#### Theoretical Issues

##### Basic Principles

^{1,2}Hence, the plasma concentration cannot be expressed in the usual way. However, the water content of whole blood reflects the dilution of solid elements such as hemoglobin.

^{3,4}Therefore, the dilution of hemoglobin may serve as an indicator of the “concentration” of an infusion fluid.

##### The Two-volume Model

*R*

_{o}increases the volume of a central body fluid space

*V*

_{c}to a larger volume,

*v*

_{c}. The rate of elimination is given by the product of the fractional volume expansion (

*v*

_{c}−

*V*

_{c})/

*V*

_{c}and the elimination clearance,

*Cl*. Thus,

*Cl*is the part of the expanded fluid volume (

*v*

_{c}−

*V*

_{c}) that is totally eliminated per unit of time.

*Cl*

_{o}, which is usually preset to 0.3–0.5 ml/min depending on the size of the subject.

^{5,6}The total elimination clearance is

*Cl*+

*Cl*

_{o}, which approaches

*Cl*

_{o}when

*v*

_{c}approaches

*V*

_{c}. If the urinary excretion is measured,

*Cl*

_{o}can be estimated and then indicates all fluid that may be allocated outside the kinetic system in the body (if any), plus the baseline fluid loss.

*V*

_{t}, which becomes expanded to

*v*

_{t}. The rate of exchange between the

*V*

_{c}and

*V*

_{t}is determined by the difference in dilution between them, multiplied by the distribution clearance,

*Cl*

_{d}. As fluid flows freely and does not bind to tissue,

*Cl*

_{d}is given the same value for flow in both directions (appendix 2).

*v*

_{c}and

*v*

_{t}, the size of which changes constantly during an experiment (table 1). In fact, their increase is what primarily exerts a therapeutic effect in sick patients.

*V*

_{c}=

*V*

_{1},

*V*

_{t}=

*V*

_{2},

*Cl*

_{d}=

*k*

_{t},

*Cl*=

*k*

_{r}, and

*Cl*

_{o}=

*k*

_{b}.

##### Physiologic Correlates

*V*

_{c}and

*V*

_{t}correspond to the plasma volume and the interstitial fluid space, respectively, and that the fractional volume expansion distributes fluid by modifying the hydrostatic and colloid pressures in these body fluid spaces.

*Cl*

_{d}is believed to reflect differences in perfusion and capillary permeability between body regions.

^{7}Because infused fluid is eliminated by the kidneys, the

*Cl*estimated by the curve-fitting procedure should correspond to the renal clearance,

*Cl*

_{r}. However, the parameter estimates are not direct measurements of physiologic variables but rather are functional trend values that indicate how the body actually handles an infusion fluid.

^{7}

*V*

_{c}is 3–4 l, and this is close to the expected

^{3,7,8}or measured

^{9}size of the plasma volume.

*V*

_{c}becomes slightly larger if calculations are based on arterial hemoglobin samples rather than on venous samples.

^{7}

*V*

_{t}is 6–8 l in adult males weighing 70–80 kg and, therefore, smaller than the expected size of the interstitial fluid space.

^{3,10,11}In contrast to tracer ions such as bromide, however, volume kinetics indicates only the body fluid spaces that can be expanded, and this is not possible in certain body regions (such as the skeleton and the skull). Moreover, some tissues have high compliance for volume expansion whereas others require a higher fluid pressure for expansion to occur.

^{12}Therefore,

*V*

_{t}may be larger for massive fluid infusions

^{13}but not for the rates and volumes normally administered to humans.

^{14}The precision of an estimate of

*V*

_{t}is usually poorer than of

*V*

_{c}.

##### The One-volume Model

^{3}The one-volume model is also appropriate for crystalloid fluid when elimination is fast, which is sometimes the case in volunteer experiments.

^{11,14}The rationale is that an increasing ratio

*Cl/Cl*

_{d}offers less time for the fluid to expand

*V*

_{t}before elimination occurs, whereby

*V*

_{c}and the partially expanded

*V*

_{t}blend into a single fluid space of intermediate size (fig. 3).

