Effects of alveolar dead-space, shunt and V/Q distribution on respiratory dead-space measurements

Y. Tang, M. J. Turner and A. B. Baker*

Department of Anaesthetics, University of Sydney, Royal Prince Alfred Hospital, Sydney, NSW 2050, Australia

* Corresponding author: Department of Anaesthetics, Royal Prince Alfred Hospital, Building 89 Level 4, Missenden Rd, Camperdown, NSW 2050, Australia. E-mail: bbaker{at}usyd.edu.au


    Abstract
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 Abstract
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 Introduction
 Methods
 Results
 Discussion
 Appendix A. Dead-space...
 Appendix B. Validation of...
 Appendix C. Calculation of...
 References
 
Background. Respiratory dead-space is often increased in lung disease. This study evaluates the effects of increased alveolar dead-space (VDalv), pulmonary shunt, and abnormal ventilation perfusion ratio (V/Q) distributions on dead-space and alveolar partial pressure of carbon dioxide () calculated by various methods, assesses a recently published non-invasive method (Koulouris method) for the measurement of Bohr dead-space, and evaluates an equation for calculating physiological dead-space (VDphys) in the presence of pulmonary shunt.

Methods. Pulmonary shunt, V/Q distribution and VDalv were varied in a tidally breathing cardiorespiratory model. Respiratory data generated by the model were analysed to calculate dead-spaces by the Fowler, Bohr, Bohr–Enghoff and Koulouris methods. was calculated by the method of Koulouris.

Results. When VDalv is increased, VDphys can be recovered by the Bohr and Bohr–Enghoff equations, but not by the Koulouris method. Shunt increases the calculated Bohr–Enghoff dead-space, but does not affect Fowler, Bohr or Koulouris dead-spaces, or VDphys estimated by the shunt-corrected equation if pulmonary artery catheterization is available. Bohr–Enghoff but not Koulouris or Fowler dead-space increases with increasing severity of V/Q maldistribution. When alveolar is increased by any mechanism, calculated by Koulouris' method does not agree well with average alveolar PCO2.

Conclusions. Our studies show that increased pulmonary shunt causes an apparent increase in VDphys, and that abnormal V/Q distributions affect the calculated VDphys and VDalv, but not Fowler dead-space. Dead-space and calculated by the Koulouris method do not represent true Bohr dead-space and respectively, but the shunt-corrected equation performs well.

Keywords: computers ; lung, shunting ; ventilation, deadspace ; ventilation, ventilation–perfusion


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PACO2=ideal alveolar carbon dioxide partial pressure; PaCO2=arterial carbon dioxide partial pressure; PAKCO2=PACO2 calculated by Koulouris method; PAXCO2=volume-weighted average of PCO2 in all the alveolar compartments in the model; PECO2=mixed expired carbon dioxide partial pressure; PE'CO2=end-tidal partial pressure of carbon dioxide; Qs/Q=pulmonary shunt fraction; VDalv=alveolar dead-space; VDanat=anatomical dead-space in the model; VDBE=Bohr–Enghoff dead-space calculated by substituting for in Bohr equation; VDBohr=Bohr dead-space; VDcorr=Bohr–Enghoff dead-space corrected for shunt; VDET=dead-space calculated by substituting for in Bohr equation; VDFowler=Fowler dead-space; VDK=dead-space calculated by Koulouris method; VDphys=physiological dead-space (total dead-space in the model); V/Q=ventilation to perfusion ratio; VT=tidal volume; VTalv=alveolar tidal volume (VT–VDFowler).


    Introduction
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 Abstract
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 Introduction
 Methods
 Results
 Discussion
 Appendix A. Dead-space...
 Appendix B. Validation of...
 Appendix C. Calculation of...
 References
 
Respiratory dead-space measurements have been used in determining surfactant efficacy in surfactant-depleted lungs,1 diagnosing pulmonary embolism,2 3 providing useful prognostic information early in the course of acute respiratory distress syndrome,4 predicting successful extubation in infants and children,5 and separating patients with asthma from patients with emphysema with the same degree of airways obstruction.6 All these clinical applications depend on accurate measurements of respiratory dead-space.

