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GENERAL ARTICLES

Estimating Alveolar Dead Space from the Arterial to End-Tidal CO₂ Gradient: A Modeling Analysis

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

Address correspondence and reprint requests to Jonathan G. Hardman, Clinical Senior Lecturer, University Department of Anesthesia, University Hospital, Nottingham, NG7 2UH, UK. Address email to j.hardman{at}nottingham.ac.uk

	Abstract

Top Abstract Introduction Methods Results Discussion Appendix References

 
Using an original, validated, high-fidelity model of pulmonary physiology, we compared the arterial to end-tidal CO₂ gradient divided by the arterial CO₂ tension (Pa-E'CO₂/PaCO₂) with alveolar dead space expressed as a fraction of alveolar tidal volume, calculated in the conventional manner using Fowler’s technique and the Bohr equation: (VDalv/VTalv)_Bohr-Fowler. We examined 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 (CO₂), venous admixture, and anatomical dead space fraction (VDanat). Pa-E'CO₂/PaCO₂ was approximately 59.5% of (VDalv/VTalv)_Bohr-Fowler. During constant alveolar configuration, the factors examined (CO₂, pulmonary shunt fraction, and VDanat) each caused variation 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 was similar during variation of venous admixture and CO₂. Pa-E'CO₂/PaCO₂ may be a useful serial measurement in the critically ill patient because all the necessary data are easily obtained and calculation is significantly simpler than for (VDalv/VTalv)_Bohr-Fowler.

IMPLICATIONS: Using an original, validated, high-fidelity model of pulmonary physiology, we have demonstrated that the arterial to end-tidal carbon dioxide pressure gradient may be used to robustly and accurately quantify alveolar dead space. After clinical validation, its use could replace that of conventionally calculated alveolar dead space fraction, particularly in the critically ill.

	Introduction

Top Abstract Introduction Methods Results Discussion Appendix References

 
Alveolar dead space (VDalv) impairs pulmonary gas exchange, increases the obligatory work of breathing and prevents or prolongs weaning from mechanical ventilation (1,2). Measurement of VDalv may facilitate estimation of disease progression, increase the efficacy of some interventions (particularly ventilatory), and improve perioperative outcome (3,4). Recent evidence suggests that pulmonary dead space fraction may be an independent predictor of mortality from acute respiratory distress syndrome (5). Therefore, its measurement may be important on the intensive care unit (ICU). However, the technical and time-consuming nature of measurement of VDalv prevents its routine use on the ICU.

Nunn and Hill (6) have suggested that there is a relationship between the arterial to end-tidal CO₂ tension gradient (Pa-E'CO₂) and the Vdalv fraction, but they did not investigate this relationship. In a previous investigation we used simple physiological modeling to examine the relationship between Pa-E'CO₂/PaCO₂ and VDalv, calculated in the conventional manner using Fowler’s technique and Enghoff’s modification of the Bohr equation, expressed a fraction of alveolar tidal volume: (VDalv/VTalv)_Bohr-Fowler. We concluded in that investigation Pa-E'CO₂/PaCO₂ had a roughly constant, linear relationship with (VDalv/VTalv)_Bohr-Fowler as follows: (VDalv/VTalv)_Bohr-Fowler = 1.14 x Pa-E'CO₂/PaCO₂ - 0.005, and that it could be substituted acceptably for the conventional calculation (7).

This investigation uses physiological models of much greater sophistication than those used previously. 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/PaCO₂ and of (VDalv/VTalv)_Bohr-Fowler to change induced by coincidental variation of physiological factors during a constant alveolar configuration (i.e., constant pulmonary ventilation and perfusion distributions). Such change is misleading because it causes the appearance of a change in alveolar configuration when such a change has not occurred. The better, independent measure of alveolar configuration is less susceptible to variation.
   

	Methods

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Physiological Models
The models used in this investigation are based on the published and validated Nottingham Physiological Simulator (9–12). Briefly, the models are iterative, mass conserving, and arithmetical (rather than calculus based). The most fundamental processes of molecular movement and physical behavior, such as the constancy of pressure x volume/temperature, were used to construct discrete program subunits, which are run repeatedly, with each iteration generating the tiny changes that have occurred since the last microsecond "time-slice." These high-fidelity models are described in greater detail separately (13). The models have been validated for the performance of this investigation, and in particular, the poly-laminar series dead space and the poly-compartmental ventilation-perfusion (VQ) aspects have been demonstrated to be robust and realistic (13).