##### Requirements for Successful Analysis

*V*

_{c},

*V*

_{t},

*Cl*, and

*Cl*

_{d}). If elimination is slow, the analysis will have difficulty differentiating between allocating fluid to

*V*

_{t}or as eliminating from the system expressed by

*Cl*. One may then replace

*Cl*by the renal clearance (

*Cl*

_{r}) as calculated from the measured urinary excretion (appendix 2).

^{6}On doing so, only three parameters have to be estimated by least-squares regression (

*V*

_{c},

*V*

_{t}, and

*Cl*

_{d}), which increases the stability of the model. The same trick is often helpful if the sampling time is shorter than 3 h.

^{10}

*V*and

*Cl*). With crystalloid fluids, the baseline hemoglobin level should be reached within 3 h (fig. 3A), whereas colloids have a much longer elimination phase.

^{3,15}

^{16}Drugs that cause diuresis or modify the adrenergic receptor activity may confuse the results if given when an experiment has already started.

##### Selection of Model

*F*test, might be applied to help decide whether the one- or two-volume model should be applied (appendix 2). Plotting the agreement between model-predicted and -measured urinary excretion may be a helpful adjunct.

^{17}In contrast, the parameters for groups of healthy volunteers may be difficult to evaluate because the two-volume model is often appropriate in some subjects but not in others.

^{10,11,14}However, all our studies published after 2003 have given the results according to only one variant of the model. For this purpose, simplifications of the two-volume model have sometimes been used.

^{18}deals with the absolute instead of the fractional volume expansion. The presence of

*V*

_{t}is acknowledged but its size is not estimated. The two-volume model then analyzes nearly all experiments, even when the urinary excretion is so large that the one-volume model would normally be appropriate.

^{7}

*V*

_{t}to a very high fixed value (like the body weight) blunts the flow from

*V*

_{t}to

*V*

_{c}which, with or without assuming that

*Cl*=

*Cl*

_{r}, yields robust estimates of

*V*

_{c}and

*Cl*

_{d}even during shorter surgery.

^{16,19–21}

^{22–24}or by pooling the data from all subjects into a single analysis

^{25–27}(fig. 3).

##### Extensions of the Kinetic Models

^{9}and volume turnover kinetics,

^{28}several modifications of the two basic kinetic models have been developed to focus on issues of special interest.

^{20,29–31}

^{11,32}

^{33}Kinetic analyses of glucose and the fluid volume are then combined so that the uptake of glucose to the cells attracts water in proportion to the osmotic strength of the glucose molecule.

^{34–36}The volume change of the body cells can then be modeled.

^{37}As their hydration is derived from

*V*

_{t}, it is difficult to find any expansion of

*V*

_{t}as long as the

*Cl*for glucose and fluid as well as perspiration are normal.

^{38,39}irreversible loss of fluid from the two functional fluid spaces to a third but “nonfunctional” space can be quantified by letting the computer estimate the zero-order constant

*Cl*

_{o}.

^{17}The loss might possibly represent accumulation in injured tissue and in the peritoneal and gastrointestinal cavities, as well as perspiration. These analyses require high-quality data on hemoglobin and urinary excretion.

*Cl*

_{o}is also a way to account for a drift in the hemoglobin baseline, which occurs during fluid therapy performed in the presence of catecholamine treatment.

^{40}

^{41}

*V*so obtained is an approximation of the size of the extracellular fluid space.

^{25,42}

^{17}The leakage is then calculated as a weight (or weight per unit of time) by multiplying the difference in fractional volume expansion by the plasma protein concentration. Mass balance calculations may be used for the same purpose, but they do not allow simulations to be performed.

^{15}

##### Presenting the Results

Table 2 Image Tools |
Fig. 5 Image Tools |

^{17,35,36,43,44}or a plot

^{19,23,44}of the fractional plasma volume expansion based on these parameter estimates (fig. 5).