Bohr7 dead-space (VDBohr) is a function of ideal alveolar partial pressure of carbon dioxide (). Because is difficult to estimate, Enghoff8 substituted arterial partial pressure of carbon dioxide () for , giving rise to the Bohr–Enghoff dead-space (Appendix A), usually referred to as physiological dead-space (VDphys). Alveolar dead-space9 is commonly defined as the difference between VDphys and the anatomical dead-space (VDFowler), which is estimated by a method proposed by Fowler10 (Appendix A). Disadvantages of the Bohr–Enghoff method are that it is invasive and cannot be used breath-by-breath when is changing rapidly. Recently, Koulouris and colleagues11 reported a new non-invasive method to calculate Bohr dead-space and based on an analysis of the expired carbon dioxide volume vs expired tidal volume curve from a single expiration (Appendix A). This technique is apparently simple and non-invasive, but has not been validated independently.

Shunt reduces the overall efficiency of gas exchange and results in arterial blood gas tensions closer to those of mixed venous blood, thus increasing the measured apparent physiological dead-space by increasing .12 A method for correcting dead-space measurements for the effects of shunt has been reported by Kuwabara and Duncalf13 (Appendix A) but its validity has not been demonstrated.

Inhomogeneity of ventilation/perfusion (V/Q) ratio increases the measured alveolar dead-space by two mechanisms. Firstly, the venous admixture is increased from lung regions with low V/Q ratio; secondly, lung units with high V/Q ratio contribute to physiological dead-space.

Although series dead-space can be altered easily in studies in vivo, it is difficult to control changes in alveolar dead-space and the V/Q distribution. Thus the effects of changes in V/Q distribution on measures of respiratory dead-space have not been studied systematically.

The aims of this study were to assess the method of Koulouris and colleagues11 for calculating Bohr dead-space and alveolar PCO2, to demonstrate the validity of the correction proposed by Kuwabara and Duncalf13 for calculating physiological dead-space in the presence of pulmonary shunt, and to evaluate the effects of varying alveolar dead-space, pulmonary shunt and abnormal V/Q distributions on and dead-space calculated by five different methods.7 8 10 11 13


    Methods
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 Appendix A. Dead-space...
 Appendix B. Validation of...
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 References
 
The computer model
We used a comprehensive, mathematical, tidally breathing computer model of the cardiorespiratory system.14 The model incorporates a 15-compartment approximation to Weibel's lung model A15 and simulates diffusive and convective transport and storage of gases in the lungs. The branching airways comprise 11 anatomical dead-space compartments and terminate in one unperfused and three perfused alveolar compartments (Fig. 1). Carbon dioxide is stored in two lung tissue compartments.16 Ventilation and V/Q heterogeneity can be simulated by varying the inspired gas distribution and the fraction of the cardiac output perfusing the three perfused alveolar compartments. A variable right-to-left shunt is provided. The model simulates alveolar-capillary diffusion; intraventricular and intravascular mixing; variable transport delays; intravascular storage; and carbon dioxide and oxygen storage, production and consumption in eight anatomically appropriate body compartments. Molar quantities of oxygen, nitrogen and carbon dioxide are conserved in the model. Non-linear blood gas dissociation curves17 include the Haldane and Bohr effects. The tissue dissociation curves are after Farhi and Rahn18 and Cherniack and Longobardo.19 The respiratory flow waveform was selected to match a mechanically ventilated subject: constant inspiratory flow followed by exponential expiratory flow. The model is implemented in Matlab and Simulink (Mathworks, Natick, MA, USA) and has been verified against published human data.14 Additional validation is described in Appendix B.



