Assessing breath-by-breath alveolar gas exchange: is the contiguity in time of breaths mandatory?

Abstract

Purpose

A new algorithm is illustrated for the determination of breath-by-breath alveolar gas exchange that neglects the contiguity in time of breaths, i.e. it allows the breaths to be partially superimposed or disjoined in time.

Methods

Traces of oxygen, carbon dioxide fractions, and ventilatory flow were recorded continuously over 20 min in 15 healthy subjects in resting conditions; at 5-min intervals, subjects voluntarily hyperventilated for ~ 30 s to induce abrupt changes in lung gas stores. Gas exchange data were calculated applying the new algorithm and were compared to those yielded by a reference algorithm, also providing values at the alveolar level.

Results

Average O₂ uptakes (V′O₂) obtained with the two algorithms were similar during quiet breathing (0.28 ± 0.06 vs. 0.29 ± 0.06 L/min; two-sided paired t test, n = 45, p = NS); during hyperventilation, average V′O₂ was significantly lower applying the new algorithm compared to the reference algorithm (0.57 ± 0.15 vs. 0.65 ± 0.17 L/min; difference − 0.077 ± 0.048 L/min; two-sided paired t test, n = 45, p < 0.001). The first breath of each hyperventilation manoeuvre showed the greatest difference in V′O₂ (− 0.25 ± 0.23 L/min, z test against zero, n = 45, p < 0.001). The volumes of O₂ considered twice (or neglected) because of the lack of contiguity of breaths were overall small (maximum of 3 mL) and, if accounted for, had only a slight softening effect on the fluctuations of the O₂ uptake.

Conclusion

The new algorithm, which assumes each breath as the leading subject, was able to effectively account for changes in lung gas stores without requiring any predetermined value or off-line optimisation procedure.

Appendix

This section is essentially the same as the Theory Section of our previous paper (Cettolo and Francescato 2015) to which the reader is referred to for further details. Main information is summarized here for the sake of convenience.

General concepts on alveolar gas exchange algorithms

During the j-th breath, the volume of oxygen taken up at the alveoli ($v{\text{O}}_{{2j}}^{{\text{A}}}$) can be determined from the volume of oxygen exchanged at the mouth ($v{\text{O}}_{{2j}}^{{\text{M}}}$) minus the volume change of pulmonary oxygen stores ($\Delta v{\text{O}}_{{2j}}^{{\text{S}}}$):

$$v{\text{O}}_{{2j}}^{{\text{A}}}=v{\text{O}}_{{2j}}^{{\text{M}}} - \Delta v{\text{O}}_{{2j}}^{{\text{S}}}.$$

(6)

According to Auchincloss et al. (1966), the volume change of pulmonary O₂ stores can be divided into two parts: the first one dependent on the variation of the lung oxygen concentration (${\text{FO}}_{2}^{{\text{A}}}$) from the j − 1 breath to the j-th breath, the second one due to the variation of lung volume ($\Delta {v^{\text{A}}}$) between the two subsequent breaths:

$$\Delta v{\text{O}}_{{2j}}^{{\text{S}}}=v_{{j - 1}}^{{\text{A}}} \cdot \left( {{\text{FO}}_{{2j}}^{{\text{A}}} - {\text{FO}}_{{2j - 1}}^{{\text{A}}}} \right)+{\text{FO}}_{{2j}}^{{\text{A}}} \cdot \Delta v_{j}^{{\text{A}}},$$

(7)

where $v_{{j - 1}}^{{\text{A}}}$ is the lung volume at the end of the preceding breath and represents the same quantity at the beginning of the current j-th breath (Auchincloss et al. 1966).

To determine $\Delta v_{j}^{{\text{A}}}$, it can be assumed that the equations above are valid also for the gas with no net exchange at the alveolar-capillary level (essentially composed of nitrogen, N₂) and, by definition, that the exchanged volume of the latter is equal to zero during any time interval. Consequently, $\Delta v_{j}^{{\text{A}}}$ can be obtained as follows:

$$\Delta v_{j}^{{\text{A}}}=\frac{1}{{{\text{FN}}_{{2j}}^{{\text{A}}}}}\left[ {v{\text{N}}_{{2j}}^{{\text{M}}} - v_{{j - 1}}^{{\text{A}}} \cdot \left( {{\text{FN}}_{{2j}}^{{\text{A}}} - {\text{FN}}_{{2j - 1}}^{{\text{A}}}} \right)} \right].$$

(8)

By substituting Eq. (8) into (7) for $\Delta v_{j}^{{\text{A}}}$ and then in Eq. (6) for $\Delta v{\text{O}}_{{2j}}^{{\text{S}}}$, rearranging and simplifying, the equation for the calculation of the volume of oxygen taken up at the alveolar level is obtained:

$$v{\text{O}}_{{2j}}^{{\text{A}}}=v{\text{O}}_{{2j}}^{{\text{M}}} - \frac{{{\text{FO}}_{{2j}}^{{\text{A}}}}}{{{\text{FN}}_{{2j}}^{{\text{A}}}}} \cdot v{\text{N}}_{{2j}}^{{\text{M}}} - v_{{j - 1}}^{{\text{A}}} \cdot {\text{FN}}_{{2j - 1}}^{{\text{A}}} \cdot \left\{ {\frac{{{\text{FO}}_{{2j}}^{{\text{A}}}}}{{{\text{FN}}_{{2j}}^{{\text{A}}}}} - \frac{{{\text{FO}}_{{2j - 1}}^{{\text{A}}}}}{{{\text{FN}}_{{2j - 1}}^{{\text{A}}}}}} \right\}.$$

(9)

In this equation, the quantity $v_{{j - 1}}^{{\text{A}}}$ is not directly measurable on a breath-by-breath basis.

