Background

Kinetics of the uptake of inhaled anesthetics have been well studied, but the kinetics of elimination might be of more practical importance. The objective of the authors’ study was to assess the effect of the overall ventilation/perfusion ratio (VA/Q), for normal lungs, on elimination kinetics of desflurane and sevoflurane.

Methods

The authors developed a mathematical model of inhaled anesthetic elimination that explicitly relates the terminal washout time constant to the global lung VA/Q ratio. Assumptions and results of the model were tested with experimental data from a recent study, where desflurane and sevoflurane elimination were observed for three different VA/Q conditions: normal, low, and high.

Results

The mathematical model predicts that the global VA/Q ratio, for normal lungs, modifies the time constant for tissue anesthetic washout throughout the entire elimination. For all three VA/Q conditions, the ratio of arterial to mixed venous anesthetic partial pressure Part/Pmv reached a constant value after 5 min of elimination, as predicted by the retention equation. The time constant corrected for incomplete lung clearance was a better predictor of late-stage kinetics than the intrinsic tissue time constant.

Conclusions

In addition to the well-known role of the lungs in the early phases of inhaled anesthetic washout, the lungs play a long-overlooked role in modulating the kinetics of tissue washout during the later stages of inhaled anesthetic elimination. The VA/Q ratio influences the kinetics of desflurane and sevoflurane elimination throughout the entire elimination, with more pronounced slowing of tissue washout at lower VA/Q ratios.

Editor’s Perspective
• Understanding the elimination kinetics of inhaled anesthetics is of more practical importance than understanding their uptake kinetics

• Normal lungs are assumed to play a major role in the elimination of inhaled anesthetics in the early rapid stages and a negligible role subsequently

• The fraction of cardiac output that is completely cleared of anesthetic in one pass is the fractional clearance

• A mathematical model of inhaled anesthetic elimination was developed in a post hoc analysis of anesthetic partial pressures measured in mixed venous and arterial blood samples after simultaneous administration of desflurane and sevoflurane to seven piglets under normal, low, and high ventilation/perfusion ratio conditions

• After a brief and rapid decline in alveolar anesthetic partial pressure, the fractional clearance of anesthetic became constant, and incomplete clearance from the lungs slowed tissue washout

• Slowing of tissue elimination by incomplete lung clearance became more pronounced at low ventilation/perfusion ratios, and was predicted to become more pronounced as blood/gas solubility increases

The time course of inhaled anesthetic uptake and elimination has been the topic of many previous studies.1–36  The majority of these studies have focused on kinetics of anesthetic uptake; the kinetics of inhaled anesthetic elimination have received less attention. An understanding of anesthetic uptake is of fundamental importance for the clinical practice of anesthesia. In some ways, however, an understanding of elimination kinetics is of more practical importance. During uptake, overpressure techniques can be very effective in speeding uptake in arterial blood and in brain. No similar option exists for speeding elimination.1,2,32  Additionally, induction of anesthesia in adults is almost universally expedited by use of intravenous drugs, again with no equivalent option to speed emergence. Finally, workflow tasks during uptake of inhaled anesthetics, for example prepping and draping, can proceed in parallel, whereas delays in emergence might directly influence operating room efficiency.

Several types of abnormalities in the distribution of lung ventilation/perfusion ratios (VA/Q) cause inefficient gas exchange for oxygen and carbon dioxide.37  Previous studies have demonstrated that several types of abnormalities in the distribution of lung VA/Q ratios can also impact the kinetics of inhaled anesthetic uptake and elimination.9,25,26,35,36,38  The traditional view, however, is that normal lungs (with efficient gas exchange and a unimodal, narrow VA/Q distribution) play a major role only in the early, rapid stages of elimination. After the completion of these early stages, lungs with normal gas exchange efficiency have generally been regarded as having little influence on later, slower stages of elimination.7,15–18

In a recent experimental study, in pigs with normal lungs, we measured anesthetic partial pressures in mixed venous (Pmv) and arterial (Part) blood samples at multiple times during uptake and elimination of desflurane and sevoflurane, for three different lung VA/Q ratios.39  In the current study, we further analyze this experimental data and develop a simplified and approximate mathematical model of anesthetic elimination that demonstrates the dependence of whole-body elimination kinetics on the global VA/Q ratio for normal lungs. The study reported here is an exploratory, post hoc analysis of a subset of a larger data set. The objective was to develop a mathematical model of inhaled anesthetic elimination that explicitly states the dependence of washout kinetics on overall VA/Q ratio.

### Experimental Measurements

Details of experimental methods and experimental results are presented in the companion experimental paper.39  In brief, seven juvenile, 2.5-month-old piglets (weight, mean ± SD 25 ± 2 kg) with normal lungs were anesthetized with intravenous anesthetics. Subanesthetic levels of sevoflurane and desflurane were administered simultaneously with an open circuit technique for 45 min. Arterial and mixed venous blood samples were collected at predetermined times during the 45 min washin and during 45 min of washout. Sevoflurane and desflurane partial pressures in the blood samples were measured with a mass spectrometer–based method.40,41  Uptake and elimination measurements were carried out for three different conditions39  of alveolar minute ventilation (VA) and cardiac output value (Q): normal VA/Q (0.91); low VA/Q (0.32); and high VA/Q (1.73). Minute ventilation was varied between these three conditions by adjustment of respiratory rate. Cardiac output was increased with dobutamine infusion and decreased by inflation of a right atrial balloon.

