Distribution of Crystalloid Fluid Infused During Onset of Anesthesia-Induced Hypotension: A Retrospective Population Kinetic Analysis

Background. Loading with crystalloid uid before induction of anesthesia is widely practiced but cannot reduce the accompanying drop in arterial pressure. The aim of this report is to provide a mathematical view of how uid distributes in this setting. Methods. Data were retrieved from three studies where 76 patients (mean age 63 years, mean body weight 66 kg) had received approximately 1.1 L of Ringer’s solution over 60 min by intravenous infusion before and during induction of spinal, epidural, or general anesthesia. A population kinetic model was used to analyze the uid distribution and its relationship to individual-specic factors. Frequent measurements of blood hemoglobin and the urinary excretion served as dependent variables. Results. Before anesthesia induction, distribution to the extravascular space was 3-fold faster than elimination by urinary excretion. Both distribution and elimination of infused uid were retarded in an exponential fashion due to the anesthesia-induced decrease in the mean arterial pressure (MAP). A decrease in MAP from 110 to 60 mmHg reduced the rate of distribution by 75% and the rate of elimination by 90%. These adaptations cause most of the infused uid to remain in the bloodstream. Age, gender, type of anesthesia, and the use of ephedrine had no statistically signicant effect on plasma volume expansion, apart from their possible inuence on MAP. Conclusion. The decrease in MAP that accompanies anesthesia induction depresses the blood hemoglobin concentration by inhibiting both the distribution and elimination of infused crystalloid uid. The report provides mathematical information about the degree of these changes.


Introduction
Crystalloid uid is usually administered intravenously (i.v.) during induction of anesthesia (McCrae and Wildsmith, 1993), although this uid is considered to have a poor plasma volume-expanding effect (Jacob et al., 2012). Pre-loading with crystalloid cannot prevent a drop in arterial pressure (Coe and Revenäs, 1993;; however, providing uid during the onset of anesthesia (coloading) can reduce the magnitude of this drop (James and Dyer, 2016). One reason for the difference might be that more volume remains in the plasma in close connection with the infusion because some time is required for crystalloid uid to distribute to the interstitial uid space (Hahn, 2010). A larger proportion of infused uid is also known to remain in the plasma when hypotension has just been induced (Hahn, 1992). A recent uid kinetic study in which the uid was given only after the induction showed an arrested distribution of the excessive uid-induced hemodilution that resolved only 20 min later (Hahn and Nemme, 2020).
The aim of the present report was to obtain a mathematical analysis of what governs the distribution of crystalloid uid when given during the onset of anesthesia. Besides being of physiological interest, the uid distribution is mirrored in the blood hemoglobin (Hb) concentration that affects oxygen delivery, the planning of deliberate hemodilution, and the blood loss allowed before initiating erythrocyte transfusion.

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The report is based on a population kinetic analysis based on plasma dilution and urinary excretion data derived from studies in which crystalloid uid was given to clinical patients as continuous infusion before, during, and after the onset of spinal, epidural, or general anesthesia. Special consideration was given to the role of the mean arterial pressure (MAP), but other factors, such as gender and the choice of uid, were also evaluated as having potential importance regarding this aspect of the adaptation to the anesthetized state.

Methods
This study is a retrospective population kinetic analysis that uses data from three previously published studies of uid distribution during induction of anesthesia. Li et al., 2007). The analysis comprised pooled data from 76 infusions in which lactated or acetated Ringer's solution had been administered by i.v. infusion in a similar way to patients during induction of spinal, epidural, or general anesthesia. All three studies had been approved by the appropriate Ethics Committee before the rst patient was enrolled. Written informed consent was obtained from all patients before starting the study.