The O₂ uptake for the j-th breath is obtained by dividing the volume of O₂ calculated with the above equation by the corresponding time interval.

With appropriate modifications (i.e. substituting FO₂ with FCO₂), alveolar CO₂ transfer can be obtained.

The directly derived algorithms

Traditionally, the start and end points of the respiratory cycle are defined as the time points where the measured flow changes orientation, i.e., as illustrated in Fig. 1a, a breath starts at the onset of inspiration (t_i,j−1) and ends when expiratory flow has entirely or largely ceased (t_i,j) (Auchincloss et al. 1966).

A common assumption is that the end-expiratory gas fractions reliably represent the “mixed” alveolar gas fractions (i.e. the fractions measured at time t_x,j on Fig. 1b) (Auchincloss et al. 1966).

All the directly derived algorithms make the volume $v_{{j - 1}}^{{\text{A}}}$ “real”, assuming that it corresponds to the end-expiratory lung volume (V_L) and that it remains constant throughout.

This quantity was set equal to resting functional residual capacity (FRC) by Auchincloss et al. (1966); 15 years later, Beaver et al. (1981) implemented this algorithm on a computer system.

Wessel et al. (1979) completely neglected the third part of Eq. (9), which is tantamount to saying that they assumed a V_L equal to 0 l.

Swanson (1980), however, proposed that the method providing the lowest variation yields the best estimate of gas exchange at the alveolar-capillary level. Accordingly, he introduced the concept of “nominal effective lung volume” (ELV), which is the end-expiratory lung volume that minimizes breath-by-breath variation (Swanson and Sherrill 1983). An off-line procedure is consequently required to determine ELV.

When the symbols of the volumes are substituted with the appropriate integrals (with the explicit start and end time points of the integration periods), the following equation is obtained for the Swanson’s approach:

$$v{\text{O}}_{{2j}}^{{{\text{SW}}}}=\int\limits_{{{t_{i,j - 1}}}}^{{{t_{i,j}}}} {{\text{ }}\dot {V} \cdot {\text{F}}{{\text{O}}_2}\,{\text{d}}t} - \frac{{{\text{F}}{{\text{O}}_2}({t_{x,j}})}}{{{\text{F}}{{\text{N}}_2}({t_{x,j}})}} \cdot \int\limits_{{{t_{i,j - 1}}}}^{{{t_{i,j}}}} {{\text{ }}\dot {V} \cdot {\text{F}}{{\text{N}}_2}\,{\text{d}}t} - {\text{ELV}} \cdot {\text{F}}{{\text{N}}_2}({t_{x,j - 1}}) \cdot \left\{ {\frac{{{\text{F}}{{\text{O}}_2}({t_{x,j}})}}{{{\text{F}}{{\text{N}}_2}({t_{x,j}})}} - \frac{{{\text{F}}{{\text{O}}_2}({t_{x,j - 1}})}}{{{\text{F}}{{\text{N}}_2}({t_{x,j - 1}})}}} \right\}.$$

(10)

The “independent breath” algorithm

To circumvent the problem that the term $v_{{j - 1}}^{{\text{A}}}$ in Eq. (9) is not directly measurable on a breath-by-breath basis, it is possible to define the start and end point of the respiratory cycle as the times in two consecutive expirations (t_1,j and t_2,j, respectively) where identical ratios between FO₂ and FN₂ are observed (Cettolo and Francescato 2015), as illustrated in Fig. 2a:

$$\frac{{{\text{FO}}_{2}^{{\text{A}}}({t_{2,j}})}}{{{\text{FN}}_{2}^{{\text{A}}}({t_{2,j}})}}=\frac{{{\text{FO}}_{2}^{{\text{A}}}({t_{1,j}})}}{{{\text{FN}}_{2}^{{\text{A}}}({t_{1,j}})}}.$$

(11)

Hence, the calculation of $v{\text{O}}_{{2j}}^{{\text{A}}}$ over the time interval from t_1,j to t_2,j is limited to the first two parts of the Eq. (9). A required assumption is that the “mixed” alveolar ratio between exchangeable and non-exchangeable gas fractions (FO₂/FN₂ for oxygen uptake) are reliably represented by the end-expiratory ratios (as determined at times t_1,j and t_2,j on Fig. 2a).

After having substituted the symbols of the volumes with the appropriate integrals (with the explicit start and end time points of the integration periods), the following equation is obtained to calculate the volume of O₂ taken up at the alveolar level:

$$v{\text{O}}_{{2j}}^{{{\text{IND}}}}=\int\limits_{{{t_{1,j}}}}^{{{t_{2,j}}}} {\dot {V} \cdot {\text{F}}{{\text{O}}_2}\,{\text{d}}t} - \frac{{{\text{F}}{{\text{O}}_{2j}}({t_{1,j}})}}{{{\text{F}}{{\text{N}}_{2j}}({t_{1,j}})}} \cdot \int\limits_{{{t_{1,j}}}}^{{{t_{2,j}}}} {\dot {V} \cdot {\text{F}}{{\text{N}}_2}\,{\text{d}}t} .$$

(12)