Estimating Alveolar Dead Space from the Arterial toEnd-Tidal CO2 Gradient: A Modeling AnalysisJonathan G. Hardman, FRCA, and Alan R. Aitkenhead, FRCA

From the University Department of Anaesthesia, University Hospital, Nottingham, NG7 2UH, UK

Using an original, validated, high-fidelity model of pul-monary physiology, we compared the arterial to end-tidal CO2 gradient divided by the arterial CO2 tension(Pa-e'co2/Paco2) with alveolar dead space expressedas a fraction of alveolar tidal volume, calculated in theconventional manner using Fowler’s technique and theBohr equation: (VDalv/VTalv)Bohr-Fowler. We exam-ined the variability of Pa-e'co2/Paco2 and of (VDalv/VTalv)Bohr-Fowler in the presence of three ventilation-perfusion defects while varying CO2 production(Vco2), venous admixture, and anatomical dead spacefraction (VDanat). Pa-e'co2/Paco2 was approximately

59.5% of (VDalv/VTalv)Bohr-Fowler. During constant al-veolar configuration, the factors examined (Vco2, pul-monary shunt fraction, and VDanat) each caused varia-tion in (VDalv/VTalv)Bohr-Fowler and in Pa-e'co2/Paco2. Induced variation was slightly larger for Pa-e'co2/Paco2 during changes in VDanat, but wassimilar during variation of venous admixture andVco2. Pa-e'co2/Paco2 may be a useful serial measure-ment in the critically ill patient because all the necessarydata are easily obtained and calculation is significantlysimpler than for (VDalv/VTalv)Bohr-Fowler.

(Anesth Analg 2003;97:1846–51)

A lveolar dead space (VDalv) impairs pulmonarygas exchange, increases the obligatory work ofbreathing and prevents or prolongs weaning

from mechanical ventilation (1,2). Measurement ofVDalv may facilitate estimation of disease progres-sion, increase the efficacy of some interventions (par-ticularly ventilatory), and improve perioperative out-come (3,4). Recent evidence suggests that pulmonarydead space fraction may be an independent predictorof mortality from acute respiratory distress syndrome(5). Therefore, its measurement may be important onthe intensive care unit (ICU). However, the technicaland time-consuming nature of measurement of VDalvprevents its routine use on the ICU.

Nunn and Hill (6) have suggested that there is arelationship between the arterial to end-tidal CO2 ten-sion gradient (Pa-e'co2) and the Vdalv fraction, butthey did not investigate this relationship. In a previousinvestigation we used simple physiological modelingto examine the relationship between Pa-e'co2/Paco2and VDalv, calculated in the conventional mannerusing Fowler’s technique and Enghoff’s modificationof the Bohr equation, expressed a fraction of alveolar

tidal volume: (VDalv/VTalv)Bohr-Fowler. We concludedin that investigation Pa-e'co2/Paco2 had a roughlyconstant, linear relationship with (VDalv/VTalv)Bohr-Fowler as follows: (VDalv/VTalv)Bohr-Fowler � 1.14 �Pa-e'co2/Paco2 � 0.005, and that it could be substi-tuted acceptably for the conventional calculation (7).

This investigation uses physiological models ofmuch greater sophistication than those used previ-ously. The advances in modeling that have made re-evaluation important are detailed in the Appendix.Our aims were:

1. To examine the relationship between Pa-e'co2/Paco2 and (VDalv/VTalv)Bohr-Fowler.

2. To determine the susceptibility of Pa-e'co2/Paco2and of (VDalv/VTalv)Bohr-Fowler to change inducedby coincidental variation of physiological factorsduring a constant alveolar configuration (i.e., con-stant pulmonary ventilation and perfusion distri-butions). Such change is misleading because itcauses the appearance of a change in alveolar con-figuration when such a change has not occurred.The better, independent measure of alveolar con-figuration is less susceptible to variation.

MethodsPhysiological Models

The models used in this investigation are based on thepublished and validated Nottingham Physiological

Accepted for publication July 21, 2003.Address correspondence and reprint requests to Jonathan G.

Hardman, Clinical Senior Lecturer, University Department of An-esthesia, University Hospital, Nottingham, NG7 2UH, UK. Addressemail to j.hardman@nottingham.ac.uk.