*V*

_{t}can be plotted,

^{44}which is not possible by other methods (fig. 5A).

Fig. 6 Image Tools |
Fig. 7 Image Tools |

^{17}For this purpose, the rate of elimination is given by

*Cl*(

*v*

_{c}−

*V*

_{c})/

*V*

_{c}. The volume expansion of

*V*

_{c}and

*V*

_{t}can be generated by multiplying the fractional expansion (

*i.e.*, the dilution) of

*V*

_{c}and

*V*

_{t}by their respective baseline volumes

^{44}(fig. 4). The distribution and elimination can also be highlighted by computer-generated plots (figs. 6 and 7).

^{22}(fig. 8). This requires that parameter values derived from several infusion volumes and rates have yielded similar plasma dilution-time curves (model linearity).

^{13,14,32,34}Glucose 2.5% solution has been most carefully validated in this respect. In one study, six volunteers received 10 and 15 ml/kg glucose 2.5% solution over 30 min and also 15 and 25 ml/kg over 60 min.

^{34}The bias (median residual error) associated with simulating plasma dilution in the 24 experiments averaged −0.009 dilution units. Two thirds of this error was due to inability of the glucose kinetics to account for rebound hypoglycemia. The inaccuracy (median absolute residual error) was 0.026 dilution units.

#### Main Results of Clinical Importance

##### Distribution Phase

^{44}and this fraction amounted to 65–70% after administration of 1.1 l over 10 min

^{7}and 2 l over 20 min.

^{9}Moreover, the retention averaged 60% when acetated Ringer's solution was infused continuously throughout transurethral resection of the prostate performed under general anesthesia.

^{45}

^{14}As a rule of thumb, however, the plasma volume expansion at the end of a brisk 30-min infusion is 50–75% larger than would expected if distribution of fluid between

*V*

_{c}and

*V*

_{t}had been immediate.

##### Low Elimination Clearance during Surgery

*Cl*) for isotonic crystalloid fluid varies greatly depending on whether a patient is conscious or anesthetized. Other factors such as hydration, stress, and trauma also seem to play a role.

*Cl*of 60–110 ml/min, and the elimination may even be so rapid that the one-volume kinetic model is appropriate.

^{3,8,10,11}The varying figures for

*Cl*in conscious healthy subjects can probably be explained by differences in body hydration before the fluid challenge. Repetitive infusions are normally followed by a slightly more efficacious elimination.

^{10,23}In contrast, hemorrhage reduces

*Cl*by 25–50% in a graded manner, even when hypovolemia is quickly restored by crystalloid fluid.

^{44}

^{17}laparoscopic,

^{16}and open abdominal surgery

^{21}(table 2). The renal clearance (

*Cl*

_{r}) is then only 5–20 ml/min, which means that only 5–15% of a volume load would be excreted within 2 h during surgery, whereas this fraction is 40–75% in conscious subjects. The half-life for crystalloid fluid during surgery (obtained as ln 2·

*V*

_{c}/

*Cl*) is even longer than the 2.5 h found for two colloid fluids, 6% dextran 70

^{3}and albumin 5%,

^{15}in volunteers. Hormonal changes are probably responsible for much of this reduction. A drift of the baseline for hemoglobin due to vasodilatation might also contribute.

*Cl*augments the plasma volume expansion and creates a risk for interstitial edema formation from infusion volumes that otherwise would be no problem for conscious healthy volunteers to excrete. This finding also implies that monitoring of urine flow is ineffective at indicating fluid overload—the urinary excretion simply increases little, despite the presence of a marked surplus of intravascular fluid.

^{16}

*Cl*had already assumed the same value as on the day before the surgery.

^{23}However, patients who had undergone surgery that was preceded by a trauma (hip fracture) had only half as high

*Cl*on the first postoperative day as compared with an age- and sex-matched control group.

^{22}

##### Role of Stress and Anesthesia in Fluid Retention

*Cl*averaged 40–60 ml/min,

^{30}and even lower values have been reported.