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Fig 1 Block diagram of the airway and alveolar compartments in the comprehensive cardiorespiratory system model. The respiratory system incorporates a 15-compartment approximation to Weibel's lung model. WG=Weibel generations; Alv=alveolar; DS=dead-space.

 
Part 1. Effect of the ratio of physiological dead-space to tidal volume on respiratory dead-space measurements
The model was configured to simulate a 70 kg healthy adult male subject with respiratory parameters, as shown in Table 1. The ratio of the tidal volume that reaches alveolar dead-space (Alv DS in Fig. 1) to the alveolar tidal volume was set in turn to 1, 10, 20, 30, 40 and 50%. Alveolar tidal volume (VTalv) is the volume of fresh inspired gas that reaches the alveoli. Corresponding ratios of physiological dead-space to tidal volume in the model were 30, 36, 43, 51, 58 and 65%, respectively. The V/Q ratios in the three ventilated and perfused lung compartments were the same, and decreased as the dead-space increased. Total anatomical dead-space and total alveolar volume were kept constant. The volumes of the alveolar dead-space compartment and the associated anatomical dead-space compartments in the model were increased in proportion to the alveolar dead-space ventilation. The volumes of the perfused alveoli and their associated anatomical dead-space compartments were decreased in proportion to the decreasing ventilation directed to those alveoli. Under each condition, the model was run for 7200 s simulation time to achieve steady-state PCO2 and PO2 in mixed venous blood, and alveolar and body compartments. The data were sampled at 100 Hz and stored for off-line analysis. The last complete respiratory cycle in each 2-h run was analysed. The simulated subject was assumed to be anaesthetized and paralysed and was allowed to reach unphysiological levels.


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Table 1 Model parameters

 
Part 2. Effect of pulmonary shunt on respiratory dead-space measurements
In this part of the study the ratio of alveolar dead-space to alveolar tidal volume was kept constant at 1%, corresponding to a ratio of physiological dead-space to tidal volume of 30%. The tidal volume in the alveolar dead-space was 5.3 ml. The pulmonary shunt was set in turn to 2, 10, 20, 30, 40 and 50% of the total pulmonary blood flow and the other respiratory parameters were the same as in Part 1 (Table 1). V/Q ratios of all the perfused compartments were the same, and increased as the shunt increased. Under each condition the model was run for 7200 s of simulation time to reach steady state PCO2 and PO2. Data from the last complete respiratory cycle in each 2-h run were recorded and analysed.

Part 3. Effect of V/Q ratio heterogeneity on respiratory dead-space measurements
The ratio of alveolar dead-space to alveolar tidal volume and the pulmonary shunt were set to 1% and 2% respectively. The minute alveolar ventilation of 5.30 litre min–1 and the pulmonary perfusion of 5.40 litre min–1 were unevenly distributed to the three ventilated and perfused alveolar compartments to create V/Q values of 0.1, 1.0 and 10 to simulate patients with chronic obstructive pulmonary disease.20 The percentage perfusion to the middle V/Q compartment (V/Q=1) was set in turn to 98, 78, 58, 38 and 18% of pulmonary blood flow to simulate increasing severity of V/Q mismatch, and the ventilation and perfusion of all three ventilated and perfused compartments were calculated by the method described in Appendix C. The model was run for 7200 s simulation time at each setting to reach steady-state PCO2 and PO2 in mixed venous blood and alveolar and body compartments, and the respiratory parameters were measured and analysed. Log standard deviations of the perfusion distributions were calculated for each condition.