DOI: 10.1213/01.ANE.0000090316.46604.89

©2003 by the International Anesthesia Research Society1846 Anesth Analg 2003;97:1846–51 0003-2999/03

Simulator (9–12). Briefly, the models are iterative,mass conserving, and arithmetical (rather than calcu-lus based). The most fundamental processes of molec-ular movement and physical behavior, such as theconstancy of pressure � volume/temperature, wereused to construct discrete program subunits, whichare run repeatedly, with each iteration generating thetiny changes that have occurred since the last micro-second “time-slice.” These high-fidelity models aredescribed in greater detail separately (13). The modelshave been validated for the performance of this inves-tigation, and in particular, the poly-laminar seriesdead space and the poly-compartmental ventilation-perfusion (VQ) aspects have been demonstrated to berobust and realistic (13).

Experimental setup. The model was configured asfollows: weight 75 kg, height 1.75 m, supine posture,inspired oxygen fraction (Fio2) 21%, inspired CO2fraction (Fico2) 0.1%, inspired gas temperature 37°C,inspired water fraction 6.2% (saturated at 37°C), car-diac index 2.73 L/min/m�2, oxygen consumption250 mL/min (Vo2), respiratory quotient 0.8, anatomi-cal dead space volume (VDanat) 65 mL (6), fixed (an-atomical) pulmonary shunt 1% of cardiac output, ven-tilatory rate 12 breaths/min, inspired tidal volume(Vt) 500 mL. The sampling interval (see Appendix)was set to 1 ms and internal mass-conservation anderror checking were enabled. The model included 500“alveolar” gas-exchanging compartments and 250 se-ries dead space laminae.

Estimation of VDalv/VTalv. VDanat was derived bygeometrically dividing the Pco2 versus time capno-gram, as per Fowler’s technique (14). Physiologicaldead space (VDphys) was calculated using Enghoff’smodification of the Bohr equation (15,16). (VDalv/VTalv)Bohr-Fowler was estimated from the output of themodel in the conventional manner: VDalv/VTalv �((1 � (PE'CO2/Paco2) � VTexh) �VDanat)/(VTexh �VDanat), where PE'CO2 represents the mixed expiredCO2 tension and VTexh is the exhaled Vt.

Patterns of VQ mismatch. The relationship between(VDalv/VTalv)Bohr-Fowler and Pa-e'co2/Paco2 was ex-amined in the presence of three patterns of VQ mis-matching. Each pattern of mismatch was created byvarying the compartmental bronchiolar resistancesand the compartmental arteriolar resistances in oppo-site directions, generating asynchronous alveolar ven-tilation and a realistic scatter of VQ ratios as follows:

• Small defect: vascular resistance (Rv) 0.5–2 timesnormal, bronchial resistance (Rb) 2–0.5 times nor-mal. Mean and standard deviation of compart-mental VQ ratios were 1.42 and 0.57, respectively.

• Moderate defect: Rv 0.25–4 times normal, Rb4–0.25 times normal. Mean and sd of compart-mental VQ ratios were 2.23 and 1.47, respectively.

• Large defect: Rv 0.1–10 times normal, Rb 10–0.1

times normal. Mean and sd of compartmental VQratios were 3.15 and 2.50, respectively.

Factor variation. Three factors were each varied in-dependently to examine their effect on (VDalv/VTalv)Bohr-Fowler and on Pa-e'co2/Paco2. The factors,each of which had been found to cause significantdisturbance in the relationship between VDalv and thePa-e'co2 gradient in our previous investigation, wereas follows:

• CO2 production (Vco2 values: 100, 200, and300 mL/min).

• VDanat (values: 33, 65, and 130 mL).• Fixed pulmonary shunt fraction (values: 1%, 15%,

and 30% of cardiac output). Pulmonary VQ con-figuration was maintained during variation ofshunt values by increasing cardiac output to keepintrapulmonary (nonshunted) blood flowconstant.

Each examination used a re-initialized scenario and(VDalv/VTalv)Bohr-Fowler and Pa-e'co2/Paco2 were re-corded after complete equilibration had been achievedfor both CO2 and O2 (defined as total body CO2 andO2 flux �0.1 mL/min each).

The distribution of variation induced in (VDalv/VTalv)Bohr-Fowler and Pa-e'co2/Paco2 are expressed as95% confidence intervals of the variation: CI95% �mean (N(i � 1 to n) � N0) � 1.95 � sd (N(i � 1 to n) � N0),where mean (N(i � 1 to n) � N0) is the mean of thevariation from baseline and where N0 is the originalvalue before coincidental physiological variation.

ResultsNormal conditions. Under normal physiological

conditions (Vco2 200 mL/min, shunt 1% of cardiacoutput, and VDanat 65 mL) while the VQ defect wasvaried, Pa-e'co2/Paco2 had a linear relationship with(VDalv/VTalv)Bohr-Fowler; it was consistently 59.5% of(VDalv/VTalv)Bohr-Fowler (Fig. 1). The CI95% of the er-ror in calculating (VDalv/VTalv)Bohr-Fowler from Pa-e'co2/Paco2 using this formula was �13.0% to 11.5%in normal physiological conditions and �32.3% to32.5% over all physiological conditions tested.