### Approximate and Simplified Mathematical Model of Elimination

We start our simplified mathematical modeling approach by considering the kinetics of anesthetic elimination from body tissues. A single well-mixed compartment, as depicted in figure 1A, has a uniform anesthetic partial pressure and is being supplied with fresh, anesthetic-free blood flowing in at rate Q, and flowing out also at rate Q, where the outflowing blood is equilibrated to the gas partial pressure of the compartment. The differential equation describing the washout of anesthetic gas from this single well-mixed compartment is

Fig. 1.

(A) Stirred, well-mixed tank of volume Volcmpt, filled with blood at anesthetic gas partial pressure Pblood. Blood flow Q is entering the tank with no anesthetic gas and exiting equilibrated to the tank gas partial pressure Pblood. λbg is the anesthetic blood-gas partition coefficient. (B) Well-mixed tank is now filled with blood and tissue. (C) Single well-mixed compartment with blood and tissue now represents the tissue for the whole body. Q flowing into and out of the body tissue compartment is the cardiac output. Gas partial pressure in the tank and equilibrated with exiting mixed venous blood is Pmv. Mixed venous blood returns to the lung, where it is completely cleared in one pass, returning to the body compartment at a gas partial pressure in arterial blood (Part) of zero. (D) The lung now only partially clears the anesthetic gas, returning a fraction (FrClr) of cardiac output (Q) to the body compartment with zero gas (Pclrd = 0), and a fraction (1 − FrClr) of cardiac output with gas at mixed venous gas partial pressure Pmv.

Fig. 1.

(A) Stirred, well-mixed tank of volume Volcmpt, filled with blood at anesthetic gas partial pressure Pblood. Blood flow Q is entering the tank with no anesthetic gas and exiting equilibrated to the tank gas partial pressure Pblood. λbg is the anesthetic blood-gas partition coefficient. (B) Well-mixed tank is now filled with blood and tissue. (C) Single well-mixed compartment with blood and tissue now represents the tissue for the whole body. Q flowing into and out of the body tissue compartment is the cardiac output. Gas partial pressure in the tank and equilibrated with exiting mixed venous blood is Pmv. Mixed venous blood returns to the lung, where it is completely cleared in one pass, returning to the body compartment at a gas partial pressure in arterial blood (Part) of zero. (D) The lung now only partially clears the anesthetic gas, returning a fraction (FrClr) of cardiac output (Q) to the body compartment with zero gas (Pclrd = 0), and a fraction (1 − FrClr) of cardiac output with gas at mixed venous gas partial pressure Pmv.

Close modal
$d(λbg · Volcmpt · P(t))dt=−λbg · Q˙ · P(t)$
(1)

where P(t) is the gas partial pressure in the compartment and in exiting blood at time t; Volcmpt is the volume of the compartment (up to this point considered to be occupied only by the flowing fluid, i.e., blood), Q is the liquid fluid flow (units of volume/time), and λbg is the Ostwald solubility of the gas in blood (units of ml gas · ml blood-1 · atm-1) that links gas partial pressure (units of pressure) to gas content of the fluid (units of volume of gas/volume of liquid). When gas partial pressure is expressed in atmospheres, the Ostwald blood gas solubility is numerically equal to the blood/gas partition coefficient (dimensionless). For constant blood flow Q, the kinetics of gas washout, starting from an initial gas partial pressure of Pi at time zero, are well known4 :

$Pt=Pi · e−(tτcmpt)$
(2)

The decay in gas partial pressure P(t) from any starting pressure Pi is a monoexponential function of time t, with time constant τcmpt given by

$τcmpt=λbg · Volcmpt · P(t)λbg · Q˙ · P(t)=VolcmptQ˙$
(2A: blood-filled tank)

The numerator of this time constant represents the total gas content in the compartment at time t, and the denominator represents how quickly this compartment is being flushed out by fresh blood flow.

The well-mixed compartment of interest here is filled not only with blood, but rather a small volume of blood (Vbld) supplying a much larger volume of tissue (Vtiss), as depicted in figure 1B. The gas content of the compartment is then (total content) = P(t) · (Volbld · λbg + Voltiss · λtg), where λtg is the Ostwald solubility coefficient for gas dissolved in tissue.5,23,42  Equation 2 still applies, but the monoexponential time constant becomes

$τcmpt=λbg · Volbld+λtg · Voltissλbg · Q˙$
(2B: tank with blood and tissue)

Blood volume in many body tissues is a small fraction of tissue volume, and most anesthetics, including desflurane and sevoflurane, partition preferentially from blood into tissue (i.e., λtg > λbg). Therefore, λtg · Voltiss >> λbg · Volbld, and as an approximation, the first term in the numerator can be neglected. Also, for a well-mixed tissue compartment, compartment gas partial pressure P(t) equals exiting venous blood partial pressure Pv(t). The mass balance for the tissue compartment then becomes a slight modification of equation 1:

$d(λtg · Volcmpt · Pv)dt=−λbg · Q˙ · Pv$
(1A)

The solution for washout from starting gas partial pressure Pi is a monoexponential decay (i.e., equation 2 still applies) with time constant

$τcmpt=λtg · Voltissλbg · Q˙$
(2C: tissue, approximate)

This approximation emphasizes that the important parameter for body tissue elimination kinetics is not the gas solubility in blood that describes how the gas partitions between a blood phase and gas phase, but rather the tissue/blood partition coefficient λtgbg that describes how gas partitions between tissue and blood.1,4,7,8,15,17,18,21,28

In the interest of arriving at simplified approximate equations that directly show the importance of parameter groups, we start by treating the entire body as one well-mixed compartment, a composite of the traditional vessel-rich, or visceral, group; the muscle group; and the fat group. In figure 1C, the blood flowing out of our compartment exits at mixed venous gas partial pressure, and the blood flow to the whole-body compartment is the entire cardiac output. Mixed venous blood is recycled back to the compartment through a gas exchanger (the lung) capable of clearing all the anesthetic gas in one pass. It is obvious that equation 2 still applies, and the time constant compared to figure 1B has not changed.

