols, the initial decline in dilution was less rapid than in either of the NS protocols. To reduce the number of unknowns, kinetic analysis of the HSD dilution-time profiles required the use of a fixed value of *k*_{r} as determined by the urinary excretion (measured in four animals) ^{(11)}. The fixed *k*_{r} was virtually identical in the two series of experiments, although the urinary excretion amounted to 861 ± 187 mL for the 2-min infusions and 584 ± 113 mL for the 20-min infusions (Table 3 [top]). In no case was the intermediate fluid space *V*_{2} statistically justified, and therefore, fluid was considered to exchange only between *V*_{1} and *V*_{3} (two-volume model).

Simulations based on these kinetic data obtained during the 2-min and 20-min experiments showed very similar results (Fig. 3, C and D). We also analyzed all dilution-time profiles according to the three-volume model to examine its stability (Table 3 [bottom]). The sum of *V*_{1} and *V*_{2} then seemed to be very similar in size to *V*_{1} obtained in the statistically justified analyses.

Plots of the differences between the dilution predicted by the model and measured dilution indicated that experiments with HSD were as well described by the model as those with NS, i.e., the differences between dilution predicted by the mean curves and that actually measured in the curves at each time point were <0.10 in all instances and usually <0.05 (Fig. 4).

Further illustrations of how kinetic variables derived in one experiment can be used for predictive purposes of another experiment are shown in Figure 5A–C. The same comparisons can be made by calculating the accuracy and grade of bias, MDAPE and MDPE, of the computer-based predictions. In general, the agreement between the measured and predicted data was very good, regardless of whether the 5-min kinetic variables were used to estimate the dilution during the 20-min experiments or *vice versa* (Table 4). Simulations of dilution-time profiles based on volume kinetic data derived from humans showed similar or better agreement with the measured dilution in sheep, the only exception being that sheep eliminated NS more rapidly than humans when infused over 20 min.

To compare the volume-expanding efficacy of the study fluids, simulations based on the mean variable estimates shown in Tables 1 and 3 (top) were used to compare the fluid volumes of NS and HSD required to obtain 3 predetermined dilution limits at the end of a theoretical 10-, 20-, 40-, or 60-min infusion of each fluid (Fig. 6 [left]). The ratio between the rates obtained by these computer simulations indicate that approximately three times more NS than HSD must be administered to achieve the same plasma dilution if the infusion time is only 10 min. In contrast, approximately 10 times more NS than HSD is required to obtain the same dilution if the infusion time is 40 min (Fig. 6 [right]).

## Discussion

These data are the first to demonstrate that analysis of small, rapidly infused boluses of a conventional crystalloid provides a reliable estimate of kinetic variables calculated from larger, slower infusions. They are also the first to demonstrate that analysis of rapid infusions of a hypertonic solution provides a reliable estimate of kinetic variables calculated from a slower infusion of equal volume. Because these experiments used extremes of volume and infusion rates, they demonstrate that volume kinetic analysis is sufficiently robust to function under typical clinical circumstances. Despite the range of durations and infusion rates, the variable estimates between isotonic fluid protocols in sheep and those from other experiments in humans are similar. When these variables were used to simulate responses to theoretical infusions of various lengths and volumes, virtually identical dilution-time curves resulted (Figs. 3 and 5). This suggests that the disposition of infused fluid can be predicted from one short or long infusion and give similar results.

Constant kinetic variables for various durations of infusions of the same volume of crystalloids have previously been found in both male and female volunteers. In men, 25 mL/kg of acetated Ringer’s solution was infused over 15 and 30 minutes ^{(9)}, and in women, 25 mL/kg of acetated Ringer’s solution was infused over 15, 30, 45, and 80 minutes ^{(13)}. Preliminary data from other studies indicate that *V*_{1} might increase during massive fluid infusions. However, no previous volume kinetic studies have compared small volumes of isotonic crystalloid infusion over a brief interval.

This study demonstrates, through simulations comparing these data in sheep with previous data in humans, that conclusions drawn from volume kinetic studies in sheep may be extrapolated to humans and *vice versa*. This permits analysis of fluid administration regimens that may be often required in clinical situations but are not practically or ethically possible in volunteers. The kinetics of isotonic or nearly isotonic fluids are dependent on the state of hydration ^{(9)}, whereas the plasma dilution in response to HSD seems to agree very closely between sheep and humans (Table 4).