Data analysis
fluctuated during respiration and was therefore averaged over a complete respiratory cycle. is the volume-weighted average of the PCO2 in the three perfused and ventilated alveolar compartments averaged over one respiratory cycle in Parts 1 and 2 of this study. is the volume-weighted average of the PCO2 in all the alveolar compartments, including the alveolar dead-space compartment, averaged over one respiratory cycle. VDFowler was calculated by the equal area method (Appendix A).10 21 22 VDBohr (Parts 1 and 2 of this study) and VDBE (Parts 1, 2 and 3 of this study) were calculated according to Equations 1 and 2 (Appendix A) respectively. Bohr–Enghoff dead-space corrected for the effects of shunt (VDCorr) was calculated using Equation 3 (Appendix A). The calculation of Bohr dead-space by the method of Koulouris and colleagues11 (VDK) is described in Appendix A. An estimate of the Bohr dead-space was also calculated by substituting end-tidal carbon dioxide partial pressure () for (Equation 4, Appendix A). We refer to this dead-space as VDET. The physiological dead-space of the model (VDphys) was calculated as follows:

where VDanat is the volume of the anatomical dead-space compartments in the model and FDalv is the fraction of the tidal volume that enters the parallel dead-space compartment.


    Results
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 Introduction
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 Appendix A. Dead-space...
 Appendix B. Validation of...
 Appendix C. Calculation of...
 References
 
Part 1. Effect of the ratio of physiological dead-space to tidal volume on respiratory dead-space measurements
The V/Q ratios in the three ventilated and perfused alveolar compartments and the arterial partial pressures of carbon dioxide and oxygen resulting from increased alveolar dead-space are shown in Table 2. Increased alveolar dead-space affects more than . Figure 2A shows simulated and calculated PCO2 as functions of the ratio of physiological dead-space to tidal volume. and , predicted by the model, do not differ greatly and increase monotonically with increasing physiological dead-space fraction. and the associated measured values ( and ) decrease slightly with increasing physiological dead-space. The maximum difference between , and is less than 0.25 kPa (2 mmHg) under all conditions studied.


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Table 2 V/Q ratios of perfused alveolar compartments and steady state arterial blood gas tensions in parts 1 and 2 of this study. V/Q is the ventilation to perfusion ratio for the three ventilated and perfused compartments with equal V/Q

 


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Fig 2 (A) Carbon dioxide partial pressures and (B) measured ratios of dead-space volumes to tidal volume as functions of VDphys/VT (model physiological dead-space to tidal volume ratio). is arterial PCO2 predicted by the model. is volume and time-averaged PCO2 in the perfused alveoli of the model. is volume and time-averaged PCO2 in all the alveoli of the model. is end-tidal PCO2. is alveolar PCO2 calculated by Koulouris' method (Appendix A). VDBE is Bohr–Enghoff dead-space calculated by using Appendix Equation 2. VDBohr is Bohr dead-space calculated by using Equation 1. VDET: dead-space calculated by using Equation 4. VDK is dead-space calculated by Koulouris' method (Appendix A). VDFowler is anatomical dead-space calculated by Fowler's graphic method. Dashed straight line in B is the line of identity.

 
Figure 2B shows calculated dead-space to tidal volume ratios as functions of true physiological dead-space to tidal volume ratio. VDBohr and VDBE increase with increasing physiological dead-space. Both VDBohr and VDBE overestimate the model physiological dead-space when alveolar dead-space is small but slightly underestimate physiological dead-space when the model alveolar dead-space is large. VDK, VDFowler and VDET are approximately independent of alveolar dead-space, shunt and V/Q ratio heterogeneity and lie between approximately 29 and 35% under all conditions studied (Figs 2Go4).



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Fig 3 (A) Carbon dioxide partial pressures and (B) measured dead-spaces as functions of pulmonary shunt. Abbreviations are the same as in Fig. 2. VDcorr is the Bohr–Enghoff dead-space corrected by the Kuwabara and Duncalf13 equation in the presence of right-to-left shunt (Appendix Equation 3).

 


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Fig 4 (A) Carbon dioxide partial pressures and (B) measured dead-spaces as functions of the combined pulmonary perfusion of the high and low V/Q compartments. , and are PCO2 in the low V/Q, middle V/Q and high V/Q compartments respectively. Other abbreviations as in Fig. 2.

