## Abstract

The lack of methodology for measuring the alveolar carbon dioxide tension (*P*_{A,CO2}) has forced investigators to make several assumptions, such as that *P*_{A,CO2} is equal to end-tidal (*P*_{ET,CO2}) and arterial CO_{2} tension (*P*_{a,CO2}).

The present study measured the mean *P*_{A,CO2} and Bohr's dead space ratio (Bohr's dead space/tidal volume (*V*_{D,Bohr}/*V*_{T})) during tidal breathing. The method used is a new, simple and noninvasive technique, based on the analysis of the expired CO_{2} volume per breath (*V*_{CO2}) *versus* the exhaled *V*_{T}. This curve was analysed in 21 normal, healthy subjects and 35 chronic obstructive pulmonary disease (COPD) patients breathing tidally through a mouthpiece apparatus in the sitting position.

It is shown that: 1) *P*_{A,CO2} is similar to *P*_{a,CO2} in normal subjects, whilst it is significantly lower than *P*_{a,CO2} in COPD patients; 2) *P*_{A,CO2} is significantly higher than *P*_{ET,CO2} in all subjects, especially in COPD patients; 3) *V*_{D,Bohr}/*V*_{T} is increased in COPD patients as compared to normal subjects; and 4) *V*_{D,Bohr}/*V*_{T} is lower than the “physiological” dead space ratio (*V*_{D,phys}/*V*_{T}) in COPD patients.

It is concluded that the expired carbon dioxide *versus* tidal volume curve is a useful tool for research and clinical work, because it permits the noninvasive and accurate measurement of Bohr's dead space and mean alveolar carbon dioxide tension accurately during spontaneous breathing.

- alveolar carbon dioxide tension
- arterial carbon dioxide tension
- Bohr's dead space
- chronic obstructive pulmonary disease
- end-tidal carbon dioxide
- physiological dead space

The respiratory dead space is the concept in gas exchange derived by the investigators in their effort to determine the effectiveness of ventilation in health and disease. After the description of the dead space by Bohr 1, numerous papers on the subject followed, in which the methodology can be divided into two categories; the noninvasive studies from gas (nitrogen (N_{2}), Helium (He), carbon dioxide (CO_{2})) concentration *versus* time or volume curves, and the invasive studies in which the arterial CO_{2} tension (*P*_{a,CO2}) instead of the alveolar CO_{2} tension (*P*_{A,CO2}) was used 2–15. In the noninvasive methods, the “anatomical” dead space, *i.e.* Fowler's dead space (*V*_{D(F)}), is determined from the expired gas concentration *versus* tidal volume or vital capacity curve, which is analysed by geometrical methods. The results obtained by this method may be doubtful since the junction of the phases II and III is difficult to define in disease, especially during tidal breathing. Furthermore, this analysis is based on the assumption that the end-tidal and alveolar CO_{2} fractions (*F*_{ET,CO2} and *F*_{A,CO2}) are identical. However, there is substantial evidence that *F*_{ET,CO2} is lower than *F*_{A,CO2} in normal subjects and patients 16–18. The invasive methods permit the measurement of the “physiological” dead space ratio (physiological dead space/tidal volume (*V*_{D,phys}/*V*_{T})), by using *P*_{a,CO2} in Bohr's equation with the assumption that *P*_{A,CO2} is equal to *P*_{a,CO2}, which is valid only in normal subjects.

In previous reports 17, 18, Bohr's dead space ratio (Bohr's dead space/tidal volume (*V*_{D,Bohr}/*V*_{T})) and *P*_{A,CO2} were not measured either simultaneously or within the volume domain. Since *V*_{D,Bohr}/*V*_{T} is in the volume domain, theoretically it appeared most appropriate to develop a new technique, *i.e.* the construction and mathematical analysis of the expired CO_{2} volume *versus* tidal volume curve (*V*_{CO2} *versus* *V*_{T} curve). This curve, recorded at the mouth during expiration, has a curvilinear shape and the CO_{2} concentrations within the airways are lower than the alveolar one as a result of the “dilution effect” due to the pre-inspired atmospheric air (Appendix 1).