^{29,31}However, a reduced

*Cl*before anesthesia might also be due to dehydration caused by preoperative fasting.

^{23,46}

*Cl*.

^{29–31}When isoflurane anesthesia was continued for 3 h in volunteers, there was an overall decrease in

*Cl*for 0.9% saline by 50%, although no surgery was conducted.

^{9}The drop was coupled with a marked increase in the serum renin and aldosterone levels. Hence, anesthesia can explain some but not all the low

*Cl*for crystalloid fluid during surgery.

*Cl*, whereas α-adrenergic stimulation by phenylephrine exerts the opposite effects.

^{40}

##### Delayed Distribution during Anesthesia

*Cl*

_{d}) drops by approximately 50% during the onset of spinal, epidural, and general anesthesia,

^{29–31}which quickly increases the plasma volume expansion resulting from an ongoing infusion.

*Cl*

_{d}correlates with the associated reduction in the arterial pressure.

^{30,31}The amount of infused fluid also seems to be of importance. Hence,

*Cl*

_{d}became slightly negative already in the average patient receiving spinal anesthesia preceded by 5 ml/kg as a bolus infusion

^{20}; this means that flow occurred against the dilution gradient between

*V*

_{c}and

*V*

_{t}. With a volume load of 20 ml/kg given slowly, distribution would be arrested (

*Cl*

_{d}= 0) if the mean arterial pressure drops by 60%,

^{30}whereas only 20% would be required when approximately 15 ml/kg is infused

^{31}(fig. 9).

*Cl*

_{d}is only slightly reduced during prolonged surgery,

^{17}which is probably due to the fact that interstitial oncotic forces eventually counteract further retention of infused fluid in the plasma.

^{47}Hence, volunteers receiving 0.9% saline had only a 25% lower

*Cl*

_{d}during experimental isoflurane anesthesia lasting for 3 h as compared with the

*Cl*

_{d}measured when they were given the same fluid in the conscious state.

^{9}

##### Small Size of *V*_{c} during Induction of Anesthesia

*Cl*and

*Cl*

_{d}vary much more than

*V*

_{c}and

*V*

_{t}depending on the physiologic situation. However, a confusing finding is that the calculated

*V*

_{c}becomes 50% smaller if volume kinetics is determined during the onset of spinal,

^{29,30}epidural,

^{31}and general anesthesia.

^{30,31}No satisfactory explanation exists at present, but the small

*V*

_{c}is mathematically due to a marked increase in plasma dilution at that time. If this dilution would be the same throughout the cardiovascular system, the calculated plasma volume expansion would exceed the infused fluid volume. Therefore, a speculation is that, with arterial hypotension, the infused fluids primarily distribute into a smaller volume, such as well-perfused vascular beds with short transit times and the central blood volume; we know that hypotension develops first and the excessive plasma dilution a few minutes later.

^{48}

##### Glucose Solutions

^{33}However, the expansion after infusion of glucose 5% does not last long because the fluid volume is cleared from the

*V*

_{c}and

*V*

_{t}by both urinary excretion and uptake to the intracellular fluid space along with the administered glucose.

^{37}

*Cl*of both glucose and the fluid load was decreased by approximately ⅔ when glucose 2.5% was infused during laparoscopic cholecystectomy.

^{35}On the first day after hysterectomy, the fluid

*Cl*was normal or high (

*Cl*= 130 ml/min), whereas the

*Cl*for glucose was still on the low side.

^{36}

*Cl*for glucose 2.5% was normal (average 99 ml/min) but patients with known impairment of renal function were not studied.

^{49}

##### Hypertonic Fluids

^{11}Hypertonic (7.5%) saline is four times more potent, and hypertonic saline in 6% dextran (HSD) is seven times more potent than 0.9% saline.

^{11}The potency of each fluid was assessed as the volume required to expand the plasma volume by 20% in 30 min.

^{50}Thereafter, 15 min is required for the infused and recruited volume to distribute throughout the extracellular fluid space.

^{11,41}

*Cl*correlates strongly with the natriuresis.