Experimental setup.
The model was configured as follows: weight 75 kg, height 1.75 m, supine posture, inspired oxygen fraction (FIO₂) 21%, inspired CO₂ fraction (FICO₂) 0.1%, inspired gas temperature 37°C, inspired water fraction 6.2% (saturated at 37°C), cardiac index 2.73 L/min/m^-2, oxygen consumption 250 mL/min (O₂), respiratory quotient 0.8, anatomical dead space volume (VDanat) 65 mL (6), fixed (anatomical) pulmonary shunt 1% of cardiac output, ventilatory rate 12 breaths/min, inspired tidal volume (VT) 500 mL. The sampling interval (see Appendix) was set to 1 ms and internal mass-conservation and error checking were enabled. The model included 500 "alveolar" gas-exchanging compartments and 250 series dead space laminae.

Estimation of VDalv/VTalv.
VDanat was derived by geometrically dividing the PCO₂ versus time capnogram, as per Fowler’s technique (14). Physiological dead space (VDphys) was calculated using Enghoff’s modification of the Bohr equation (15,16). (VDalv/VTalv)_Bohr-Fowler was estimated from the output of the model in the conventional manner: VDalv/VTalv = ((1 - (PE'CO₂/PaCO₂) x VTexh) -VDanat)/(VTexh - VDanat), where PE'CO₂ represents the mixed expired CO₂ tension and VTexh is the exhaled VT.

Patterns of VQ mismatch.
The relationship between (VDalv/VTalv)_Bohr-Fowler and Pa-E'CO₂/PaCO₂ was examined in the presence of three patterns of VQ mismatching. Each pattern of mismatch was created by varying the compartmental bronchiolar resistances and the compartmental arteriolar resistances in opposite directions, generating asynchronous alveolar ventilation and a realistic scatter of VQ ratios as follows:

• Small defect: vascular resistance (R_v) 0.5–2 times normal, bronchial resistance (R_b) 2–0.5 times normal. Mean and standard deviation of compartmental VQ ratios were 1.42 and 0.57, respectively.
  
• Moderate defect: R_v 0.25–4 times normal, R_b 4–0.25 times normal. Mean and SD of compartmental VQ ratios were 2.23 and 1.47, respectively.
  
• Large defect: R_v 0.1–10 times normal, R_b 10–0.1 times normal. Mean and SD of compartmental VQ ratios were 3.15 and 2.50, respectively.
  

Factor variation.
Three factors were each varied independently 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 significant disturbance in the relationship between VDalv and the Pa-E'CO₂ gradient in our previous investigation, were as follows:

• CO₂ production (CO₂ values: 100, 200, and 300 mL/min).
  
• VDanat (values: 33, 65, and 130 mL).
  
• Fixed pulmonary shunt fraction (values: 1%, 15%, and 30% of cardiac output). Pulmonary VQ configuration was maintained during variation of shunt values by increasing cardiac output to keep intrapulmonary (nonshunted) blood flow constant.
  

Each examination used a re-initialized scenario and (VDalv/VTalv)_Bohr-Fowler and Pa-E'CO₂/PaCO₂ were recorded after complete equilibration had been achieved for both CO₂ and O₂ (defined as total body CO₂ and O₂ flux <0.1 mL/min each).

The distribution of variation induced in (VDalv/VTalv)_Bohr-Fowler and Pa-E'CO₂/PaCO₂ are expressed as 95% confidence intervals of the variation: CI_95% = mean (N_{(i = 1 to n)} - N₀) ± 1.95 x SD (N_{(i = 1 to n)} - N₀), where mean (N_{(i = 1 to n)} - N₀) is the mean of the variation from baseline and where N₀ is the original value before coincidental physiological variation.

	Results

Top Abstract Introduction Methods Results Discussion Appendix References

 
Normal conditions.
Under normal physiological conditions (CO₂ 200 mL/min, shunt 1% of cardiac output, and VDanat 65 mL) while the VQ defect was varied, 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 CI_95% of the error 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% to 32.5% over all physiological conditions tested.

View larger version (13K): [in this window] [in a new window]   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’s technique and the Bohr equation - (VDalv/VTalv)Bohr-Fowler. Solid lines show the behavior of Pa-E'CO2/PaCO2 and (VDalv/VTalv)Bohr-Fowler during conditions that were constant other than changing alveolar configuration (i.e., changing VDalv). Dashed lines show the effect on Pa-E'CO2/PaCO2 and (VDalv/VTalv)Bohr-Fowler of independently varying CO2 while alveolar configuration remained constant. The vertical displacement along the dashed line reflects the change in alveolar configuration misleadingly implied by Pa-E'CO2/PaCO2 and the horizontal displacement reflects the change in alveolar configuration misleadingly implied by conventionally calculated VDalv/VTalv.