This technique allowed the simultaneous measurement of *V*_{D,Bohr}/*V*_{T} and *P*_{A,CO2}. This simple and noninvasive method was applied in 21 normal subjects and 35 chronic obstructive pulmonary disease (COPD) patients breathing tidally through a mouthpiece apparatus. *V*_{D,Bohr}/*V*_{T} was compared to *V*_{D,phys}/*V*_{T}, and *P*_{A,CO2} to *P*_{a,CO2} and end-tidal carbon dioxide tension (*P*_{ET,CO2}).

## Methods

### Theoretical considerations

The *V*_{CO2} *versus* *V*_{T} curve was derived from the expiratory flow and CO_{2} concentration *versus* time tracings measured at the mouth. It was constructed by plotting the exhaled *V*_{CO2} (the integral of CO_{2} fraction and flow with respect to time (*V*_{CO2}=∫*F*_{CO2} *V*′_{dt})) *versus* the tidal volume (the integral of flow with respect to time (*V*_{T}=∫*V*′_{dt})). The *F*_{ET,CO2} was determined, by computer analysis, from the mean of 10 points of the last segment on the *F*_{CO2} *versus* time curve, at which the positive slope of the tangent with the horizontal line becomes zero. Beyond these points the curve started to have a consistent negative slope. The height of the mean of these points from the zero line of the curve represents the *F*_{ET,CO2}. The mixed expired CO_{2} fraction (*F*_{E,CO2}) is the ratio of the total expired *V*_{CO2} per breath over *V*_{T} (Appendix 1). The analysis of the *V*_{CO2} *versus* *V*_{T} curve is described in detail in the Appendix section.

### Study design

The experimental set-up consisted of a flanged semirigid plastic mouthpiece connected in series to a Fleisch No. 2 flow transducer head (Fleisch, Lausanne, Switzerland) *via* a metal piece (monitoring ring), on which the CO_{2} probe was attached (mouthpiece apparatus). The pneumotachograph (transducer and amplifier: Gould, Godart BV; No. 17212, Bilthoven, Holland) was connected with the Fleisch head *via* two semirigid plastic tubes 50 cm in length. The pneumotachograph system (rise time 10–90%=13 ms) was linear over the range of flows used. Volume was obtained by integration of the flow signal. An infrared capnograph (Jaeger; CO_{2} test III, Wuerzburg, Germany) (rise time 10–90%=100 ms) was connected to the monitoring ring through a thin polythene tube (length 50 cm, internal diameter 1.2 mm). The resistance of the mouthpiece apparatus to airflow was negligible. The rise time (10–90%) of the capnograph measured at the mouthpiece was ≥4.5 times faster than that of the fastest *F*_{CO2} *versus* time curve (*F*_{CO2}/*t*), in normal subjects and COPD patients breathing at a frequency of 10–25·min^{−1}. Calibration of the CO_{2} analyser was made using a standard mixture of CO_{2} (4.0%) in N_{2}. The phase lag between the *F*_{CO2} *versus* *t* and *V*′ *versus* *t* signals was determined by an abrupt change in flow of the above gas mixture generated through the experimental set-up. The measurement of the phase lag and the calibration of the CO_{2} analyser were repeated three times and the mean values were used. Airflow and CO_{2} signals were monitored on-line on a computer screen and sampled simultaneously at a rate of 150 Hz using a computer data acquisition system with a built-in 12-bit analogue-to-digital converter (National Instruments, AT-M10, Austin, Texas, USA). Collected data were stored on computer disk for subsequent analysis with custom-made computer analysis software. *V*_{CO2} and *V*_{T} were expressed in mL body temperature and pressure, saturated (BTPS).

The study was performed in 21 normal subjects and 35 ambulatory COPD patients. Lung function data were obtained in the seated position with a flow-sensing spirometer (Fukuda; Spiroanalyzer ST300, Tokyo, Japan). Anthropometric and routine lung function data are given in table 1⇓. Predicted values were those of Morris *et al*. 19. The subjects were studied while seated, breathing room air through the mouthpiece apparatus with a noseclip on, at their own resting *V*_{T} and respiratory frequency. Each subject had an initial 10–15 min trial run to become accustomed to the apparatus and procedure. After regular breathing had been achieved, a series of breaths over a period of 1 min were recorded. At the end of the recording time, while the subject was still connected to the mouthpiece, an arterial blood sample (>1 mL) was taken for gas analysis. An expert physician using a 21 G needle, performed a quick (5–10 s) and direct puncture of the brachial artery. It is highly unlikely that a change in blood gases took place in such a short time interval. If the procedure of gas sampling was not successful after one single effort, the experiment was cancelled. The cancelled experiments were <7.5%. The *P*_{a,CO2} was measured with a blood gas analyser (CIBA-CORNING; 288 Blood gas system, MA, USA) in 12 normal subjects and in all COPD patients.