Anesthesia procedures
The infusions were initiated between 7 and 9 AM. The patients had fasted overnight and were placed on a bed to rest for 30 min to reach a hemodynamic steady state. In the rst two studies, a cannula was placed in the cubital vein of each arm, one for blood sampling and the other for infusion of uid. The arm used for blood sampling was placed on a body-warm heating pad. In the third study, arterial blood was sampled.
Anesthesia was usually induced when 1/3 of the infusions had been administered. The types of anesthesia used in the three studies are shown in Table 1. Spinal anesthesia was induced in the left lateral decubitus position using a 25G Whitacre needle. The subarachnoid space was punctured in the L 3 and L 4 interspace or, if that was unsuccessful, in the L 2 -L 3 interspace. Plain bupivacaine (2.4-3.4 mL; Marcain-Spinal, 5 mg mL -1 ; AstraZeneca, Södertälje, Sweden) was injected as required to achieve surgical anesthesia.
Epidural anesthesia was applied through a 17-gauge Tuohy needle with the patient in the same body position. Increments of 3-5 mL 0.5% ropivacaine (Pharmacia, AstraZeneca, Germany) were injected every 5 min until a surgical block was achieved.

Measurements
The plasma volume was expanded by infusing the Ringer solution i.v. using infusion pumps. During and after these infusions, venous blood (3-4 mL) was withdrawn to measure the Hb concentration and the hematocrit (Hct) on the apparatus used for routine measurements in the hospital's Clinical Chemistry Laboratory. The samples were withdrawn every 3-5 min in a standardized manner to ensure a coe cient of variation (CV) of about 1%. The baseline sample was drawn in duplicate, and the mean of the two concentrations was used in subsequent calculations.
The excreted urine was collected via an indwelling catheter, which had been inserted into the bladder under topical anesthesia before the studies started.
MAP was measured non-invasively with an automatic device (Datex AS3, Datex, Helsinki, Finland) , while invasive measurements were performed and displayed on a multifunction Datex-Ohmeda instrument (Hoevelaken, The Netherlands) when an arterial line had been established ).
An intravenous bolus dose of 5 mg of ephedrine, which could be repeated if necessary, was given if the systolic arterial pressure fell to 60% of baseline or if the patient experienced near-fainting symptoms (nausea, sweating, and bradycardia). No other vasopressor was used.

Kinetic analysis
Population (mixed effects) kinetics is an industry-standard tool for evaluating and recommending dosing regimens for drugs with regard to individual-speci c factors, such as age, gender, and body weight (Heeremans et al., 2010). The volume kinetic method is a modi cation of drug pharmacokinetics for the study of infusion uids, but it differs from conventional pharmacokinetics in that the uid compartments have expandable walls. A bene t of this approach is that it allows analysis of dynamic events, as this is di cult to achieve with radioactive tracer methods.
Volume kinetics is based on repeated measurement of the blood Hb concentration, which is the inverse of the blood water concentration (Hahn, 2020). Infusion uids contain almost exclusively water; therefore, Hb changes are an index of the water volume that rapidly equilibrates with the circulating blood.
A two-volume kinetic model with micro-constants was simultaneously tted to all data for the dependent variables, which were the frequently measured plasma dilution and the urinary excretion. The appropriateness of all xed parameters was challenged one by one to arrive at an optimal base model. Thereafter, the in uence of covariates on the xed parameters was tested sequentially, as guided by a reduction in the residual error for the model (Owen and Fiedler-Kelly, 2014).

Base model.
In the nally used base model, uid is infused at rate R o to expand the volume of the central body uid space V c to v c . The uid distribution to a peripheral body uid space is governed by k 12 . The elimination is given as the product of the volume expansion of V c at any time and an elimination rate constant, k 10 ( Fig. 1A). The differential equation is: To stabilize the model, k 10 is obtained as the urinary excretion divided by (v c -V c ). This rate constant is still estimated for each iteration, as (v c -V c ) changes constantly during the curve-tting process.
The plasma dilution is used to indicate the volume expansion of V c resulting from the infusion. Hence: Symbols in capital letters denote baseline values. A minor correction is made for the effects of blood sampling on the plasma dilution (Hahn, 2010).