These data also suggest the possibility that small, rapid boluses can be used as diagnostic tests to define whether additional crystalloid fluid is likely to remain in the circulation, based on the previous observation that mild or moderate hypovolemia in volunteers markedly delays the loss of infusion-induced dilution of [Hb]^{(4)}. Smaller boluses of NS could provide diagnostic information because a larger proportion of an infused volume of isotonic crystalloids such as NS remains in the circulation at the conclusion of a shorter infusion than at the conclusion of a longer infusion. Fluids such as HSD, which are less rapidly eliminated from the circulation, are less well suited for small, diagnostic infusions.

If a predetermined plasma dilution (e.g., 10%) is desired, kinetic variables derived from bolus studies can be used to simulate the infusion rates of NS and HSD that are required to obtain and maintain that degree of dilution. In effect, the simulations calculate the volumes and infusion rates required to offset excretion and redistribution. Such simulations have previously been accomplished using crystalloid solutions in volunteers ^{(4,14)}. As expected, faster infusion rates of both fluids would be required to obtain greater dilution than we achieved in this study, and the rates must also be faster to allow for the predetermined dilution to be reached within a short period of time. Empirical testing of these concepts is required.

This comparative study between NS and HSD expands the observations from a previous comparison of lactated Ringer’s solution with HSD in sheep ^{(8)}. In that report, analyzed by the indicator-dilution technique, 30-minute infusions of lactated Ringer’s solution—a solution approximately 90% of the tonicity of NS—were compared with 30-minute infusions of HSD in sheep. The initial volume expanding efficiency HSD was seven-fold more than that of lactated Ringer’s solution (i.e., for each milliliter of lactated Ringer’s solution infused, intravascular volume increased only 0.27 mL). In contrast, for each milliliter of HSD infused, intravascular volume increased 1.8 mL. At 30 minutes after completing the infusion, for each milliliter of lactated Ringer’s solution infused, intravascular volume increased by only 0.07 mL. In contrast, for each milliliter of HSD infused, intravascular volume increased 1.3 mL—a 20-fold difference in volume expanding efficiency. Ninety minutes after completing the infusions, for each milliliter of lactated Ringer’s solution infused, intravascular volume increased only by 0.07 mL, whereas for each milliliter of HSD infused, intravascular volume increased by 0.8 mL—again, a 20-fold difference.

In a simulation based on the present study, the ratios of NS to HSD that were required to dilute plasma equally seemed to be independent of the target dilution used but were strongly dependent on infusion time. The difference in volume expanding efficiency is smallest when the infusion time is short; conversely, the difference is many times larger if both fluids are infused more slowly (Fig. 6) as they were in the previous study ^{(8)}. Simulations indicate that the volume expanding efficiency of the two fluids would differ only by a factor of two if an instantaneous bolus (infusion time = 0) could be administered. This time dependence is explained by excretion of a larger proportion of infused Ringer’s solution during a prolonged infusion, whereas differences in urinary excretion minimally influence plasma dilution during a short bolus infusion. Furthermore, the plasma dilution becomes pronounced early on during a bolus infusion of NS because one third of the animals showed signs of fluid accumulation in a small central fluid space, *V*_{1}. These animals are the ones in which a two-volume model applies better than a one-volume model.

However, theoretically, the volume expanding efficiency of HSD should exceed that of an isotonic crystalloid because of the ability of acute hypertonicity to osmotically attract intracellular fluid into the extracellular and vascular space. Possible reasons for the difference between predicted and actual plasma dilutions include expansion of a larger body space (between 3 L and 4 L) by HSD in comparison to expansion of a body space of only 1.5 L to 2.5 L by NS.

We speculate that plasma dilution by hypertonic fluids would be greater in hypovolemic animals or humans. Previous mass balance studies demonstrate greater plasma volume expansion with hypertonic solutions ^{(15,16)}. No kinetic data for hypertonic saline solutions have been reported during hypovolemic conditions, but in volunteers subjected to moderate hemorrhage, the *V*_{1} and *k*_{r} for acetated Ringer’s solution were reduced in proportion to the degree of hypovolemia ^{(4)}. As indicated above, a smaller *V*_{1} promotes a more pronounced dilution in response to a bolus.