The method was experimentally verified in three normal subjects during tidal breathing through different tubes of a known capacity. The dead space of the added tube (*V*_{tube}) was calculated from the difference *V*_{D}−*V*_{D(o)}, where: *V*_{D} and *V*_{D(o)} are *V*_{D,Bohr} of the subject breathing through the mouthpiece apparatus with and without the added tube, respectively. Three tubes were used, the capacities (*V*_{cap}) of which were 180, 337 and 504 mL calculated from the equation π×r^{2}×l (π=3.14, r=radius and 1=length of the tube). The capacity of the tube deviated from the measured volume by <2.3% (table 2⇓).

The study had the approval of the local ethics committee and all subjects gave informed consent.

## Results

*P*_{A,CO2} and *V*_{D,Bohr}/*V*_{T} were measured by analysis of the *V*_{CO2} *versus* *V*_{T} curve obtained from 21 normal subjects and 35 COPD patients during tidal breathing. It is noted that cardiogenic oscillations had no effect on the *V*_{CO2} *versus* *V*_{T} curve, as this was consistently smooth in all subjects (Appendix 1). *V*_{D,Bohr}/*V*_{T}, *V*_{T}, *P*_{ET,CO2} and *P*_{A,CO2}, were obtained for each subject by averaging all breaths during a 1-min data recording period.

The mean within-study, within-day and day-to-day coefficient of variation for *V*_{D,Bohr}/*V*_{T} was 6.5, 6.85 and 7.25% and for *P*_{A,CO2} 1.57, 3.06, and 3.05%, respectively. These were determined in three normal subjects in whom measurements were repeated three times per day for 3 consecutive days.

*V*_{D,Bohr}/*V*_{T} and *V*_{D,phys}/*V*_{T} were not significantly different in the 12 normal subjects. In contrast, this difference was statistically significant in the COPD patients (p<0.001). The *V*_{D,Bohr}/*V*_{T} ratio in COPD patients was significantly higher than in normal subjects (p<0.001), (table 3⇓; fig. 1⇓). *V*_{D,Bohr}/*V*_{T} was higher than the dead space ratio measured from the *F*_{CO2} *versus* *V*_{T} curve by Fowler's geometrical method of orthogonal projection (*V*_{D(F)}/*V*_{T}). In normal subjects, mean±sd *V*_{D(F)}/*V*_{T} was 28±8% and in COPD patients, 37±8%. The mean difference between *V*_{D,Bohr}/*V*_{T} and *V*_{D(F)}/*V*_{T} is 5±1% in normal subjects (p<0.001) and 7±2% in COPD patients (p<0.001).

*P*_{A,CO2} and *P*_{a,CO2} were compared in 12 normal subjects and in all patients. In the COPD patients, *P*_{a,CO2} was significantly higher than *P*_{A,CO2} (p<0.001). In the 12 normal subjects, the difference between *P*_{a,CO2} and *P*_{A,CO2} was not statistically significant (table 4⇓; fig. 2⇓). In all normal subjects and patients, *P*_{A,CO2} was significantly higher than *P*_{ET,CO2} (table 4⇓, fig. 3⇓). The relationship between *P*_{ET,CO2} and *P*_{A,CO2} is shown in figure 3⇓.

The alveolar-end-tidal *P*_{CO2} ((A-ET) CO_{2}) and the arterial-alveolar *P*_{CO2} ((a-A) CO_{2}) differences were also related to the *V*_{D,Bohr}/*V*_{T} ratio. In all subjects, no statistical relationship was found between (a-A) CO_{2} and *V*_{D,Bohr}/*V*_{T}. In contrast, the (A-ET) CO_{2} was significantly related to the *V*_{D,Bohr}/*V*_{T} ratio in both groups of subjects, *i.e.* (A-ET) CO_{2}=−0.050+(*V*_{D,Bohr}/*V*_{T}) (kPa) (r=0.79, SEE=0.072, p<0.001) in normal subjects and (A-ET) CO_{2}=−0.261+1.852×(*V*_{D,Bohr}/*V*_{T}) (kPa) (r=0.83, SEE=0.122, p<0.001) in COPD patients.