Covariates.
A search for trends in plots of random effects (eta:s) was used to identify covariate candidates. A trend in an eta plot could suggest the existence of a signi cant covariate. The most promising candidate parameters were then tested, one by one, by adding them to the base model in a diagonal design, which assumes no correlation between random effects. The threshold for accepting one curve-t as signi cantly better than another was guided by a reduction of > 3.8 points in the goodness-of-t (-2 LL = log likelihood) for the model; where > 6.6 points represents P < 0.01. The base model with the signi cant covariates added to it constituted the nal model and had the lowest residual error of all runs.
Gender, age, body weight, general or regional anesthesia (spinal/epidural), type of Ringer's (lactated or acetated), the use of ephedrine, and the rate of infusion per kilo body weight, as well as various time factors, were sequentially tested as covariates to the three xed parameters in the model. The mean arterial pressure (MAP) and the heart rate were evaluated as time-varying covariates, which means that a new value was considered for each point of measurement. Both the crude value of MAP and the change in MAP from baseline were tested.
The xed parameters in the model and the statistically signi cant covariates, if any, were estimated simultaneously using the Phoenix software for nonlinear mixed effects (NLME), version 1.3 (Pharsight, St. Louis, MO) with the First-Order Conditional Estimation Extended Least Squares as a search routine and an additive model for the random-error variability (Owen and Fiedler-Kelly, 2014). While the covariates were added with a diagonal design, the nally reported parameter values represent a "Full Block Model" that considers correlations between random effects and is considered more accurate for simulation purposes.
The goodness-of-t of the model was studied by residual plotting, where the dependent variables were recreated from the xed parameters with and without consideration of the covariates.
The performance of the model was illustrated by predictive checks (1,000 iterations) and bootstrap analysis (500 runs, with random sampling and replacement), using the built-in features of the Phoenix software.
Demographic data were reported as the mean (standard deviation), and the kinetic data were reported as the mean (95% con dence interval). Table 1 shows the characteristics of the 76 infusion experiments. The patient mean age was 63 (standard deviation, 15) years, the mean body weight was 66 (17)  Base model.

Results
The search strategy used to nd the nal population kinetic model is shown in Table 2. First, a number of modi cations of the conventional two-volume model were applied before arriving at the nally used base model (Fig. 1A). The search resulted in removal of the rate constant k 21 for the return of distributed uid to V c (which was not statistically signi cant), the dismissal of a second elimination function unrelated to the peripheral uid space, and the dismissal of an absorption function where uid from a peripheral uid space could be absorbed to V c . These parameters did not improve the model. Hence, the presented variant with one expandable uid space (V c ), one distribution (k 12 ), and one elimination (k 10 ) function was found to be optimal. Covariates.
The covariate search then showed that the body weight (BW) affected V c and that the absolute (crude) value of MAP signi cantly in uenced both k 12 and k 10 .
Several potential covariates were tested that did not reach statistical signi cance. These included age; gender; ratio and change of MAP from baseline; the use of ephedrine (n = 13); use of regional/general anesthesia; and the time period before versus after the induction. Acetated Ringer's seemed to distribute more rapidly than lactated Ringer's, but this difference lost signi cance in the nal model. In accordance with a previous study (Hahn & Nemme, 2020), the distribution (k 12 ) was hypothesized to be turned off for a period of time after the induction; therefore, time periods of 3, 5-6, 9-10, 15, 20, and 25 min were tested. The amount of uid infused up to those timepoints was also evaluated, but none of those variables was strong enough for inclusion in the model. Illustrations. Figure 1B-E show the ability of the model to recreate the dependent variables with and without consideration of the covariates. Comparison of Fig. 1B and 1E illustrates the great importance of MAP to the urinary excretion.
The distributions of the data on plasma dilution and MAP are given in Fig. 2A-B, and their relationship is shown in Fig. 2C.
The in uence of MAP on k 12 and k 10 is shown graphically in Fig. 2D.
The performance of the model in the form of a predictive check based on 1,000 simulations is illustrated in Fig. 2E.
The model parameters in Table 3 were used to perform secondary calculations. These serve to illustrate the relevance of the kinetic data. Figure 3 shows the magnitude of the uid retention occurring when MAP decreases by entering the model parameters into a simulation program.
The rate constants can be used to calculate the estimate distribution and elimination half-lives for different MAP levels. For example, the intravascular half-life T 1/2 = natural logarithm of 2 (= 0.693) / (k 10 + k 12 ) was 15 min at a MAP of 110 mmHg and 80 min at a MAP of 60 mmHg. The corresponding values for the elimination T 1/2 , obtained as ln 2 / k 10 , were 77 min and 737 min, respectively.