^{41}

^{32}but not in humans.

^{11}Figure 8 depicts that the difference in potency between HSD and 0.9% saline is strongly dependent on the infusion time.

^{29}Besides explaining why the potency of HSD is reported differently in various studies, such computer simulations indicate that HSD is not a bad choice for longer infusions, although it is recommended to be administered as a bolus. The increasing difference in potency with time can be understood from the fact that the body does not very easily excrete dextran and a surplus of sodium.

##### Colloid Fluids

^{3,15}

*V*

_{c}to

*V*

_{t}, probably because of the presence of dextran, but

*Cl*

_{r}was similarly small for 3% dextran and acetated Ringer's solution (8–16 ml/min).

^{29}

^{27}

*Cl*

_{r}also increased, but not as much. This study shows that, when preceded by the colloid fluid, the postoperative infusion of acetated Ringer's solution was of little value for plasma volume expansion as it merely promoted tissue edema and urinary excretion.

##### Isoflurane and “Nonfunctional” Fluid Spaces

^{51}The aberrant handling of fluid is not caused by mechanical ventilation but by the use of isoflurane

*per se*.

^{52}

^{17}Approximately 25% of this rate can be accounted for by insensible water loss.

^{53}

##### Alternative Kinetic Models

*et al.*

^{54}predicted that 88% of infused 0.9% saline is retained in the plasma at the end of a 6-min bolus,

^{55}which is consistent with volume kinetic calculations (fig. 6A). Their model can also estimate certain microvascular parameters and the urinary excretion during volume loading

^{55}and hemorrhage.

^{56}The urinary excretion is indeed governed by the fractional plasma volume expansion, although the reported

*Cl*is higher than that in most volume kinetic studies.

^{57}

^{50}based on data from dogs predicted well the relatively slow distribution of fluid between

*V*

_{c}and

*V*

_{t}during infusion of 0.9% saline.

^{11}As in volume kinetics, distribution occurs relatively faster after infusion of hypertonic saline,

^{41}which is due to an vasodilatation-associated increase in capillary filtration capacity.

^{58}

*et al.*

^{59}predicted that edema would develop in injured tissues if the operating time is more than 3 h and that there is a risk of interstitial edema if the operating time is more than 6 h.

#### Conclusions and Future Views

*per se*has been known for a long time.

*V*

_{c}to

*V*

_{t}. This delayed distribution effect is slightly more pronounced during general anesthesia than in the conscious state. However, it is most apparent during the onset of spinal, epidural, and general anesthesia. Then, the distribution of fluid from the plasma to the interstitium might even be arrested. The effect is dependent on the decrease of the arterial pressure and boosts the plasma volume expansion in response to infused fluid.

*Cl*

_{r}, which then becomes similar to the

*Cl*of a colloid fluid. The urinary excretion then increases little even in the presence of marked plasma volume expansion. This remarkable lowering of

*Cl*

_{r}makes it inappropriate to extrapolate findings made with crystalloid fluids in volunteers to the operating room.

*Cl*

_{r}, not to an increased plasma volume expansion. The second modification results in a fraction of the infused crystalloid fluid becoming unavailable for excretion, perhaps by accumulating outside the two functional spaces

*V*

_{c}and

*V*

_{t}rather than allowing for the free exchange of fluid between them. Such allocation to a third but nonfunctional space might give rise to longstanding edema. Attempting to normalize the situation by drugs acting on adrenergic receptors is a current line of research. In such work, quantification of the allocation of fluid to nonfunctional fluid spaces by volume kinetic analysis is an essential tool.

##### Appendix 1

##### Hemoglobin-derived Plasma Dilution

*v*

_{c}(

*t*) −

*V*

_{c})/

*V*

_{c}. The reference equation for this relationship is

*v*

_{c}is the size of the expanded central body plasma fluid space,

*V*

_{c}is the same body fluid space at baseline, Hct is the hematocrit, and Hgb is the hemoglobin concentration in whole blood at baseline or at time (

*t*). Symbols without an index denote baseline values and (

*t*) those obtained at a later point in time.