The validity of the analysis of the *V*_{CO2} *versus* *V*_{T} curve was also examined by calculating the expired *V*_{CO2} per breath from 1) the product *F*_{A,CO2}×alveolar ventilation (*V*′_{A}) (=*V*_{CO2(A)}), and 2) the equation *V*_{CO2(B)}=*F*_{sl}×*V*_{d}+*F*_{ET,CO2}×alveolar volume (*V*_{A}). The mean error between *V*_{CO2(A)} and *V*_{CO2(B)} was −0.02±1% in the normal subjects and 0.4±1% in the COPD patients (Appendix 3). The area A_{(A)} differed from the area A_{(B)} by 4±0.5% in normal subjects and 3±2% in COPD patients (Appendix 3).

## Discussion

The present study used the expired *V*_{CO2} *versus* the exhaled *V*_{T} curve for the noninvasive measurement of *V*_{D},Bohr/*V*_{T} and mean *P*_{A,CO2} in normal subjects and COPD patients during tidal breathing. According to the results: 1) *V*_{D,Bohr}/*V*_{T} is increased in COPD patients as compared to normal subjects; 2) *V*_{D,Bohr}/*V*_{T} is lower than *V*_{D,phys}/*V*_{T} in COPD patients; 3) *P*_{A,CO2} is closely similar to *P*_{a,CO2} in normal subjects, whilst it is significantly lower than *P*_{a,CO2} in COPD patients; and 4) *P*_{A,CO2} is significantly higher than *P*_{ET,CO2} in all subjects, especially in COPD patients. This curve overcomes: 1) the assumption inherent in the analysis of the *F*_{CO2} *versus* *V* curve, *i.e.* that *F*_{A,CO2} is equal to *F*_{ET,CO2}; and 2) the difficulty in drawing the extrapolated line of the sloping alveolar plateau in disease, especially during tidal breathing.

At a steady state, regardless of the actual value of the *V*′_{A}/perfusion (*Q*′) ratio, the CO_{2} molecules within the residual air define *F*_{A,CO2}. This is dependent on the dynamic equilibrium within the alveolar space between the inflow and outflow of the CO_{2} molecules, and the overall *V*′_{A}/*Q*′ ratio of the lungs at the existing functional residual capacity. A portion of the CO_{2} molecules is exhaled with the *V*_{T} constituting the expired *V*_{CO2} per breath. During expiration, these molecules move out from the alveolar space mainly by bulk movement but also by diffusion, and at the same time they are replaced by CO_{2} molecules originating from blood through the alveolar membrane. The concentrations of the CO_{2} molecules within the airways, diluted by pre-inspired atmospheric air, are lower than the alveolar concentration. As a result of the “dilution effect”, the *V*_{CO2} *versus* *V*_{T} curve gets the curvilinear shape recorded at the mouth (Appendix 1).

The analytical procedure of the *V*_{CO2} *versus* *V*_{T} curve was verified, both theoretically and practically *i.e.* 1) The measurement of the volume of the added tubes with a deviation of <2.3% from the capacity of the tubes denotes the validity and the accuracy of the described method. The added tube affects the entire *V*_{CO2} *versus* *V*_{T} curve and its volume was calculated from the change of *V*_{D,Bohr}. 2) The nonsignificant difference in normal subjects between *P*_{A,CO2} and *P*_{a,CO2} is strong evidence for the accuracy of the method. 3) The Equations 9 and 10 in Appendix 1, derived from the *V*_{CO2} *versus* *V*_{T} curve, are identical to those widely accepted in the literature 1–18. 4) The mean error between *V*_{CO2(A)} and *V*_{CO2(B)} is <0.4%, *i.e.* *V*_{CO2(A)}=*V*_{CO2(B)}. This equality means that the points *b*, *d* and *a* are correctly positioned and not arbitrarily taken (Appendix 1, 3). Furthermore, the mean deviation of the area A_{(A)} from the area A_{(B)} is very small (Appendix 3).

The repeatability of the measurements for *V*_{D,Bohr}/*V*_{T} was ∼7% and for *P*_{A,CO2} ∼3%. However, the increase of the coefficient of variation of the *V*_{T} *per se* beyond the value of 16% reduces the repeatability especially of *V*_{D,Bohr}/*V*_{T}.