For fluids infused over a longer interval, the elimination rate is a strong factor governing plasma dilution or volume expansion. For NS, the effectiveness of urinary excretion is evidenced by the fact that most infusions in normovolemic sheep were statistically most consistent with a one-volume model, which has a high elimination rate constant (*k*_{r}) in comparison to a two-volume model ^{(7)}. The ratios of 0.51 and 0.36 for urinary excretion, divided by the infused fluid volume at 60 minutes for the one- and two-volume models, respectively, are similar to those found after infusions of acetated Ringer’s solution at various rates to women ^{(13)} and those noted after urological irrigating fluids were infused in male volunteers ^{(17)}.

The increased urinary excretion associated with fluid infusion in experiments in which the one-volume model is statistically preferable (the majority of experiments) supports the conjecture that the determinant of whether a crystalloid infusion results in a one- or two-volume model may be related to small differences in baseline extracellular volume. In other words, if an experimental animal or volunteer is normovolemic, infused fluid will be excreted rapidly, and a one-volume model will result; if extracellular volume is slightly reduced, fluid will be eliminated more slowly, and some fluid will accumulate extravascularly.

These hypertonic saline data were analyzed according to a three-volume model for which the intermediate fluid space *V*_{2} could not be distinguished from the central space *V*_{1}^{(3)}. As in humans ^{(5)}, the kinetics of HSD should therefore be reported according to a distribution of fluid between two fluid spaces, *V*_{1} and *V*_{3}, in which the peripheral space physiologically resembles intracellular volume (Table 3 [top]). Previously, we analyzed hypertonic saline infusions in volunteers without considering the osmotic fluid shift between *V*_{1} and *V*_{3}^{(3)}, but the three-volume model applied here better represents physiological responses to HSD.

After infusion of the hypertonic solutions, the plasma volume expansion is reduced by two mechanisms: urinary excretion and return of previously osmotically translocated water back to the second peripheral space *V*_{3}. The return of water to *V*_{3} is reflected by the variable *k*_{13}. Although in this study, *k*_{13} was highly variable, and simulations using a range of values for this variable demonstrated that the reentry of fluid to *V*_{3} contributed relatively little to the fluid elimination (Appendix). Interestingly, urinary excretion during and after the hypertonic infusions exceeded the infused fluid volume by 400–700 mL, most of which presumably is derived from *V*_{3}, given that plasma dilution had nearly returned to baseline for *V*_{1} by 180 minutes after the beginning of the infusions.

Calculations of distribution volumes using volume kinetics suggest that both isotonic saline solutions and hypertonic solutions dilute a space that is clearly smaller than total extracellular volume, which is conventionally considered to be approximately 20% of total BW or approximately 200 mL/kg ^{(4,9,18)} and which is considered on theoretical grounds to be the distribution volume for isotonic or hypertonic saline solutions. The total body fluid space (*V*_{1} +*V*_{2} or *V*_{1} +*V*_{2} +*V*_{3}) expanded by the hypertonic solution in these experiments was approximately 100 mL/kg, which is similar to the volume obtained in male volunteers receiving acetated Ringer’s solution over 15 or 30 minutes ^{(3,9)}. In sheep receiving NS, the total distribution volume was 100 mL/kg or less with either fluid infusion and with either the one- or two-volume models.

The use of volume kinetics in experimental animals or humans to analyze the temporal response to fluid infusion is based on the observation that serial analysis of endogenous substances (e.g., water, albumin, and [Hb]) after fluid infusion can provide information about the disposition of infused fluid. For estimates of plasma dilution, [Hb] is particularly suitable, presumably because it is a nondiffusable tracer ^{(3)}. The use of volume kinetics overcomes many of the limitations with earlier methods used to estimate physiologic spaces such as isotope dispersal ^{(19)} or measurement of hemodynamic end-points ^{(20)}. Isotopes distribute within physiological spaces but may not accurately reflect the effects of an infused fluid, particularly under dynamic, non–steady-state circumstances. Hemodynamic end-points provide useful information but do not provide information about volume shifts, functional body space volumes, or mechanisms behind differences in fluid dynamics.