Main ndings
This report provides a mathematical description of what happens to the uid distribution when anesthesia is induced in the middle of a continuous crystalloid infusion. The most apparent effect consists of a marked reduction in the rate constant that governs the distribution of infused uid (k 12 ) from the central uid space (V c , the plasma) to the extravascular space.
The change is probably an illustration of the classical Starling equation, which holds that the transcapillary exchange of uid is determined by the balance between hydrostatic and oncotic forces across the capillary wall. A lowered crude MAP is likely to decrease the intravascular hydrostatic pressure.
This, in turn, would reduce the capillary ltration because the interstitial hydrostatic pressure remains unchanged.
The elimination rate constant that describes urinary excretion (k 10 ) also decreased and to an even greater extent than was observed for the distribution. Both reductions were proportional to the anesthesiainduced decrease in MAP, as shown in Fig. 2D and illustrated in Fig. 3.

Clinical implications
A dependency of the plasma volume expansion on MAP was observed previously in the papers underlying this work Li et al., 2007). However, the present population kinetic analysis of the pooled data provides a more precise understanding of this relationship. This is achieved by allowing simulations to be made that predict how variations in MAP and the amount and rate of infused uid affect changes in Hb and the plasma volume expansion.
The following example, based on mass balance calculations (Ho et al., 2017), illustrates the in uence of the reported uid retention as compared to the conventional view of how uid affects the blood Hb concentration. A widely cited relationship holds that infusion of 1 L of crystalloid increases the blood volume by 150 mL (Jacob et al., 2012), which would reduce blood Hb from 150 g/L to 144 g/L if the baseline blood volume is 5 L. Based on the relationships presented here, infusion of the same amount over 30 min during the induction of anesthesia would reduce Hb to 129 g/L if the MAP is 70 mmHg; i.e., 3.5 times more. Hence, marked Hb changes occur even without hemorrhage when MAP is modulated by regional or general anesthesia. Some of this difference is not due to MAP but to the fact that crystalloid uid shows a distribution function that requires 25-30 min for completion. In the example above, as much as 83% of the infused volume remains in the blood, and the patient will be close to being anuric if the anesthesia reduces MAP to 70 mmHg (Fig. 3). However, induction of general or regional anesthesia with unchanged MAP was still associated with a plasma volume expansion amounting to 50% of a Ringer's infusion over 30 min. This con rms previous ndings in volunteers (Hahn, 2010) and is 3 times greater than after the infusion is completed (Jacob et al., 2012).
The excessive intravascular accumulation of infused uid during induction of anesthesia reduces the blood Hb level more than is expected, and this is relevant for perioperative medicine. For example, the drop in Hb will decrease oxygen delivery if it is unmatched by an increase in the stroke volume. Furthermore, a pre-set Hb used as transfusion trigger will be reached more rapidly than is indicated by the surgical blood loss.
The increased plasma volume expansion due to the MAP-dependent decrease in k 12 is likely to remain until the intravascular hydrostatic pressure has increased su ciently to reach a new Starling equilibrium, and this increase requires a vasoconstrictor, capillary re ll, or additional infusion of uid. The k 10 value is known to remain low, despite adequate volume compensation, as long as MAP is low, but the normal value is resumed when the patient awakens from the anesthesia (Hahn, 2020).