In figure 1D, we now consider a lung that does not clear all of the anesthetic in one pass. Partial clearance in the lung for the entire cardiac output can be divided conceptually into a partial lung blood flow that is not cleared at all and is recycled into the body compartment, and a partial lung blood flow that is cleared completely and is returned to the body tissue compartment as gas-free blood. It is obvious in figure 1D that the part of blood flow that is not cleared at all is simply recycled into the well-mixed compartment and has no role in elimination kinetics. It is also obvious that the effective blood flow washing out the well-mixed compartment is the fraction of cardiac output that is cleared in one pass. Consistent with traditional pharmacokinetic terminology, we denote the amount of blood flow that is completely cleared of anesthetic as “clearance,” with units of milliliters blood/min. The fraction of cardiac output that is cleared in one pass will then be called “fractional clearance” and denoted as FrClr (dimensionless). The whole-body elimination time constant becomes

$τbody=(λtgλbg) · VoltissFrClr · Q˙$
(3)

If, for example, the lung fractional clearance (FrClr) was 20% or one fifth, the overall elimination time constant would be increased, by the incomplete lung clearance, fivefold compared to the intrinsic tissue time constant of equation 2C.

An illustration of the potential effect of incomplete lung clearance on the tissue washout time constant is shown in figure 2. Tissue parameter data for the plots in figure 2 were chosen to mimic the muscle tissue compartment for the population averages for the pigs of our experiments.39  A muscle compartment volume of 13,960 ml was estimated based the “standard 30-kg dog” presented by Cowles et al.,23  scaled to our average pig weight of 25 kg. Average cardiac output for our piglets, for the “normal” VA/Q condition, was measured at 3,300 ml/min. The fraction of cardiac output to muscle in the Cowles 30-kg dog of 0.31 was applied to estimate muscle blood flow at 1,023 ml/min. The tissue/gas partition coefficient λtg for desflurane in pig muscle of 0.56, and the blood/gas partition coefficient for desflurane in pig blood of 0.40, were taken from Zhou and Liu.43  For purposes of illustration, the single muscle compartment was connected in figure 2 to a lung with fractional clearance assumed to be constant throughout the elimination. Under these assumptions, the intrinsic muscle tissue time constant (equation 2c or equation 3 with FrClr = 1.0) is estimated as 19.1 min. The marked effect on the muscle washout kinetics of progressively decreasing lung clearance from 1.0 to 0.2 is readily apparent in figure 2.

Fig. 2.

Hypothetical plots of desflurane washout from tissue to illustrate the potential magnitude of the effect of incomplete lung clearance on tissue washout kinetics. Tissue desflurane partial pressure (scaled to the initial pressure at the beginning of washout) is plotted versus time for a single muscle tissue compartment connected to a normal lung as in figure 1D, and the lung maintains constant fractional clearance (1, solid line; 0.7, dashed line; 0.4, dotted line; 0.2, dash-dot line) throughout elimination. The washout is monoexponential in all cases, with an intrinsic muscle time constant (equation 2C) of 19.1 min.

Fig. 2.

Hypothetical plots of desflurane washout from tissue to illustrate the potential magnitude of the effect of incomplete lung clearance on tissue washout kinetics. Tissue desflurane partial pressure (scaled to the initial pressure at the beginning of washout) is plotted versus time for a single muscle tissue compartment connected to a normal lung as in figure 1D, and the lung maintains constant fractional clearance (1, solid line; 0.7, dashed line; 0.4, dotted line; 0.2, dash-dot line) throughout elimination. The washout is monoexponential in all cases, with an intrinsic muscle time constant (equation 2C) of 19.1 min.

Close modal

We next consider behavior of the lung during anesthetic elimination, and we divide this consideration into early stages of elimination, and later stages. We consider a normal, homogeneous lung with a single, narrow mode in the VA/Q distribution. A mass balance on the lung provides the key differential equation4 :

$d(PalvPb · Vlungeff)dt=[λbg · Q˙ · Pmv−Part · P0Pb]−(PalvPb · V˙A)$
(4)

Palv is alveolar anesthetic gas partial pressure (atm), Pb is barometric pressure, Vlungeff is effective lung volume, and P0 is standard pressure (1 atm). Effective lung volume Vlungeff is the total gas capacity of the lung, given by Vlungeff = Vfrc + Vtave + λbg · Vlungbld · P0 + λlungtiss · Vlungtiss · P0,5,23,29  where Vfrc is functional residual capacity (milliliters gas); Vtave is time averaged tidal volume in the alveolus (for example, one half tidal volume for a sinusoidal breathing pattern); Vlungbld is the lung blood volume; λlungtiss is the gas solubility in lung tissue; and Vlungtiss is the tissue volume of the lung. Equation 4 states that the time rate of change of the amount of gas in the lung is equal to the net delivery by blood into the alveolus (positive for elimination since Pmv > Part) minus removal in the gas phase by tidal ventilation.

Two assumptions are commonly made to simplify equation 4. The first is that alveolar gas and arterial blood, for any homogenous lung unit, are equilibrated to the same gas partial pressure.4,5,10,11,23,29,42,44,45  The second, applicable to later stages of elimination, is the pseudo steady-state assumption that the term in brackets (net delivery to the alveolus by blood) is approximately equal to the term in parentheses (net loss in expired gas).44,45  Therefore, as Pmv, and Part (= Palv), are all changing slowly together, the net result is that dPalv/dt becomes small enough to be neglected, and therefore the left term in brackets is approximately equal to the right term in parentheses. Equation 4 under these assumptions simplifies to the retention equation44,45 :

$PartPmv=11+V˙Aλbg · Q˙ · P0$
(5)

The retention equation is the underlying basis for the multiple inert gas elimination technique that has been used successfully hundreds of times in describing lung gas exchange.37  During these later stages of elimination, the retention (defined as Part/Pmv, the ratio of arterial to mixed venous anesthetic partial pressure) is predicted to become constant as both Part and Pmv continue to change together. We refer in this manuscript to these later stages of elimination as the “retention equation plateau.”