*V*_{D,Bohr}(=segment *ia*; Appendix 1) is higher than *V*_{D(F)}, measured from the *F*_{CO2} *versus* *V* curve (Fowler's method), by the amount of the volume segment *V*_{da} (=*V*_{CO2(d)}/*F*_{ET,CO2}), if the analysis of the *F*_{CO2} *versus* *V* curve is performed according to the method of orthogonal projection of the curve (Appendix 2). So, the ratio *V*_{D,Bohr}/*V*_{T} is higher than usually referred to in the literature 1–18. If, however, the *F*_{CO2} *versus* *V* curve is analysed by the method of the “sloping alveolar plateau”, the difference between *V*_{D,Bohr} and *V*_{D(F)} is even higher (Appendix 2).

The alveolar CO_{2} and the mixed expired CO_{2} concentrations are the two ends of a spectrum of gas fractions influenced by several mechanisms affecting the homogeneity of ventilation. According to the law of conservation of mass, the product *F*_{E,CO2}×*V*_{T} is equal to *F*_{A,CO2}×*V*_{A} or to the intermediate products *F*_{sl}×*V*_{be} or *F*_{ET,CO2}×*V*_{de} (Appendix 1). That is, *V*_{CO2} per breath is exhaled with the *V*_{T} at the mixed expired gas concentration or with smaller volumes at increased gas concentrations until the minimal volume (alveolar, *V*_{A}) at the highest gas concentration (*F*_{A,CO2}). Mean *F*_{A,CO2} cannot be greater than a maximal value determined by the overall *V*′_{A}/*Q*′ ratios of the lungs.

The real *V*_{CO2} *versus* *V*_{T} curve functionally can be represented by the line *iac* (Appendix 1). The segment *ia* is *V*_{D,Bohr}, which if considered without CO_{2} gas, then the segment *ae* (=projection of the line *ac* on the horizontal axis with the angle *cae* (=*F*_{A,CO2})) is the mean *V*_{A} transferring out all the *V*_{CO2} per breath with the maximal concentration (*F*_{A,CO2}).

*V*_{D,Bohr}/*V*_{T} is considered as an index of maldistribution of the expired air within the lungs, *i.e.* within the space between the inner surface of the alveolar membrane and the mouth. The *V*_{D,phys}/*V*_{T} is influenced not only by the mechanisms of uneven ventilation, but also by the mechanisms of inhomogeneous distribution of *Q*′. In normal subjects, in whom *P*_{a,CO2} is approximately equal to *P*_{A,CO2}, the difference between *V*_{D,phys}/*V*_{T} and *V*_{D,Bohr}/*V*_{T} was not statistically significant. This may be a true result or is more likely due to the small number of observations (power of paired t-test=0.201). In the COPD patients, in whom *P*_{a,CO2} was higher than *P*_{A,CO2}, the difference between the two dead space ratios was considerable (table 3⇓).

*P*_{a,CO2} in normal subjects was not significantly different from *P*_{A,CO2}. However, in COPD patients (a-A) CO_{2} was significantly higher than in normal subjects (table 4⇓). This may be explained as follows. An increased *P*_{A,CO2} in regions with low *V*′_{A}/*Q*′ ratio is followed by an increase of the end-capillary *P*_{CO2} (*P*_{c,CO2}) locally, while in regions with a high *V*′_{A}/*Q*′ ratio, the decrease of *P*_{A,CO2} is accompanied by a local decrease of *P*_{c,CO2}. If the arterial blood is composed mainly from blood perfusing regions with a low *V*′_{A}/*Q*′ ratio, *P*_{a,CO2} will be increased. At the same time, when the exhaled *V*_{T} contains air coming mostly from regions with high *V*′_{A}/*Q*′ ratio, *P*_{A,CO2} will be decreased. The combination of these two conditions probably results in the increased (a-A) CO_{2} in patients with COPD. This is compatible with the results obtained, *i.e.* that the (a-A) CO_{2} is not statistically related to *V*_{D,Bohr}/*V*_{T} in the subjects studied. In contrast, (A-ET) CO_{2} is significantly related to *V*_{D,Bohr}/*V*_{T} in all subjects, due to the existing inhomogeneity of ventilation, especially in the COPD patients. The (A-ET) CO_{2} was higher in COPD patients than in normal subjects (table 4⇓). In addition, *P*_{A,CO2} was linearly related to *P*_{ET,CO2}, and *P*_{A,CO2} was higher than *P*_{ET,CO2} in all subjects (fig. 3⇓). The (A-ET) CO_{2} difference was strongly related to *V*_{D,Bohr}/*V*_{T} in normal subjects and COPD patients, as is described in Equation 11 (Appendix 1).