View larger version (14K): [in this window] [in a new window]   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’s technique and the Bohr equation - (VDalv/VTalv)Bohr-Fowler. Solid lines show the behavior of Pa-E'CO2/PaCO2 and (VDalv/VTalv)Bohr-Fowler during conditions that were constant other than changing alveolar configuration (i.e., changing VDalv). Dashed lines show the effect on Pa-E'CO2/PaCO2 and (VDalv/VTalv)Bohr-Fowler of independently varying fixed pulmonary shunt fractio, while alveolar configuration remained constant. The vertical displacement along the dashed line reflects the change in alveolar configuration misleadingly implied by Pa-E'CO2/PaCO2 and the horizontal displacement reflects the change in alveolar configuration misleadingly implied by conventionally calculated VDalv/VTalv.

View larger version (14K): [in this window] [in a new window]   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’s technique and the Bohr equation - (VDalv/VTalv)Bohr-Fowler. Solid lines show the behavior of Pa-E'CO2/PaCO2 and (VDalv/VTalv)Bohr-Fowler during conditions that were constant other than changing alveolar configuration (i.e., changing VDalv). Dashed lines show the effect on Pa-E'CO2/PaCO2 and (VDalv/VTalv)Bohr-Fowler of independently varying anatomical dead space volume while alveolar configuration remained constant. The vertical displacement along the dashed line reflects the change in alveolar configuration misleadingly implied Pa-E'CO2/PaCO2 and the horizontal displacement reflects the change in alveolar configuration misleadingly implied by conventionally calculated VDalv/VTalv.

	Discussion

Top Abstract Introduction Methods Results Discussion Appendix References

Our methodology included the maintenance of nonshunted cardiac output while fixed shunt fraction varied. Clearly, reality is far less simple, and one cannot expect a patient to maintain their nonshunted pulmonary blood flow; this was necessary. If shunt is increased while cardiac output is unchanged then nonshunted blood flow decreases, automatically increasing the mean VQ ratio and thereby increasing VDalv. In our study, nonshunted blood flow was maintained during increasing venous admixture to keep alveolar configuration (and VDalv) constant, allowing examination of the robustness of measures representing VDalv during changing venous admixture.

The dependence of the apparent VDalv on the VDanat has been noted previously in an elegant investigation using a four-compartment lung model (21). This model predicted that various combinations of serial and parallel dead spaces that should add up to identical VDphys values as calculated using Enghoff’s modification of the Bohr equation in fact produced differing VDphys values. This area requires further investigation using high fidelity modeling.

The dependence of both Pa-E'CO₂/PaCO₂ and (VDalv/VTalv)_Bohr-Fowler on VDanat does not imply the superiority of either method of representing VDalv, but highlights a potential problem with both. Large changes in VDanat are rare in clinical practice, and when such variations occur they are usually easily noted, and a change in the trend in Pa-E'CO₂/PaCO₂ may be anticipated. The most important scenario in this context is probably that of VDanat conforming to a fraction of changing VT (6).

Pa-E'CO₂/PaCO₂ and (VDalv/VTalv)_Bohr-Fowler differ numerically, and although a conversion formula may be used as described in Results, it is probably unnecessary. Indeed, it is probably inappropriate because large variation was observed in the relationship between the two measures during physiological variation.

It is clear that, despite constant alveolar configuration, (VDalv/VTalv)_Bohr-Fowler is susceptible to variation while other physiological factors vary. Therefore, (VDalv/VTalv)_Bohr-Fowler is not an independent representation of alveolar configuration. The question of whether (VDalv/VTalv)_Bohr-Fowler or Pa-E'CO₂/PaCO₂ is closer to the "truth" is difficult to answer. If lungs were constructed in a fashion similar to Riley’s original lung model (22), consisting of a single dead space volume, a shunted volume, and an optimally ventilated and perfused volume, then there would be a simple, correct answer, but in the presence of a continuous distribution of variably perfused and ventilated alveoli the answer is less clear. The most clinically applicable measure is probably that whichever is most independent of coincidental physiological variation. As (VDalv/VTalv)_Bohr-Fowler is marginally more robust in the presence of variation in VDanat then it may be considered the superior measure. However, calculation of (VDalv/VTalv)_Bohr-Fowler requires the collection and analysis of expired gas (over a significant time period), the calculation of VDanat using a partial pressure versus volume capnogram and arterial gas tension analysis. Calculation of Pa-E'CO₂/PaCO₂, however, requires only measurement of Pa-E'CO₂ tension. Additionally, Pa-E'CO₂/PaCO₂ may be updated in real time. The appropriate use of Pa-E'CO₂/PaCO₂ may include its daily calculation on the ICU as an estimate of disease progression or to assess the efficacy 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 may allow easier clinical application of VDalv estimation. Indeed, its ease of use may encourage the more widespread use of VDalv monitoring to quantify disease progression and to assess the effects of interventions.