Volume kinetic analysis also complements mass balance analysis, which is a useful but traditional approach to describe the disposition of infused fluid ^{(8)}. In mass balance analysis, dilution of [Hb] can be used to calculate plasma volume (PV) expansion based on the assumption that added fluid is evenly mixed throughout blood volume and that the sum of PV expansion, urinary output, and extravascular expansion equals infused volume. However, differences between tissues in the mixing rate of infused fluid with blood volume could influence the accuracy of the calculations. By reporting results in terms of plasma dilution rather than PV expansion, volume kinetic analysis avoids the necessity of assuming that fluid is uniformly distributed throughout blood volume and also provides estimates of clearance and intercompartmental transfer. These estimates can be used to predict responses during similar physiologic circumstances and also can be used to simulate the outcome of other rates and volumes of infusion of similar fluids. The reliability of the models and the variables calculated from those models are evident in the residual plots and performance curves (Table 4, Figs. 4 and 5). However, the kinetic approach interprets the data only in terms of the applied model and is therefore highly dependent on the assumptions inherent in that model. Because the analysis describes the average behavior of the fluid, problems with interpretation may arise if major physiological changes occur in the course of an experiment.

Both mass balance and kinetic analysis offer important insights into the application of current practices related to IV infusion of fluids, especially by illustrating the duration of plasma dilution or PV expansion produced by various fluids and by demonstrating the effects of pathophysiologic disturbances on distribution and elimination of infused fluids.

In summary, we have demonstrated that analysis of small, rapidly infused boluses of a conventional crystalloid provides a reliable estimate of kinetic variables calculated from larger, slower infusions and that analysis of rapid infusions of a hypertonic solution provides a reliable estimate of kinetic variables calculated from a slower infusion of the same volume of solution. Despite the range of durations and infusion rates, the analysis generated similar estimates of kinetic variables. We interpret these data as suggesting that the disposition of infused fluid can be adequately described using a short bolus infusion.

### Appendix

#### Kinetics for Isotonic Fluid

For NS, the volume change of the single expandable body fluid space is then indicated by the dilution of the venous plasma according to equation 1:

The existence of a peripheral body fluid space *V*_{2} is said to be statistically justified if the lowest possible average difference between the model-predicted and measured data points (mean square error) is significantly reduced by fitting the solution to Equation 2 to the measured data points instead of the solution to Equation 1. In the absence of osmotic shifts, the situation in the central body fluid space *V*_{1} and the peripheral body fluid space *V*_{2} are as follows:MATH

Solutions to these differential equations have been published in previous work ^{(3)}.

#### Kinetics for Hypertonic Fluid

An osmotic shift of water, **f** (t), occurs when hypertonic fluid is infused IV. The osmotic shift occurs across the cell membrane and exchanges water from the intracellular to the extracellular fluid space, which amount to 20% and 40% of the BW, respectively (18). Using a baseline osmolality of 291 and a calculated osmolality of 2458 mOsm/kg for HSD, the translocated volume can be estimated from the following equation:MATH

This equation indicates that the first milliliter of infused HSD translocates 4.9 mL of water and that the osmotic gradient becomes progressively reduced for each subsequent volume of infused fluid as the osmolality of all body fluids gradually increases. Therefore, **f** (t) is entered as linear function in the analysis process where **f** (t) at each point in time is governed by the volume of previously infused fluid.

If an intermediate fluid space (*V*_{2}) is not statistically significant, and if **f** (t) denotes the osmotic fluid shift and *k*_{13} denotes the return of water to *V*_{3} (intracellular space), the volume changes in *V*_{1} and *V*_{3} can be expressed as:MATHMATH

The solutions to these differential equations and also to the ones describing the three-volume model in which *V*_{2} is statistically significant have recently been published ^{(5)}.

#### Limited Importance of *k*_{13}

Simulations in which hypothetical values of *k*_{r} and *k*_{13} were entered into the equations demonstrated that *k*_{r} had a greater influence than *k*_{13} on the overall rate of fluid elimination from *V*_{1}. Setting *k*_{13} to negligible values increased the area under the dilution-time curve by 6% and 19% during the 2-min and 20-min experiments, respectively, whereas the corresponding increases obtained by setting *k*_{r} to negligible values were 45% and 73%. Thus, the power of *k*_{13} to eliminate fluid was only between 13% and 26% of the power of *k*_{r}.

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