 
Part 2. Effect of pulmonary shunt on respiratory dead-space measurements
The ventilation and perfusion to the three ventilated and perfused alveolar compartments and the arterial partial pressure of carbon dioxide and oxygen when the shunt is increased are shown in Table 2. Shunt affects more than . increases with increasing pulmonary shunt, while , and the volume-averaged and fall slightly (Fig. 3A). The effect of pulmonary shunt on the partial pressure of carbon dioxide in the alveoli and in the arterial blood is less marked than the effect of directing similar proportions of tidal volume to alveolar dead-space.

The Bohr–Enghoff dead-space increases with increasing pulmonary shunt (Fig. 3B). VDCorr and VDBohr are greater than VDFowler but smaller than VDK and are approximately independent of shunt.

Part 3. Effect of V/Q ratio heterogeneity on respiratory dead-space measurements
The perfusion and ventilation of each compartment, the respiratory variables and the arterial partial pressures of carbon dioxide and oxygen are shown in Table 3. In the V/Q ratio heterogeneity study it was not appropriate to calculate Bohr dead-space due to the variation of PCO2 among the ventilated and perfused alveoli. and the PCO2 in each individual alveolar compartment increase with increasing heterogeneity of the V/Q ratio (Fig. 4A). The PCO2 of each alveolar compartment is inversely related to the V/Q ratio of the compartment. The VDBE/VT ratio increases from 30.5% at optimal V/Q distribution to 64.9% when 78% of the pulmonary perfusion is distributed to the compartments with V/Q of 0.1 and 10 (Fig. 4B).


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Table 3 Ventilation and perfusion parameters of perfused alveolar compartments in Part 3 of this study

 

    Discussion
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 Appendix B. Validation of...
 Appendix C. Calculation of...
 References
 
This study found that the Koulouris method for calculating Bohr dead-space overestimates slightly when alveolar dead-space is small but substantially underestimates Bohr dead-space when alveolar dead-space is increased. VDK appears to be no better than VDET as an estimate of Bohr dead-space under the conditions in this study. The Koulouris method for calculating alveolar PCO2 does not respond to changes in the PCO2 of ventilated and perfused alveoli caused by increasing alveolar dead-space. As respiratory dead-space has been widely studied and used in anaesthesia and in emergency and intensive care medicine, it is important that new methods for dead-space measurement should be validated independently before they are used clinically. Bohr dead-space as calculated by the Koulouris method is not accurate and thus the Koulouris method is not validated by this study.

Pulmonary embolism results in lung units that are poorly perfused but maintain approximately normal ventilation.23 Although pulmonary embolism is a complex pathological entity with mixed presentation of shunt and dead-space, the increased pulmonary dead-space in Part 1 of this study approximately simulates the main features of gas exchange in patients with pulmonary embolism. The Bohr–Enghoff dead-space accurately follows the increase in the model alveolar dead-space, while the Fowler dead-space is unaffected (Fig. 2B). Hence, calculated alveolar dead-space, one of the diagnostic markers of pulmonary embolism,23 increases. Dead-space calculated by the Koulouris method does not correlate well with alveolar dead-space or Bohr dead-space, suggesting that the Koulouris method cannot contribute to the diagnosis of pulmonary embolism.

Pulmonary shunt increases VDBE but does not affect VDFowler or VDBohr. The equation of Kuwabara and Duncalf13 (Equation 3, Appendix A) calculates model dead-space correctly in the presence of substantial shunt. Use of this correction equation requires pulmonary artery catheterization due to the need for measurement of shunt, mixed venous PCO2 and .