CO2 production. Variation in CO2 production (from200 mL/min down to 100 mL/min and up to 300 mL/min) had very small effects on Pa-e'co2/Paco2 and on(VDalv/VTalv)Bohr-Fowler (Fig. 1). The CI95% of the in-duced variation are shown in Table 1.

Fixed shunt fraction. Increasing fixed, pulmonaryshunt fraction (from 1% of cardiac output to 15%, then30%) in the presence of a constant alveolar configura-tion caused similar increases in Pa-e'co2/Paco2 and(VDalv/VTalv)Bohr-Fowler (Fig. 2). The CI95% of the in-duced variation are shown in Table 1.

ANESTH ANALG HARDMAN AND AITKENHEAD 18472003;97:1846–51 ALVEOLAR DEAD SPACE AND END-TIDAL CO2

Anatomical dead space volume. An increase in ana-tomical dead space volume in the presence of a con-stant alveolar configuration significantly increased Pa-e'co2/Paco2 but reduced (VDalv/VTalv)Bohr-Fowler(Fig. 3). The CI95% of the induced variation are shownin Table 1.

DiscussionAlthough increasing fixed shunt fraction did not sig-nificantly alter the relationship between Pa-e'co2/Paco2 and (VDalv/VTalv)Bohr-Fowler, it created a “vir-tual” dead space in both. This is in agreement with ourprevious study (7) and with previous clinical investi-gations (17–19). Correction may be made for the influ-ence of changing fixed shunt fraction by estimatingshunt fraction using iso-shunt diagrams (20) or aphysiological model (10).

Our methodology included the maintenance of non-shunted cardiac output while fixed shunt fraction var-ied. Clearly, reality is far less simple, and one cannotexpect a patient to maintain their nonshunted pulmo-nary blood flow; this was necessary. If shunt is increasedwhile cardiac output is unchanged then nonshuntedblood flow decreases, automatically increasing the meanVQ ratio and thereby increasing VDalv. In our study,nonshunted blood flow was maintained during increas-ing venous admixture to keep alveolar configuration

(and VDalv) constant, allowing examination of the ro-bustness of measures representing VDalv during chang-ing venous admixture.

The dependence of the apparent VDalv on theVDanat has been noted previously in an elegant in-vestigation using a four-compartment lung model(21). This model predicted that various combinationsof serial and parallel dead spaces that should add upto identical VDphys values as calculated using Eng-hoff’s modification of the Bohr equation in fact pro-duced differing VDphys values. This area requiresfurther investigation using high fidelity modeling.

The dependence of both Pa-e'co2/Paco2 and(VDalv/VTalv)Bohr-Fowler on VDanat does not implythe superiority of either method of representingVDalv, but highlights a potential problem with both.Large changes in VDanat are rare in clinical practice,and when such variations occur they are usually easilynoted, and a change in the trend in Pa-e'co2/Paco2may be anticipated. The most important scenario inthis context is probably that of VDanat conforming toa fraction of changing Vt (6).

Pa-e'co2/Paco2 and (VDalv/VTalv)Bohr-Fowler differnumerically, and although a conversion formula maybe used as described in Results, it is probably unnec-essary. Indeed, it is probably inappropriate becauselarge variation was observed in the relationship be-tween the two measures during physiologicalvariation.

It is clear that, despite constant alveolar configura-tion, (VDalv/VTalv)Bohr-Fowler is susceptible to varia-tion while other physiological factors vary. Therefore,(VDalv/VTalv)Bohr-Fowler is not an independent repre-sentation of alveolar configuration. The question ofwhether (VDalv/VTalv)Bohr-Fowler or Pa-e'co2/Paco2is closer to the “truth” is difficult to answer. If lungswere constructed in a fashion similar to Riley’s origi-nal lung model (22), consisting of a single dead spacevolume, a shunted volume, and an optimally venti-lated and perfused volume, then there would be asimple, correct answer, but in the presence of a con-tinuous distribution of variably perfused and venti-lated alveoli the answer is less clear. The most clini-cally applicable measure is probably that whichever ismost independent of coincidental physiological varia-tion. As (VDalv/VTalv)Bohr-Fowler is marginally morerobust in the presence of variation in VDanat then itmay be considered the superior measure. However,calculation of (VDalv/VTalv)Bohr-Fowler requires thecollection and analysis of expired gas (over a signifi-cant time period), the calculation of VDanat using apartial pressure versus volume capnogram and arte-rial gas tension analysis. Calculation of Pa-e'co2/Paco2, however, requires only measurement of Pa-e'co2 tension. Additionally, Pa-e'co2/Paco2 may beupdated in real time. The appropriate use of Pa-e'co2/Paco2 may include its daily calculation on the ICU as