*P*_{A,CO2} is *V*_{CO2}/*V*_{A} times the factor (barometric pressure-47), as conventionally referred to in the literature. It is mentioned that the measured value of *P*_{A,CO2} per breath is a mean value from all the regions of the lungs with different *V*′_{A}/*Q*′ ratios. Furthermore, the values of *P*_{A,CO2} shown in the Results section are mean values from all breaths during the 1-min sampling period.

It is concluded that the carbon dioxide output *versus* tidal volume curve obtained during tidal breathing with minimal cooperation on the patient's part, is useful in clinical practice and research work. It allows, with accuracy and precision, the noninvasive measurement and monitoring of the mean alveolar carbon dioxide tension and Bohr's dead space volume. The alveolar carbon dioxide tension can be safely used instead of the arterial one in normal subjects, but not in chronic obstructive pulmonary disease patients. In all subjects, end-tidal carbon dioxide tension cannot be used instead of alveolar carbon dioxide tension.

## Appendix

1. The expired carbon dioxide volume *versus* tidal volume curve

The simplified analysis of the *V*_{CO2} *versus* *V*_{T} curve, presented in geometrical terms, is as follows. The total area under the *V*_{CO2} *versus* *V*_{T} curve (E) is equal to the area of the triangle *bce.* In either side of the line *bc* (hypotenuse) the areas K and M are equal to each other (fig. 4⇑). Accordingly, the volume segment *be* (*V*_{be}) on the *V*_{T} axis is equal to (fig. 5⇑):

The angle *cbe* represents the average slope (*F*_{sl}=*V*_{CO2}/*V*_{be}) of the *V*_{CO2} *versus* *V*_{T} curve (figs. 4 and 5⇑⇑). *F*_{ET,CO2} is measured directly at the end of the *F*_{CO2} *versus* time curve. The ratio *V*_{CO2}/*F*_{ET,CO2} determines the volume segment *de* (*V*_{de}) on the horizontal axis (fig. 5⇑), *i.e.*

The line *cd*, the volume segment *id* and the curve itself confine the one-sided area D, which is equal to the area of the triangle *bcd* (fig. 5⇑), *i.e.*where: (*V*_{be}-*V*_{de}) is the base (volume segment *bd*=*V*_{d}) of the triangle *bcd*. The area D denotes that a part of *V*_{CO2} (*V*_{CO2(d)}) is exhaled at smaller concentrations than *F*_{ET,CO2}. The gas volume *V*_{CO2(d)} (segment *xd*) is calculated from the average slope of the curve (*F*_{sl}) and the volume segment *V*_{d} (fig. 5⇑), *i.e.*

The gas volume *V*_{CO2}, as already described, is expired in two parts, the initial one (*V*_{CO2(d)}) with a mean concentration *F*_{d} and the rest (*V*_{CO2}-*V*_{CO2(d)}) with concentration equal to *F*_{ET,CO2} (fig. 6⇑). The meeting point (y) of these two slopes (*F*_{d} and *F*_{ET,CO2}) lies on the line *cd* and the gas volume segments *xd*, *ya* and *ee*′ are equal to each other (=*V*_{CO2(d)}) (figs. 5 and 6⇑⇑). The volume segment *ye*′* is* equal to the segment *ae*, and represents the alveolar part of the *V*_{T} with which the gas volume (*V*_{CO2}-*V*_{CO2(d)}) is expired at the end-tidal concentration (*F*_{ET,CO2}), *i.e.**V*_{D,Bohr} is equal to:

This is divided into two portions, the initial volume *V*_{o} (=volume segment *iu*) and the transitional volume *V*_{tr} (=volume segment *ua*) (figs. 5 and 6⇑⇑). The volume *V*_{o} is the initial part of the *V*_{T} with no CO_{2} gas in it. The transitional volume contains the gas volume *V*_{CO2(d)} and is equal to (figs. 5 and 6⇑⇑):and

The volume *V*_{o} is directly measured by the computer as the volume segment from the beginning of expiration (point *i*) to point *u*, at which CO_{2} gas starts appearing in the expired air.