The invalidity of the Koulouris method can be demonstrated theoretically. The Bohr dead-space equation (Equation 1, Appendix A) assumes that expired gas emanates from two compartments: a perfused alveolar compartment and an unperfused dead-space. The Bohr equation makes no assumptions regarding the sequence in which gas from the two compartments is expired. In contrast, the Koulouris method (Fig. A1B in Appendix A) assumes that an expiration comprises two sequential volumes: a dead-space (VDK=ia) containing a volume of carbon dioxide [VCO2(d)=ay] is expired first at a mean carbon dioxide concentration of Fd=ya/ia, and the remaining carbon dioxide (ce') is assumed to be expired in a subsequent volume ae of alveolar gas at end-tidal carbon dioxide concentration FE'CO2=ce'/ae=ya/da. In Figure A1B in Appendix A the lines ia and id represent VDK and VDET respectively. Hence the line da, which represents the difference between VDK and VDET, can be expressed as VCO2(d)/FE'CO2. VDK and VDET are therefore related by:

where VCO2(d) is the volume of carbon dioxide expired in VDK. Similarly, it can be shown that and are related by:

where VCO2 is the total amount of carbon dioxide expired in one breath. It can be concluded that VDK and are always larger than VDET and respectively.



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Fig A1 Diagrammatic representations of the Fowler method for calculating anatomical dead-space and the method of Koulouris and colleagues11 for the calculation of respiratory dead-space and alveolar partial pressure of carbon dioxide2. (A) Expired carbon dioxide concentration as a function of expired volume. Areas p' and q' are equal and the perpendicular line intersects the x-axis at the Fowler dead-space. (B) Expired carbon dioxide volume as a function of expired volume. Areas p and q are equal. According to Koulouris and colleagues,11 the slope of line ca is an estimate of alveolar carbon dioxide concentration and the lengths ia and ae are estimates of Bohr dead-space and alveolar volume respectively.

 
Alveolar PCO2 increases during expiration and reaches a peak shortly after end-expiration. In vivo, is dominated by end-expiratory alveolar gas and can be greater than , particularly during exercise or when tidal volumes are large.24 26 Fletcher and colleagues27 found zero or negative arterial-to-end-tidal PCO2 differences in 12% of non-pregnant patients during anaesthesia in which large tidal volumes and low respiratory rates were used. In the method of Koulouris and colleagues,11 the relationship between VDK and VDET is fixed and independent of tidal volume, respiratory rate and alveolar dead-space, so end-tidal PCO2 can never exceed .

In this study the simulated subject was assumed to be anaesthetized and paralysed and was allowed to rise as high as 8.1 and 9.2 kPa when alveolar dead-space and V/Q ratio heterogeneity respectively were increased. In a study of COPD patients with severe V/Q heterogeneity, Conti and colleagues28 reported values as high as 11.6 kPa in mechanically ventilated patients. Breen and colleagues29 studied the effects of large pulmonary embolism on carbon dioxide kinetics and physiological dead-space. They found that after 70 min of occlusion of a large pulmonary artery, the increased from 5.5 to 7.3 kPa and decreased by 13% of baseline value while physiological dead-space increased from 31 to 52%. Our simulation results are consistent with the in vivo observations of Conti and colleagues28 and Breen and colleagues.29

We used a mathematical model for this study to facilitate the controlled variation of alveolar dead-space, anatomical dead-space, pulmonary shunt and V/Q ratio distribution, which are difficult to change prospectively in in vivo studies. A computer model study also avoids the confounding effects associated with biological variations and measurement errors. The main limitations of our computer model include the lumped approximation of the respiratory tree and alveoli, the approximations used to estimate diffusion and convection in the airways and the assumption of equal respiratory time constants and consequent simultaneous emptying of alveolar compartments. The model does not automatically redistribute ventilation or perfusion when these parameters are perturbed and does not simulate hypoxic pulmonary vasoconstriction. These approximations and limitations may affect the shape of the expirogram and hence the calculated dead-space and PCO2 values. We expect, however, that the limitations of the model affect only the magnitude of the results, not their form or direction.

In conclusion, our simulation results suggest that while the physiological dead-space is estimated well by the Bohr–Enghoff equation when alveolar dead-space and V/Q ratio distribution vary, respiratory dead-space and alveolar carbon dioxide partial pressure calculated by the Koulouris method do not represent the true Bohr dead-space or alveolar carbon dioxide partial pressure. Increasing pulmonary shunt can cause an apparent increase in VDphys, and abnormal V/Q distributions affect calculated VDphys and VDalv, but not Fowler dead-space. The equation suggested by Kuwabara and Duncalf13 for the calculation of dead-space in the presence of shunt performs well, but requires invasive measurements.