Figure 1. The relationship between arterial-end-tidal CO2 gradient/arterial CO2 tension (Pa-e'co2/Paco2) and alveolar deadspace/alveolar tidal volume, calculated conventionally using Fowler’stechnique and the Bohr equation - (VDalv/VTalv)Bohr-Fowler. Solidlines show the behavior of Pa-e'co2/Paco2 and (VDalv/VTalv)Bohr-Fowler during conditions that were constant other than changingalveolar configuration (i.e., changing VDalv). Dashed lines show theeffect on Pa-e'co2/Paco2 and (VDalv/VTalv)Bohr-Fowler of indepen-dently varying Vco2 while alveolar configuration remained con-stant. The vertical displacement along the dashed line reflects thechange in alveolar configuration misleadingly implied by Pa-e'co2/Paco2 and the horizontal displacement reflects the change in alve-olar configuration misleadingly implied by conventionally calcu-lated VDalv/VTalv.

1848 HARDMAN AND AITKENHEAD ANESTH ANALGALVEOLAR DEAD SPACE AND END-TIDAL CO2 2003;97:1846–51

an estimate of disease progression or to assess theefficacy of interventions. It is obvious that use of Pa-e'co2/Paco2 should not replace the use of (VDalv/VTalv)Bohr-Fowler, but its simplicity of calculation mayallow easier clinical application of VDalv estimation.Indeed, its ease of use may encourage the more wide-spread use of VDalv monitoring to quantify diseaseprogression and to assess the effects of interventions.

It is probably inappropriate to refer to Pa-e'co2/Paco2 as the “alveolar dead space fraction” becausethis is widely accepted as being represented by theconventional calculation. It may be more appropriateto refer to it by its formula if it is recorded in trends indaily ICU patient management. It is widely recog-nized that the VDphys, calculated using Enghoff’smodification of Bohr’s equation, does not refer to anydiscrete part of the respiratory system, and has beentermed by some the “Bohr” dead space (23).

The limitations of this investigation include thefollowing:

1. The use of only three discrete VQ defects.Clearly, the patient population includes a near-infinite number of discrete VQ defects. Our mod-els were chosen to represent large, heterogeneousgroups rather than individuals, and we expectthat each defect modeled will adequately repre-sent patients whose VQ defects are similar. Theresults of this investigation may not be applicableto those patients whose VQ defects are grosslydissimilar to those examined in this investiga-tion. This includes patients with the most severelung pathology, whose VQ configurations wehave not reproduced in this study.

2. The use of a limited number of alveolar compart-ments (500) and series dead space laminae (250).This is unlikely to represent a serious flaw, andwill achieve greater accuracy than any other cur-rently investigated pulmonary model.

3. The use of a mathematical model rather than apatient group. This criticism may be directed at

Table 1. Variation Induced in Arterial-End-Tidal CO2 Gradient/Arterial CO2 Tension (Pa-e�co2/Paco2) and in AlveolarDeadspace/Alveolar Tidal Volume (Calculated Conventionally Using Fowler’s Technique and the Bohr Equation) DuringCoincidental Variation of Physiological Factors

Varying factor

CI95% of induced misleading variation

Pa-e� co2/Paco2 Conventional VDalv/VTalv

Vco2 �0.03% � 0.20% 0.33% � 0.34%Pulmonary shunt fraction 3.00% � 2.40% 2.80% � 2.37%VDanat 2.50% � 1.87% �1.00% � 1.67%

Figure 2. The relationship between arterial-end-tidal CO2 gradient/arterial CO2 tension (Pa-e'co2/Paco2) and alveolar deadspace/alveolar tidal volume, calculated conventionally using Fowler’stechnique and the Bohr equation - (VDalv/VTalv)Bohr-Fowler. Solidlines show the behavior of Pa-e'co2/Paco2 and (VDalv/VTalv)Bohr-Fowler during conditions that were constant other than changingalveolar configuration (i.e., changing VDalv). Dashed lines show theeffect on Pa-e'co2/Paco2 and (VDalv/VTalv)Bohr-Fowler of indepen-dently varying fixed pulmonary shunt fractio, while alveolar con-figuration remained constant. The vertical displacement along thedashed line reflects the change in alveolar configuration mislead-ingly implied by Pa-e'co2/Paco2 and the horizontal displacementreflects the change in alveolar configuration misleadingly impliedby conventionally calculated VDalv/VTalv.