In equations 4 and 5, λbg is the Ostwald solubility coefficient, and the standard barometric pressure P0 of 1.0 atm appears in these equations to maintain dimensional consistency. If instead we use the numerically equal value of the blood/gas partition coefficient (dimensionless) for λbg, equation 5 takes the slightly simpler form:

$PartPmv=11+V˙Aλbg · Q˙$
(6: retention equation)

Fractional clearance of anesthetic gas from pulmonary blood is defined as the fraction of gas removed from mixed venous blood and can be expressed as

$FrClr=Pmv−PartPmv=1−PartPmv$
(7: definition of FrClr)

It is clear that if retention is constant in the latter stages of elimination, fractional clearance FrClr will also be constant. Fractional clearance can also be expressed by substituting the expression in the retention equation for Part/Pmv, and rearranging:

$FrClr=11+λbg · Q˙V˙A$
(8)

This equation for fractional clearance has been presented previously.1,2,34,46  Retention and fractional clearance curves for a desflurane blood/gas partition coefficient in pig blood of 0.4043  are shown in figure 3, with three points on each curve corresponding to our experimental values of VA/Q . Figure 3 also shows the retention and fractional clearance curves, and the corresponding three points, for sevoflurane with a blood/gas partition coefficient in pig blood of 0.48.43

Fig. 3.

Plots of the retention (defined as Part/Pmv, calculated from retention equation; solid lines) and the corresponding fractional clearance (1 − Part/Pmv; dashed lines) as a function of ventilation/perfusion ratio (VA/Q) for a gas with blood/gas partition coefficient of 0.40 (matching the partition coefficient for desflurane in pig blood; in blue) and for a gas with blood/gas partition coefficient of 0.48 (matching the partition coefficient for sevoflurane in pig blood; in red). The three points on each curve correspond to the mean VA/Q ratios in the three conditions of our experiments: 0.32, 0.91, and 1.73.

Fig. 3.

Plots of the retention (defined as Part/Pmv, calculated from retention equation; solid lines) and the corresponding fractional clearance (1 − Part/Pmv; dashed lines) as a function of ventilation/perfusion ratio (VA/Q) for a gas with blood/gas partition coefficient of 0.40 (matching the partition coefficient for desflurane in pig blood; in blue) and for a gas with blood/gas partition coefficient of 0.48 (matching the partition coefficient for sevoflurane in pig blood; in red). The three points on each curve correspond to the mean VA/Q ratios in the three conditions of our experiments: 0.32, 0.91, and 1.73.

Close modal

Constant fractional clearance in the later stages of elimination, where “later” is yet to be defined, will directly affect the terminal time constant (often called the “beta” constant in pharmacokinetics) for elimination for a single whole-body compartment interacting with the lungs, as discussed above. Substituting the expression for fractional clearance in equation 8 into equation 3, we obtain the late stage terminal time constant:

$τlate=λtgλbg · VoltissFrClr · Q˙=λtgλbg · Voltiss11+λbg · Q˙V˙A · Q˙ =λtgλbg · VoltissQ˙ · (1+λbg · Q˙V˙A)$
(9)

This simple connection between lung gas exchange and whole-body washout time constant shows that during later stages of elimination, lung gas exchange efficiency, even for normal lungs with a narrow unimodal VA/Q distribution, directly affects the whole-body elimination time constant, because incomplete clearance directly reduces the effective blood flow that is washing out the body compartment. In the rightmost version of equation 9, the term in left parentheses is recognized as the intrinsic tissue time constant from equation 2C. The term in right parentheses is the impact of the lung in slowing whole body washout, and this impact of the lung is determined by (1) the overall lung VA/Q ratio and (2) the blood/gas partition coefficient. Equation 9 can also be rearranged to more directly show the individual roles of cardiac output Q and alveolar minute ventilation Valv:

$τlate=λtgλbg · Voltiss · (1Q˙+λbgV˙A)$
(10)

Equation 10 tells us that the direct effect of increased cardiac output in speeding the tissue compartment washout (the Q in equation 2C) always dominates over the indirect effect of increased cardiac output in decreasing clearance and slowing the late-stage whole-body washout (the Q in equation 8), and therefore the net effect of an increase in cardiac output is to speed late-stage elimination. This result, however, is restricted to the assumption of a single body compartment where a change in cardiac output cannot be accompanied by a change in the organ level distribution of blood flow.2,11,28  Equation 10 also helps to clarify the concepts of perfusion-limited elimination and ventilation-limited elimination.4,21,44,47  For a gas that has a solubility coefficient much greater than 1, or when VA is much less than Q, the second term in parentheses will dominate, changes in cardiac output will have little effect, changes in ventilation will have a large effect, and the late-stage elimination is ventilation-limited. For a gas that has a solubility coefficient much less than 1, or when Q is much smaller than VA, the 1/ Q term in parentheses will dominate, changes in ventilation will have little effect, changes in cardiac output will have a large effect, and the late-stage elimination is perfusion-limited.