Kinetic analysis
Several variables were evaluated that did not receive su cient strength to be included in the kinetic model. For example, ephedrine administration had only an indirect effect via MAP on the kinetic parameters. Previous work has shown that buffered Ringer's solution undergoes a more rapid turnover in young subjects than in aged subjects, but the current age span was probably too narrow to distinguish that relationship. No differences in uid kinetics were found between patients who received spinal, epidural, or general anesthesia. The crude MAP, and not the change from baseline, governed the uid kinetics, just as occurs during ongoing surgery (Hahn, 2017).
A thorough evaluation was made to determine whether a normal k 12 was resumed after a certain amount of uid had been infused. However, no such "turning point" was found. The reason is probably that the infusion of 1.1 L of Ringer's did not ll up the vasodilated cardiovascular system su ciently to allow a resumption of the normal exchange of uid with the extravascular space. In a previous study, this "turning point" was reached when 16.6 mL kg − 1 of Ringer's (1.25 L) had been administered (Hahn and Nemme, 2020). This probably corresponds to the anesthesia-induced expansion of the part of the blood volume that is sometimes called "unstressed" and which denotes the amount of venous blood that does not increase the transmural pressure (Gelman, 2008).

Current versus previous models
The excessive accumulation of infused Ringer's solution during the onset of regional anesthesia was rst studied by linear regression and reported in the early 1990s (Hahn, 1992). A later study showed that the decrease in MAP appears a few minutes before the increased hemodilution, and this nding clari ed the order of events (Drobin and Hahn, 1996). Subsequent analyses of the uid kinetics during induction of anesthesia applied a clearance version with a single inter-compartmental clearance parameter , but this is problematic because the Starling forces are changed in the middle of the experiment. The present population kinetic model separates the ows in and out of the plasma volume and shows clearly that no return of uid to the plasma occurs during the onset of anesthesia as long as uid loading is ongoing.
The current model also uses micro-constants instead of clearances, which makes it independent of plasma volume and body uid volumes. The micro-constant model detects a "wall" between a central space, where uid equilibrates very rapidly with the site of infusion, and a more remote peripheral space. The space with this fast equilibration is very likely to represent the plasma volume contained in blood vessels that are allowed to expand. The exchange of infused uid between these two body uid spaces is determined by rate constants (k 12 and k 21 ). The volume of the infused uid residing in the two body uid spaces is obtained directly in the micro-constant model, while their dilution must be multiplied by the volume of distribution to obtain volume expansion in the clearance model.

Limitations
The limitations of the present study include that the data were obtained from three previously published works, although all three used a similar protocol and sampling had been performed in the same way.
The fairly low values of k 12 and k 10 before the induction may re ect preoperative anxiety, which has been observed in adults and children alike (Li et al., 2009). Higher values would be expected if the uid had been infused in volunteers not scheduled for surgery (Hahn, 2020).
The strength of this study is that a modern and robust kinetic approach was used that allows simultaneous analysis of all studied patients, as well as comparisons of several different covariances that could potentially in uence the uid kinetics.

Conclusion
The induction of regional and general anesthesia was followed by a MAP-dependent decrease in both the distribution and elimination of an infused crystalloid uid. Both decreases markedly increased the hemodilution.     Predictive check showing the original data (blue points) with their con dence limits (red). The con dence limits based on 1,000 simulations using the model parameters in the nal model are superimposed (green lines). Hatched pattern is due to different sampling times. A small difference between observed and predicted con dence limits is a sign of good model performance.¬ The difference between observed and predicted dilution averaged 0.03 (0.02) at the 5% level, -0.01 (0.05) at the 50% level, and -0.06 (0.06) dilution units at the 95% level.