    Appendix A. Dead-space calculation methods and symbols
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 Abstract
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 Methods
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 Appendix A. Dead-space...
 Appendix B. Validation of...
 Appendix C. Calculation of...
 References
 
Fowler dead-space
On an expired carbon dioxide concentration vs expired volume curve (Fig. A1A), a straight line is fitted to the alveolar plateau between 60 and 90% of expired volume by linear regression.6 A vertical line is drawn from the regression line to the x-axis to divide the rapidly rising part of the carbon dioxide expirogram into two equal areas (p' and q'). The intersection of the perpendicular line and the x-axis is the anatomical or Fowler dead-space.10 22

Bohr dead-space
Bohr dead-space (VDBohr) is the dead-space calculated by the original Bohr equation:7 21

(1)
Where is the mixed expired carbon dioxide partial pressure and is the ideal alveolar carbon dioxide partial pressure. In Parts 1 and 2 of this study, is the volume-weighted average of the PCO2 in the three perfused and ventilated alveolar compartments averaged over one respiratory cycle.

Bohr–Enghoff dead-space
Because of the controversy concerning the definition and estimation of ,21 Enghoff8 suggested substituting for in the Bohr equation. The dead-space so calculated is termed the Bohr–Enghoff dead-space (VDBE):

(2)

Shunt correction method
In the presence of right-to-left shunt, venous blood mixes with pulmonary capillary blood and raises , thus increasing the difference between and . Hence the Bohr–Enghoff dead-space calculated by Equation 2 is increased. Kuwabara and Duncalf13 applied simple mass balance principles and derived an equation to estimate a corrected physiological dead-space (VDcorr) in the presence of right-to-left shunt.

(3)
where QS/QT is the shunt fraction, and is the mixed venous PCO2.

VDET
Dead-space calculated by using in place of in the Bohr equation (Equation 1) is termed VDET in this paper:

(4)

Koulouris dead-space
Figure A1B shows expired carbon dioxide volume vs expired volume. Line cb is drawn such that areas p and q are equal. Point d is chosen such that the slope of line cd is end-tidal concentration. The volume de is the volume the expired carbon dioxide would occupy at end-tidal carbon dioxide concentration (total volume of carbon dioxide divided by end-tidal carbon dioxide concentration), hence id represents VDET as calculated by Equation 3 above. Line dx is perpendicular and intersects cb at x. Line xy is parallel to the x-axis and intersects cd at y. Line ya is perpendicular and intersects the x-axis at a. According to Koulouris and colleagues,11 lines ee' and e'c represent the quantities of carbon dioxide expired in the dead-space and alveolar gas respectively. The line ae represents alveolar tidal volume, line ia represents Bohr dead-space and the slope of line ac is the alveolar concentration of carbon dioxide.


    Appendix B. Validation of the cardiorespiratory model
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 Appendix B. Validation of...
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 References
 
The model was validated in a clinical study and by comparison of predicted PCO2 with published measurements.

Figure A2A shows the realistic airway, arterial, alveolar and mixed venous PCO2 changes with time in the tidally breathing model. The model simulated a 70-kg male subject who was mechanically ventilated with a tidal volume of 9 ml kg–1 at a respiratory rate of 10 bpm. Other parameters are the same as shown in Table 1. Both arterial and alveolar PCO2 fluctuate during the respiratory cycle but arterial PCO2 lags alveolar PCO2. Also demonstrated is that alveolar PCO2 peaks shortly after the end of expiration. The average mixed venous and arterial PCO2 are in agreement with the literature.30 31



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Fig A2 Validation of the computer model. (A) Simulation of airway, arterial, alveolar and mixed venous PCO2 during respiratory cycles. The circles and triangles are mixed venous and arterial PCO2 recorded from in vivo studies.30 31 (B) Dynamic changes with different combinations of tidal volume and respiratory rate. The dots are recorded from the patient and the solid line is the model prediction. (C) Effects of tidal volume and respiratory rate on the measured and predicted and . The carbon dioxide values predicted by the model closely follow those recorded in the patients. RR and VT are respiratory rate and tidal volume respectively.