Early in elimination, in contrast to the later stages of elimination, Palv is changing rapidly, and the retention equation cannot be applied to predict Part/Pmv. Some insight into the early stages of elimination can be gained, however, by considering the limiting case of an anesthetic gas with a solubility in blood approaching zero. For sparingly soluble gases, the term describing net delivery in blood in equation 4 approaches zero, and equation 4 reduces to

$d(PalvPb · Vlungeff)dt=−(PalvPb · V˙A)$
(11: sparingly soluble gas)

For a gas that is sparingly soluble in tissue (λtg ≈ 0) as well as in blood (λbg ≈ 0), effective lung volume becomes Vfrc + Vtave, and the differential equation describing washout becomes

$d(Palv)dt=−V˙AVfrc+Vtave · Palv$
(12: sparingly soluble gas)

The solution is a monoexponential decay in alveolar partial pressure from starting pressure Pi:

$Pt=Pi · e−tτfrc τfrc=Vfrc+VtaveV˙A$
(13: sparingly soluble gas)

For our piglets, functional residual capacity is estimated as 669 ml by scaling the data of Ludwigs et al.48  to our pig weight of 25 kg and linearly interpolating to our set positive end-expiratory pressure of 5 cm H2O. An average tidal volume for our experiments, globally for all conditions, estimates tidal volume as 268 ml, giving functional residual capacity + ½ tidal volume as 803 ml. For the three different minute ventilation settings in our experiments, the three time courses predicted for alveolar washout for a hypothetical, very low solubility gas are graphed in figure 4. Of note, all of the functional residual capacity time constants, even for low minute ventilation, are small, with this part of washout completing in about 1 min or less.

Fig. 4.

Predicted washout of a very low solubility gas from the alveolar gas spaces for a normal, homogeneous lung. Alveolar gas partial pressure (scaled to the beginning partial pressure) is plotted versus time for a fixed effective alveolar volume (functional residual capacity + ½ tidal volume) of 803 ml for our 25-kg pigs at 5 cm H2O of positive end-expiratory pressure. The three alveolar minute ventilations correspond to the average VA for our experiments (high, 3.7 l/min, dotted line; normal, 2.9 l/min, solid line; low, 2.0 l/min, dashed line).

Fig. 4.

Predicted washout of a very low solubility gas from the alveolar gas spaces for a normal, homogeneous lung. Alveolar gas partial pressure (scaled to the beginning partial pressure) is plotted versus time for a fixed effective alveolar volume (functional residual capacity + ½ tidal volume) of 803 ml for our 25-kg pigs at 5 cm H2O of positive end-expiratory pressure. The three alveolar minute ventilations correspond to the average VA for our experiments (high, 3.7 l/min, dotted line; normal, 2.9 l/min, solid line; low, 2.0 l/min, dashed line).

Close modal

For more soluble gases, the approach from Part/Pmv = 1 to the retention equation plateau is not easily predicted with simplified models, because the kinetic behavior is governed by the behavior of the lung kinetics interacting with the tissue kinetics. It can be appreciated qualitatively that the approach to the retention equation plateau will be slower than predicted by τfrc because gas exiting the alveolus in arterial blood speeds the decay, but gas entering from mixed venous blood slows the decay, and Pmv during elimination will be larger than Part. More quantitative descriptions would require simultaneous solution of both differential equations for lung and body tissue (equations 1A and 4), either numerically5,10,11,13,21,23,29,42  or analytically.4,28  Alternatively, the early stages of anesthetic washout can be described with experimental data. In the current study, the experimental data for Part and Pmv from our companion experimental study39  are analyzed by calculating the retention at each time point and plotting retention versus time.

### Graphic and Statistical Analysis of Experimental Measurements

Data for Part and Pmv were taken from our companion experimental study.39  For each individual animal, and each of the three VA/Q conditions, arterial (Part) and mixed venous (Pmv) mass spectrometer signals were scaled to that individual’s arterial signal at the end of the 45-min anesthetic (sevoflurane and desflurane) administration. For each VA/Q condition and each subject, the scaled Part was divided by scaled Pmv to calculate the retention for that individual, VA/Q condition, and time. Then, for each VA/Q condition, Part/Pmv was averaged for each time over the seven animals, 99% CIs were determined using the Student’s t distribution, and values were plotted as means and CIs versus time (fig. 5 for desflurane and fig. 6 for sevoflurane). Scaled desflurane mixed venous partial pressure means and 99% CIs were plotted as spline-smoothed curves on a semi-log plot (fig. 7). Desflurane elimination kinetics in mixed venous blood from 20 to 45 min were compared to two theoretical monoexponential washout kinetics, the intrinsic time constant for the muscle compartment, and the fractional clearance–corrected time constant (fig. 7). A similar analysis for sevoflurane was performed, with a similar plot in figure 8. The rationale for focusing on the muscle compartment, when we restrict our attention to the time period from 20 to 45 min, is presented in the appendix. The intrinsic muscle compartment time constants (equation 2C, or equation 3 with FrClr = 1) were calculated from the following: measured cardiac output times the fractional flow to muscle for the Cowles “standard 30-kg dog”23 ; muscle tissue volume from the Cowles 30-kg dog, scaled to our pig weight of 25 kg; pig tissue/gas partition coefficients for desflurane (0.56) and sevoflurane (1.17) in muscle from Zhou and Liu.43 ; and pig blood/gas partition coefficients of 0.40 for desflurane and 0.48 for sevoflurane from Zhou and Liu.43  Fractional clearance for the clearance-corrected muscle time constant (equation 3) was calculated (equation 8) from measured cardiac output and alveolar minute ventilation, and λbg from Zhou and Liu.43

Fig. 5.