*V*_{A} is calculated from Equation 5. If *F*_{CO2} is considered as zero in *V*_{D,Bohr} (*V*_{o}+*V*_{tr}), then all the *V*_{CO2} per breath should be expired with the *V*_{A} (fig. 6⇑). So, the mean alveolar *F*_{CO2} is calculated from the equation:By substituting in Equation 9 the term *V*_{CO2} by its equal *F*_{E,CO2}×*V*_{T}, the equation for the *V*_{A}/*V*_{T} ratio is the following:where *F*_{E,CO2} is the mixed expired CO_{2} fraction (=(Σo^{n}(d*V*_{CO}_{2}/d*V*)]/n=angle *cie*) (fig. 4⇑). Since the *V*_{A} is smaller than the volume segment *V*_{de} by the volume segment *da* (*V*_{da}=*V*_{CO2(d)}/*F*_{ET,CO2}), *F*_{A,CO2} is greater than *F*_{ET,CO2} (fig. 6⇑), *i.e.*Equation 11 is derived from Equations 5 and 9.

2. Relationship between the expired carbon dioxide *versus* tidal volume and carbon dioxide fraction *versus* tidal volume

The gas volume *versus* *V*_{T} curve (lower curve) and the corresponding gas concentration *versus* *V*_{T} curve (upper curve) are obtained from a single breath of a COPD patient (fig. 7⇑). In the upper curve (*F*_{CO2} *versus* *V*_{T} curve), the vertical line *dd*′ corresponds to the point *d* of the lower curve. The line *dd*′ separates the *F*_{CO2} *versus* *V*_{T} curve to the areas A and B, which are equal to each other. It is evident that *V*_{D(F)}, measured by Fowler's technique of orthogonal projection, is smaller than *V*_{D,Bohr} by the volume segment *V*_{da}. It is mentioned that if *V*_{CO2(d)} is zero, *V*_{D(F)} is equal to *V*_{D,Bohr}. In the upper curve (*F*_{CO2} *versus* *V*_{T}), the drawing of the line of the “sloping alveolar plateau” is very difficult. However, if the last part of the *F*_{CO2} *versus* *V*_{T}, which by no means is a straight line, is extrapolated, *V*_{D(F)} becomes even smaller as compared to *V*_{D,Bohr} by the volume segment *V*_{da} (fig. 7⇑).

3. Verification of the method

The analysis of the *V*_{CO2} *versus* *V*_{T} curve was verified as follows: 1) In three normal subjects breathing tidally through tubes of known capacity (*V*_{cap}) interposed between the mouthpiece and the monitoring ring, the dead space volume was measured without (*V*_{D(o)}) and with the added tube (*V*_{D}). The difference *V*_{D}-*V*_{D(o)} was compared with the capacity of the added tube (Results). 2) The gas volume *V*_{CO2(A)} (=*F*_{A,CO2}×*V*_{A}) was compared to the volume *V*_{CO2(B)} (=*F*_{sl}×*V*_{d}+*F*_{ET,CO2}×*V*_{A}) (fig. 6⇑). The error between *V*_{CO2(A)} and *V*_{CO2(B)} was calculated from (l-(*V*_{CO2(B)}/*V*_{CO2(A)}))×100 (Results). 3) The area A(A) (=^{1}/_{2}*V*_{A}×*V*_{CO2}) was compared to the area A_{(B)} (=^{1}/_{2}*V*_{D,Bohr}×*V*_{CO2(d)}+^{1}/_{2}*V*_{A}×(*V*_{CO2}-*V*_{CO2(d)})) with an error calculated from [1-(A_{(B)}/A_{(A)})]×100 (Results) (fig. 6⇑). 4) In normal subjects, *P*_{A,CO2} did not differ significantly from *P*_{a,CO2} (Results).

## Acknowledgments

The authors are grateful to N.B. Pride, M. Hughes, and P.T. Macklem for their most constructive criticism.

- Received July 13, 2000.
- Accepted January 30, 2001.

- © ERS Journals Ltd