##### Appendix 2

##### The Two-volume Model

*v*

_{c}is given by the rate of infusion (

*R*

_{o}) minus the baseline fluid losses (

*Cl*

_{o}), the elimination (

*Cl*· plasma dilution) and the distribution of fluid to

*v*

_{t}in which the rate is governed by a clearance,

*Cl*

_{d}(fig. 2). The differential equation is

*v*

_{t}are determined only by the balance in dilution between

*V*

_{c}and

*V*

_{t}and the rate of equilibration is governed by

*Cl*

_{d}. The differential equation is

*v*

_{t}increases faster if

*Cl*

_{d}is high and also decreases more promptly when fluid is eliminated from

*v*

_{c}by the mechanisms

*Cl*

_{o}and

*Cl*. As fluid does not bind to tissue,

*Cl*

_{d}is given the same value for translocation of fluid in both directions. There is evidence that the interstitial fluid compliance cannot be greatly modified by a modest volume load,

^{10}but the finding that a computer estimate of

*Cl*

_{o}is often higher than the known insensible water loss increases the suspicion that

*Cl*

_{d}is lower when fluid is returned from

*v*

_{t}to

*v*

_{c}as compared with when fluid is translocated from

*v*

_{c}to

*v*

_{t}.

^{17}Alternatively, fluid accumulates in a third “nonfunctional” space.

*V*

_{t}and

*Cl*may become apparent if the experiment is short or the elimination is slow. If we assume that nearly all elimination occurs by virtue of renal excretion and no accumulation of fluid in nonfunctional spaces (“third-spacing”) occurs,

*Cl*may be set equal to the renal clearance (

*Cl*

_{r}) of the infused fluid

^{8}:

##### The One-volume Model

*V*is governed by the rate of infusion (

*R*

_{o}) minus the baseline fluid losses (

*Cl*

_{o}) and the elimination (

*Cl*· plasma dilution). The differential equation is

*Cl*than the value of

*Cl*

_{r}determined by the urinary excretion strongly suggests the existence of a peripheral fluid compartment (

*V*

_{t}). Discrimination between the two models can also be made by statistics, based on the squared differences between best model-predicted and measured data points.

^{2–4,7–10}

##### Least-squares Regression

^{2}and matrix

^{3,8,11}solutions have been published. Some mathematical software is able to estimate the model parameters using only the crude differential equations.

##### The *F* Test

*F*test might be applied to help decide whether the one- or two-volume model should be applied.

^{2–4}This test holds that the use of a more complex model must markedly reduce the squared deviations between computer-generated and measured data points or else the simpler model should be preferred. An

*F*value is obtained as follows:

*i.e.*, the number of data points used in fitting the function minus the number of parameters fitted. The calculated value for

*F*is compared with the critical value for significance in a standard statistical

*F*table. Cited Here...

##### Appendix 3

##### Correction for Blood Loss and Sampled Volume

^{2–4,10,11}The total hemoglobin mass (MHgb) is first obtained and the expanded blood volume at a later time is then obtained (BV(

*t*))

^{60}:

Equation (Uncited) Image Tools |
Equation (Uncited) Image Tools |
Equation (Uncited) Image Tools |

*t*]):

Equation (Uncited) Image Tools |
Equation (Uncited) Image Tools |
Equation (Uncited) Image Tools |

*t*) in the reference equation, whereas the inverse relationship is used when the hematocrit is corrected for dilution.

^{17}

^{44}The error associated with applying an erroneous blood sampling volume is larger because blood sampling is done frequently in volume kinetic studies. Cited Here...

##### Appendix 4

##### Osmotic Fluid Shift

^{61}Using the baseline serum osmolality, which is approximately 295 mosmol/kg, the translocated volume

*f*

_{t}can be obtained from

^{11,32}:

*f*

_{t}be entered as a linear function in the analysis process in which

*f*

_{t}at each point in time is governed by the total amount of infused fluid. Cited Here...