Experimental measurements of Part/Pmv (retention, defined as anesthetic gas partial pressure in arterial blood/anesthetic gas partial pressure in mixed venous blood) for desflurane, plotted versus time for the three ventilation/perfusion ratio (VA/Q) conditions as means and 99% CIs for the seven animals. High VA/Q, dotted line and open squares; normal VA/Q, solid line and filled circles; low VA/Q, dashed line and open circles. For comparison to theoretical calculations of retentions for each condition (the retention equation applied to mean VA/Q for that condition), the middle section of the retention curve from figure 3, for desflurane, is reproduced and aligned to the right of the experimental data plots.

Fig. 5.

Experimental measurements of Part/Pmv (retention, defined as anesthetic gas partial pressure in arterial blood/anesthetic gas partial pressure in mixed venous blood) for desflurane, plotted versus time for the three ventilation/perfusion ratio (VA/Q) conditions as means and 99% CIs for the seven animals. High VA/Q, dotted line and open squares; normal VA/Q, solid line and filled circles; low VA/Q, dashed line and open circles. For comparison to theoretical calculations of retentions for each condition (the retention equation applied to mean VA/Q for that condition), the middle section of the retention curve from figure 3, for desflurane, is reproduced and aligned to the right of the experimental data plots.

Close modal
Fig. 6.

Experimental measurements of Part/Pmv (retention, defined as anesthetic gas partial pressure in arterial blood/anesthetic gas partial pressure in mixed venous blood) for sevoflurane, plotted versus time for the three ventilation/perfusion ratio (VA/Q) conditions as means and 99% CIs for the seven animals. High VA/Q, dotted line and open squares; normal VA/Q, solid line and filled circles; low VA/Q, dashed line and open circles. To separate the overlapping CIs, at each time point the three means and CIs are plotted with a slight time offset. For comparison to theoretical calculations of retentions for each condition (the retention equation applied to mean VA/Q for that condition), the middle section of the retention curve from figure 3, for sevoflurane, is reproduced and aligned to the right of the experimental data plots.

Fig. 6.

Experimental measurements of Part/Pmv (retention, defined as anesthetic gas partial pressure in arterial blood/anesthetic gas partial pressure in mixed venous blood) for sevoflurane, plotted versus time for the three ventilation/perfusion ratio (VA/Q) conditions as means and 99% CIs for the seven animals. High VA/Q, dotted line and open squares; normal VA/Q, solid line and filled circles; low VA/Q, dashed line and open circles. To separate the overlapping CIs, at each time point the three means and CIs are plotted with a slight time offset. For comparison to theoretical calculations of retentions for each condition (the retention equation applied to mean VA/Q for that condition), the middle section of the retention curve from figure 3, for sevoflurane, is reproduced and aligned to the right of the experimental data plots.

Close modal
Fig. 7.

Scaled desflurane mixed venous partial pressure means and 99% CIs for the seven animals, plotted with a smoothed spline curve on a semi-log plot (dashed line, open circles). ln(Scaled Pmv) is the natural logarithm of the scaled mixed venous partial pressure Pmv. Also plotted are the monoexponential washouts, from the time = 20 min point, predicted by a single muscle compartment connected to the lungs, for the intrinsic muscle group time constant (dot-dash line) and for the effective, or clearance-corrected (FrClr-corrected), time constant (solid line). VA/Q, ventilation/perfusion ratio.

Fig. 7.

Scaled desflurane mixed venous partial pressure means and 99% CIs for the seven animals, plotted with a smoothed spline curve on a semi-log plot (dashed line, open circles). ln(Scaled Pmv) is the natural logarithm of the scaled mixed venous partial pressure Pmv. Also plotted are the monoexponential washouts, from the time = 20 min point, predicted by a single muscle compartment connected to the lungs, for the intrinsic muscle group time constant (dot-dash line) and for the effective, or clearance-corrected (FrClr-corrected), time constant (solid line). VA/Q, ventilation/perfusion ratio.

Close modal
Fig. 8.

Scaled sevoflurane mixed venous partial pressure means and 99% CIs for the seven animals, plotted with a smoothed spline curve on a semi-log plot (dashed line, open circles). ln(Scaled Pmv) is the natural logarithm of the scaled mixed venous partial pressure Pmv. Also plotted are the monoexponential washouts, from the time = 20 min point, predicted by a single muscle compartment connected to the lungs, for the intrinsic muscle group time constant (dot-dash line) and for the effective, or clearance-corrected (FrClr-corrected), time constant (solid line). VA/Q, ventilation/perfusion ratio.

Fig. 8.

Scaled sevoflurane mixed venous partial pressure means and 99% CIs for the seven animals, plotted with a smoothed spline curve on a semi-log plot (dashed line, open circles). ln(Scaled Pmv) is the natural logarithm of the scaled mixed venous partial pressure Pmv. Also plotted are the monoexponential washouts, from the time = 20 min point, predicted by a single muscle compartment connected to the lungs, for the intrinsic muscle group time constant (dot-dash line) and for the effective, or clearance-corrected (FrClr-corrected), time constant (solid line). VA/Q, ventilation/perfusion ratio.

Close modal

Figure 5 shows the time course of calculated Part/Pmv (retention) during desflurane elimination, as means (over the seven animals at each time point) and 99% CIs. In all three VA/Q conditions, the values reached the retention equation plateau within 5 min. The ratio of scaled Part to scaled Pmv is initially greater than 1 in all cases, reflecting the fact that not all body tissues were completely equilibrated at the end of a 45 min administration. The experimental values of retention on this retention equation plateau are compared graphically to the values calculated from the retention equation (applied to the average values of VA/Q of 0.91, 0.32, and 1.73), with the central section of the graph of figure 3 reproduced and aligned to the right of the data plot in figure 5.