 
Figure A2B shows the dynamic change of measured and predicted following changes in tidal volume and frequency. After ethics committee approval and informed consent from the patient, a 59-yr-old 77-kg male patient undergoing vascular surgery was studied. The tidal volume and respiratory rate were set to: 950 and 15; 750 and 7; 500 and 6; 850 ml and 10 bpm respectively and in sequence. The was recorded for each expiration. The model was set to simulate a 75-kg male subject undergoing mechanical ventilation with the same combinations of tidal volume and respiratory rate. The results show that the model simulation closely represents the dynamic change of recorded from the patient (Fig. A2B).

Figure A2C shows the model's prediction of and and comparison with a published study.32 In that study, 12 patients were ventilated with tidal volumes of 10, 7.5, 5 and 2.5 ml kg–1 and respiratory rates of 10, 13, 21 and 40 bpm respectively. Each setting was maintained for 10 min and and were measured and recorded.32 The model was set to simulate a 75-kg male subject ventilated with the same tidal volumes and respiratory rates as the patients. Other parameters are the same as in Table 1. The anatomical dead-space was adjusted as a function of tidal volume.33 At each setting, the model was run for 10 min and the and were recorded.32 The average absolute error of the model predictions was 0.94% of the mean and 9.1% of the SD for measured , and 4.4% of the mean and 28.4% of the SD for measured . This result is comparable with a model study by Hardman and Aitkenhead.34

Our results show that the model simulates realistically the gas exchange of a human lung both dynamically and in steady state.


    Appendix C. Calculation of ventilation and perfusion to the three perfused and ventilated compartments in the model
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 Symbols used in the...
 Introduction
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 Appendix A. Dead-space...
 Appendix B. Validation of...
 Appendix C. Calculation of...
 References
 
In Part 3 of this study the ventilation and perfusion of the three ventilated and perfused alveolar compartments, V1, V2 and V3 and V3 and Q1, Q2 and Q3 litre min–1 respectively, were varied to simulate lungs with various degrees of V/Q inhomogeneity. The V/Q ratios of the ventilated and perfused compartments were fixed as follows:

(5)

(6)

(7)

The total ventilation V, total perfusion Q, shunt blood flow (Qs) and alveolar dead-space ventilation (VDalv) were known, therefore:

(8)

(9)

There were six unknown parameters in five equations (Equations 5 GoGoGo9). We assigned values to Q2 and calculated the remaining five unknown parameters (V1, V2, V3, Q1 and Q3) by solving Equations 5GoGoGo9 simultaneously. We used software written in Matlab (Mathworks, Natick, MA, USA) to solve the equations and hence determine the fractions of ventilation and perfusion directed to each alveolar compartment.


    Acknowledgments
 
The authors are pleased to acknowledge the financial support of an Australian Research Council ‘Strategic Partnership with Industry—Research and Training’ (ARC-SPIRT) grant, Dräger Australia Pty Ltd, The Joseph Fellowship, The University of Sydney and the Australian National Health and Medical Research Council (NHMRC).


    References
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 Abstract
 Symbols used in the...
 Introduction
 Methods
 Results
 Discussion
 Appendix A. Dead-space...
 Appendix B. Validation of...
 Appendix C. Calculation of...
 References
 
1 Wenzel U, Rudiger M, Wagner MH, Wauer RR. Utility of deadspace and capnometry measurements in determination of surfactant efficacy in surfactant-depleted lungs. Crit Care Med 1999; 27: 946–52[CrossRef][ISI][Medline]

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