It is probably inappropriate to refer to Pa-E'CO₂/PaCO₂ as the "alveolar dead space fraction" because this is widely accepted as being represented by the conventional calculation. It may be more appropriate to refer to it by its formula if it is recorded in trends in daily ICU patient management. It is widely recognized that the VDphys, calculated using Enghoff’s modification of Bohr’s equation, does not refer to any discrete part of the respiratory system, and has been termed by some the "Bohr" dead space (23).

The limitations of this investigation include the following:

1. The use of only three discrete VQ defects. Clearly, the patient population includes a near-infinite number of discrete VQ defects. Our models were chosen to represent large, heterogeneous groups rather than individuals, and we expect that each defect modeled will adequately represent patients whose VQ defects are similar. The results of this investigation may not be applicable to those patients whose VQ defects are grossly dissimilar to those examined in this investigation. This includes patients with the most severe lung pathology, whose VQ configurations we have not reproduced in this study.
   
2. The use of a limited number of alveolar compartments (500) and series dead space laminae (250). This is unlikely to represent a serious flaw, and will achieve greater accuracy than any other currently investigated pulmonary model.
   
3. The use of a mathematical model rather than a patient group. This criticism may be directed at any investigation using mathematical modeling. However, the stratification of physiological factors that was crucial to this investigation could not be performed in vivo. The modeling used in this investigation included dynamic and static lung inhomogeneity. Viscoelasticity, nonsynchronous alveolar exhalation, poly-laminar dead space, poly-compartmental lung and micro-timeslicing all contributed to a very credible, high-fidelity model of pulmonary physiology. In addition, performance of this investigation in vivo would be very difficult because achieving CO₂ equilibrium after changes in physiological factors would take too long for the investigation to be feasible (24). Finally, it is impossible in vivo to vary a physiological value independently. Without this independent variation, clear conclusions of causal relationships cannot be drawn, particularly within a heterogeneous patient group.
   
4. Other techniques of VDalv estimation. A further criticism that may be leveled at this investigation is that VDalv may be measured almost in real-time using a technique of continuous expired gas analysis, such as the single-breath CO₂ test (25). Therefore, why should we require a further method of quantifying alveolar configuration? First, most ICUs do not have a single-breath CO₂ analyzer, and thus cannot use that method. Second, the technique of single-breath capnographic VDalv quantification is validated only in animals and has not been demonstrated to be independent of variation in the factors presented here, such as VDanat. VDalv measured by the single-breath test may, in fact, be just as variable as either of the measures presented in this investigation.
   

Several conclusions drawn from this modeling investigation contrast with conclusions based on our previous investigation in this area (7). These differences are explained by the increase in complexity and fidelity of the modeling. However, several of our previous conclusions are supported by this investigation, and this includes the recommendation Pa-E'CO₂/PaCO₂ may be useful in clinical practice, particularly as a monitor of trends in pulmonary condition.

	Appendix

Top Abstract Introduction Methods Results Discussion Appendix References

 
The following advances in physiological modeling make re-evaluation of this topic important:

• Nonsynchronous alveolar exhalation. This is responsible for much of the variation in Pa-E'CO₂ in vivo. In contrast to the previous modeling, the current model includes alveolar units with unique time constants generated by independent inlet resistances and compliance curves. Thus, the full spectrum of respiratory disturbance may be accurately recreated.
  
• Poly-compartmental lung. Each of the 500 alveolar compartments has a unique ventilation-perfusion (VQ) ratio. The modeling used in the previous investigation used only 3 compartments: a true shunt, a true dead space and an optimally ventilated and perfused compartment. The large number of compartments allows the smooth distribution of VQ ratios, avoiding the production of "step-artifacts." The validated modeling used in this investigation more closely resembles in vivo lung anatomy and physiology.
  
• Inclusion in the model of a poly-laminar anatomical dead space. The previous modeling used a single, immediate-mixing, fixed-volume compartment. This modeling was expedient and computationally efficient, but with the advent of greater computer power a more sophisticated and credible model is possible.
  
• The use of micro-timeslicing (1 ms) for far greater accuracy (8). The fidelity of the model is greatly increased by considering, during each iteration of the model algorithms, the changes that occur in the system each millisecond. The modeling used in the previous investigation used a time slice, or sampling interval, of 250 ms. Any physiological changes that occur in the system during the time slice are averaged into the net change for that period. The result is analogous to damping and causes the loss of potentially important data.
  

	References

Top Abstract Introduction Methods Results Discussion Appendix References

 

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Anesthesia & Analgesia® is published for the International Anesthesia Research Society® by Lippincott Williams & Wilkins with the assistance of Stanford University Libraries' HighWire Press®. Copyright 2006 by the International Anesthesia Research Society. Online ISSN: 1526-7598   Print ISSN: 0003-2999