Figure 6 shows the corresponding analysis for the sevoflurane data. There is a similar trend for retentions to reach a plateau after the first 2 to 5 min, especially notable in the time period from 2 to 10 min. Later time periods in figure 6 are difficult to interpret; the noise in the sevoflurane data becomes large as the signals approach zero late in washout. As a consequence of the mass spectrometer setup and the chosen inspired concentrations of sevoflurane and desflurane, the signal to noise ratio for the sevoflurane data analysis was on the order of 20 to 50 times smaller than the corresponding desflurane signal to noise ratios. The result is the large error bars, compared to desflurane, in the sevoflurane retention plots late in washout, where two very small quantities are divided as they both approach zero. The very small signal for sevoflurane in late stages of washout also makes the calculation of the ratio of two small numbers susceptible to systematic, nonrandom measurement errors—for example, any small amount of drift in the mass spectrometer baseline.

Figure 7 presents the measured Pmv values for desflurane during elimination, connected by a spline-smoothed curve on a semi-log plot (means with 99% CIs). Also plotted are the monoexponential washouts (linear on a semi-log plot) from the 20-min time point that are predicted for the muscle compartment alone. The more rapid washout (steeper slope of a straight line on the semi-log plot; dot-dash line) is predicted by the intrinsic time constant of the muscle compartment (intrinsic tissue compartment time constant, methods), i.e., the washout that is predicted for complete lung fractional clearance. The slower linear, monoexponential washout (solid line) is predicted by the intrinsic muscle time constant corrected for the fractional clearance by the lung (late-stage effective time constant; see Materials and Methods, equation 9). Figure 8 presents the corresponding analysis for sevoflurane.

Our study develops a simplified mathematical model of inhaled anesthetic elimination that explicitly shows the dependence of elimination on overall lung VA/Q, in contrast to both of the two most prominent previous approaches to mathematical modeling of elimination. One prominent approach to previous mathematical modeling of elimination has used numerical (or, in early studies, analog electrical) solutions to the system of differential equations that arise from compartmental modeling.2,5,9–13,21–24,27–29,34,42,49,50  Each compartment included in the model (for example, lungs/central, visceral, muscle, and fat compartments in a four-compartment model)2,23,42  is represented by a differential equation, and the resulting uptake and elimination curves can be compared to the experiment. Because this approach includes a differential equation for the lungs, equivalent or identical to equation 4, the effect of overall lung VA/Q is implicitly included. The numerical solutions, however, do not provide any equations that explicitly show the functional form of the dependence of kinetics on VA/Q .

Another prominent approach to mathematical modeling of washout data is the empiric fitting of kinetic data for Part (or for Pet) to sums of multiple exponential terms of the form

$Part=∑i=1n Ai · e−ki · t$
(14)

Terms are empirically added to the model according to whether or not they improve the fit of the multiexponential curve to the data points. Carpenter et al. used five terms fit to prolonged (several days) washout data for multiple anesthetics.15,16  Because the recovered rate constants ki were substantially separated in magnitude, it was assumed as an approximation that each coefficient Ai and rate constant ki corresponds to a distinct tissue group, resulting in the five-compartment model15–18  for anesthetic elimination: lungs/central compartment, vessel-rich group, muscle group, fat group, and the “fourth compartment,” attributed to intertissue diffusion.1,15–18  Earlier work interpreted uptake kinetics in a similar way for a four-compartment model.7

The lungs/central compartment has the fastest kinetics compared to all the other compartments, and it has been assumed as an approximation7,15–18  that after the early period of washin or washout, the role of the lungs in later stages of kinetics can be neglected. The expectation is then that the empirically recovered time constants for each compartment (the time constants τi = 1/ki) should reasonably match the intrinsic tissue time constants7,15–18  as calculated from the equation for “intrinsic tissue compartment time constant,” equation 2C in the Materials and Methods. Our study shows that this concept is incorrect. The lung overall VA/Q ratio, and more specifically the dimensionless parameter group VA/(λbgQ), continues to directly influence the terminal elimination constant for the entire elimination. During these later stages of elimination, it is true that the influence of Vfrc/ VA kinetics has decayed to a negligible role. The fractional clearance, however, takes on a constant value and plays a major role in modulating the intrinsic tissue time constants to determine the overall late-stage kinetics.

Figures 7 and 8 illustrate several features of late-stage desflurane and sevoflurane washout. First, although there is still some curvature in the time period from 20 to 45 min, the washout during this time is a close approximation to a monoexponential washout that would plot as a straight line on a semi-log plot. This is consistent with the anticipation that this period of washout can be approximately represented by a single muscle compartment connected to the lungs, as described in the appendix. Second, the clearance-corrected time constant for a monoexponential muscle compartment washout is generally a better predictor of the experimental data than the intrinsic time constant, supporting our point that incomplete lung clearance slows washout during the entire elimination. Third, the clearance-corrected time constant makes a very good prediction of the experimental washout kinetics. This fit of theory to experimental data is striking, considering that this match did not use any adjusted parameters for “best fit.” Finally, the correction for lung clearance makes little difference when lung fractional clearance approaches 1.0 (figs. 7C and 8C, high VA/Q), i.e., when VA/Q is high and/or solubility is small. When clearance is not close to 1.0, however, the uncorrected washout time constant substantially overestimates the speed of washout (figs. 7B and 8B, low VA/Q). Of note, desflurane and sevoflurane are two of the least soluble inhaled anesthetics. These effects will be even more pronounced for higher solubility gases.

The dimensionless group VA/(λbgQ), highlights two important factors in anesthetic elimination: the role of overall lung VA/Q, and the role of the blood/gas partition coefficient. Even for normal lungs with efficient gas exchange for oxygen and carbon dioxide, and a narrow unimodal VA/Q distribution, the overall lung VA/Q ratio has a direct effect on anesthetic elimination kinetics. In general, higher VA/Q ratios produce higher fractional clearance in the lung (fig. 3) and therefore less slowing of tissue washout kinetics (equation 9 for τlate).