Figure 3. The relationship between arterial-end-tidal CO2 gradient/arterial CO2 tension (Pa-e'co2/Paco2) and alveolar dead space/alveolar tidal volume, calculated conventionally using Fowler’stechnique and the Bohr equation - (VDalv/VTalv)Bohr-Fowler. Solidlines show the behavior of Pa-e'co2/Paco2 and (VDalv/VTalv)Bohr-Fowler during conditions that were constant other than changingalveolar configuration (i.e., changing VDalv). Dashed lines show theeffect on Pa-e'co2/Paco2 and (VDalv/VTalv)Bohr-Fowler of indepen-dently varying anatomical dead space volume while alveolar con-figuration remained constant. The vertical displacement along thedashed line reflects the change in alveolar configuration mislead-ingly implied Pa-e'co2/Paco2 and the horizontal displacement re-flects the change in alveolar configuration misleadingly implied byconventionally calculated VDalv/VTalv.

ANESTH ANALG HARDMAN AND AITKENHEAD 18492003;97:1846–51 ALVEOLAR DEAD SPACE AND END-TIDAL CO2

any investigation using mathematical modeling.However, the stratification of physiological fac-tors that was crucial to this investigation couldnot be performed in vivo. The modeling used inthis investigation included dynamic and staticlung inhomogeneity. Viscoelasticity, nonsyn-chronous alveolar exhalation, poly-laminar deadspace, poly-compartmental lung and micro-timeslicing all contributed to a very credible,high-fidelity model of pulmonary physiology. Inaddition, performance of this investigation invivo would be very difficult because achievingCO2 equilibrium after changes in physiologicalfactors would take too long for the investigationto be feasible (24). Finally, it is impossible in vivoto vary a physiological value independently.Without this independent variation, clear conclu-sions of causal relationships cannot be drawn,particularly within a heterogeneous patientgroup.

4. Other techniques of VDalv estimation. A furthercriticism that may be leveled at this investigationis that VDalv may be measured almost in real-time using a technique of continuous expired gasanalysis, such as the single-breath CO2 test (25).Therefore, why should we require a furthermethod of quantifying alveolar configuration?First, most ICUs do not have a single-breath CO2analyzer, and thus cannot use that method. Sec-ond, the technique of single-breath capnographicVDalv quantification is validated only in animalsand has not been demonstrated to be indepen-dent of variation in the factors presented here,such as VDanat. VDalv measured by the single-breath test may, in fact, be just as variable aseither of the measures presented in thisinvestigation.

Several conclusions drawn from this modeling in-vestigation contrast with conclusions based on ourprevious investigation in this area (7). These differ-ences are explained by the increase in complexity andfidelity of the modeling. However, several of our pre-vious conclusions are supported by this investigation,and this includes the recommendation Pa-e'co2/Paco2 may be useful in clinical practice, particularly asa monitor of trends in pulmonary condition.

AppendixThe following advances in physiologicalmodeling make re-evaluation of this topicimportant:

• Nonsynchronous alveolar exhalation. This is re-sponsible for much of the variation in Pa-e'co2 invivo. In contrast to the previous modeling, the

current model includes alveolar units with uniquetime constants generated by independent inletresistances and compliance curves. Thus, the fullspectrum of respiratory disturbance may be accu-rately recreated.

• Poly-compartmental lung. Each of the 500 alveo-lar compartments has a unique ventilation-perfusion (VQ) ratio. The modeling used in theprevious investigation used only 3 compartments:a true shunt, a true dead space and an optimallyventilated and perfused compartment. The largenumber of compartments allows the smooth dis-tribution of VQ ratios, avoiding the production of“step-artifacts.” The validated modeling used inthis investigation more closely resembles in vivolung anatomy and physiology.

• Inclusion in the model of a poly-laminar anatom-ical dead space. The previous modeling used asingle, immediate-mixing, fixed-volume compart-ment. This modeling was expedient and compu-tationally efficient, but with the advent of greatercomputer power a more sophisticated and credi-ble model is possible.

• The use of micro-timeslicing (1 ms) for far greateraccuracy (8). The fidelity of the model is greatlyincreased by considering, during each iteration ofthe model algorithms, the changes that occur inthe system each millisecond. The modeling usedin the previous investigation used a time slice, orsampling interval, of 250 ms. Any physiologicalchanges that occur in the system during the timeslice are averaged into the net change for thatperiod. The result is analogous to damping andcauses the loss of potentially important data.

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