It has been taught intuitively for many years that lower blood gas solubility leads to faster overall uptake and elimination kinetics,1–3,6,9–11,13,24  a fact completely consistent with experimental data,6,14–20,23,24  mathematical models of kinetics solved numerically2,5,9–11,13,23,24,34  or analytically,4,28  and routine clinical experience. Surprisingly, however, it is hard to find in previous literature any equation that directly shows a connection between the overall terminal elimination time constant and blood gas solubility. Equation 9 for τlate directly makes a connection between the beta elimination half-life in the later stages of elimination, and blood gas solubility.

Our two companion studies, i.e., the current modeling study and the experimental study that provided the data for analysis, have several limitations. First, the current study was an exploratory post hoc analysis of a subset of a larger data set. This type of analysis is recognized as a way to generate hypotheses, but is not appropriate for testing of hypotheses. Second, we did not perform a complete factorial design with three levels each of cardiac output times three levels of ventilation. Rather, our experiments explored a limited subset of all nine possible conditions, with cardiac output and ventilation both changing between conditions. Third, cardiac output was varied in the desired directions, but there was no way to assess or influence the distribution of blood flow that could have accompanied these changes. Redistribution of blood flow between body compartments can impact elimination kinetics beyond the change in cardiac output alone. Both of the maneuvers to manipulate cardiac output (dobutamine to increase cardiac output; atrial obstruction to decrease cardiac output) could have changed distribution of flow as well as total flow. Additionally, both hypocarbia and hypercarbia can change the blood flow distribution. Fourth, an optimal condition for studying the kinetics of anesthetic elimination would be a starting point of complete equilibration of all of the body tissues to the same anesthetic partial pressure. Our 45-min administration obviously did not completely equilibrate all body tissues. Fifth, we did not directly measure the VA/Q distributions in our piglets. Previous measurements of VA/Q distributions in this model, however, have demonstrated normal distributions with a single narrow VA/Q mode, minimal shunt, and minimal alveolar dead space.36  Based on the close matching of VA and Q for normal lungs, our mathematical model further makes the approximation that matching between VA and Q is perfect, i.e., that the distribution is a single, idealized spike in both VA and Q . Finally, in the interests of arriving at relatively simple equations for kinetic time constants, we recognize that many approximations were made that do not represent the full complexity of anesthetic kinetics. In particular, representation of the whole body with a single time constant does not address the known complexity of multiple tissues and multiple compartments.

### Conclusions

The ratio of alveolar minute ventilation, VA, to cardiac output, Q, influences the kinetics of inhaled anesthetic elimination throughout the entire elimination. After a brief and rapid decline in alveolar anesthetic partial pressure, the fractional clearance of anesthetic by the normal lung becomes constant, and incomplete clearance from the lung slows the anesthetic washout from tissues. The increase in the elimination time constant for body tissues is a function of the dimensionless group VA/(λbg Q) that combines VA, Q, and the blood/gas partition coefficient λbg. Slowing of tissue elimination by incomplete lung clearance becomes more pronounced at low VA/Q ratios, and is predicted to become more pronounced as blood/gas solubility increases.

### Research Support

The work was supported by the Swedish Research Council (grant No. 2018-02438), Stockholm, Sweden, the Swedish Heart and Lung Fund, Stockholm, Sweden, and institutional sources of Uppsala University, Uppsala, Sweden, and the Otto-von-Guericke University Magdeburg, Magdeburg, Germany.

### Competing Interests

Dr. Baumgardner is president of Oscillogy LLC (Pittsburgh, Pennsylvania), the manufacturer of the multiple inert gas elimination technique by Micropore Membrane Inlet Mass Spectrometry system. The other authors declare no competing interests.

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### Appendix

#### Rationale for the Approximation that the 20- to 45-Min Washout Period Reflects a Single Compartment, the Muscle Compartment, Connected to the Lung

The functional residual capacity–dominated early time constant produces a very rapid early decrease in desflurane and sevoflurane elimination, decaying in less than 5 min as shown by the rapid approach to the retention equation plateau for all VA/Q (figs. 5 and 6). The visceral group time constant can be estimated from Cowles et al.23 and Mapleson's data5 on compartment volumes and fractional flows, combined with tissue partition coefficient data for the various central organs from Zhou and Liu.43 We estimate, for desflurane, in our pigs the intrinsic visceral time constants for the normal, low, and high VA/Q conditions, respectively, at 2.5, 1.3, and 3.7 min, and the clearance-corrected time constants at 3.5, 2.8, and 4.5 min. The corresponding estimates for sevoflurane for intrinsic visceral time constants are 3.8, 1.9, and 5.7 min, and the clearance-corrected time constants are estimated as 5.8, 4.8, and 7.3 min. Thus, after the first 20 min of elimination and approximately two to three effective visceral group time constants, the low arterial partial pressures (figs. 2 through 5 of the companion manuscript39) will be roughly matched in venous visceral blood with low venous partial pressures, and the visceral group will contribute little to the mixed venous washout kinetics. That leaves the muscle and fat groups (and possibly the intertissue diffusion group). Fat fractional flow, however, is about one tenth of the muscle fractional flow, limiting its contribution to mixed venous partial pressures. In addition, in our experiment, fat was very poorly equilibrated after 45 min, making it even less effective as a gas source for the mixed venous blood. It is therefore reasonable that the washout during 20 to 45 min is approximately monoexponential, since the washout is approximately described by a single muscle compartment.