Fitting of single-exhalation profiles using a pulmonary gas exchange model—application to carbon monoxide

Real-time detection of CO in breath and ambient air was achieved using a home-built mid-infrared tunable diode laser absorption spectroscopy (TDLAS) system [30]. The spectrometer employed a distributed-feedback interband cascade laser (ICL, Nanoplus GmbH) in combination with a circular, low-volume multipass cell (MPC, IR Sweep, IRcell-4M) for absorption path length enhancement (4 m) and 2f-wavelength modulation spectroscopy for noise-reduction. Breath samples were analyzed in the MPC at a pressure of 100 Torr and at close to room temperature (23 °C). The absorption spectra were scanned with a frequency of 140 Hz and averaged 10 times. This sensor provided selective, interference-free eCO quantification down to 9 ppb at a precision of 5 ppb and time-resolution of 0.1 s. The rapid gas exchange in the MPC (<0.1 s) required for real-time detection was guaranteed by the low MPC volume (38 ml) and the pumping speed (360 ml s⁻¹ at 100 Torr) of the vacuum pump (Leybold, Divac 1.4HV3C). True real-time capability of the system was previously confirmed by direct comparison of eCO₂ profiles measured by capnography and TDLAS [4]. The main components of the experimental setup are shown schematically in figure 1.

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Pulmonary gas exchange and exhaled biomarker concentrations strongly depend on the breath sampling conditions, including hyperventilation, body position and inhaled biomarker concentrations [32]. To minimize these effects and ensure repeatability, the sampling process was standardized using an advanced breath sampling system that controlled the breathing frequency and the inhalation and exhalation flow rates (IFR and EFR, respectively).

The online breath sampler (figure 1) comprised a flow meter (Phillips Respironics, FloTrak Elite Module), a mainstream capnograph (Phillips Respironics, Capnostat 5) and a Teflon buffer tube of length 15 cm and volume 30 ml. A mouthpiece made of Teflon and a disposable anti-bacterial filter (GVS, Eco Maxi Electrostatic Filter, 4222/701) were mounted at the inlet. The flow meter was a fixed orifice differential pressure type, which potentially can evaluate more than 60 respiratory parameters on a breath by breath basis including the flow rate (range −5000 to +5000 ml s⁻¹, resolution 0.2 ml s⁻¹), volume (range −1000 to +4000 ml, resolution 1 ml) and airway pressure (range −150 to +150 cmH₂O, resolution 0.05 cmH₂O). The capnograph was a NDIR single beam optical device with a CO₂ measurement range of 0%–19.7% and a resolution of 0.1%.

A two-way non-rebreathing valve (Rudolph Inc.) was connected to the outlet of the buffer tube to separate the inspiration and expiration routes, as subjects performed both inhalation and exhalation through the breath sampler. In order to restrict IFR and EFR close to desired values, suitable orifices of different diameters were installed at the inlet and outlet ports of the two-way valve. A LabVIEW computer interface with audiovisual indicators helped the subjects to maintain the intended IFR/EFR and breathing frequency (6 or 3 breaths/min for normal breathing and breath-holding, respectively) according to the protocol specified in section 2.6. A portion of the inhaled and exhaled breath was continuously extracted from the buffer tube and led to the MPC of the TDLAS sensor at a flow rate of 50 ml s⁻¹ (set by the vacuum pump speed).

Figure 2 shows typical respiratory data sets recorded over a single breath-cycle during normal breathing for the two healthy, non-smoking subjects. For a breathing frequency of 6 breaths/min and an IFR/EFR of around 250 ml s⁻¹ (average indicated by a dashed line), the inhaled/exhaled volumes were close to 1250 ml. A clear inter-individual difference in airway pressure and end-tidal CO₂ concentration can be observed. In all experiments, the respiratory data and ambient air CO (C_amb) were continuously recorded during the breath cycles and later used as input parameters to the mathematical model of pulmonary CO gas exchange dynamics.

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The TMAD is based on a one-dimensional, trumpet-shaped representation of the respiratory tract following Weibel's symmetrically bifurcating lung structure [33, 34] rescaled for a total airspace volume of 3700 ml [35] and using the latest anatomical data [31]. Figure 3 shows a schematic drawing of the TMAD with the main model parameters indicated.

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The governing equation, which accounts for the axial gas transport and radial sources and sinks as a function of time and axial distance z along the trumpet, can generally be written as

$$\begin{eqnarray}&&\begin{array}{l}\left\{{A}_{{\rm{c}},\mathrm{aw}}(z)+\left[\displaystyle \frac{{N}_{{\rm{alv}}}(z)}{{N}_{{\rm{t}}}}\right]{A}_{{\rm{c}},{\rm{A}}}\right\}\displaystyle \frac{{\rm{d}}{{\rm{C}}}_{{\rm{C}}{\rm{O}}}}{{\rm{d}}t}\\ \,=\,T(\mathop{V}\limits^{},{D}_{\mathrm{CO},\mathrm{air}},z)\\ \,\,+\,S(J{{\prime} }_{{\rm{awCO}}},D{{\prime} }_{{\rm{awCO}}},J{{\prime} }_{{\rm{ACO}}},D{{\prime} }_{{\rm{ACO}}},z),\end{array}\end{eqnarray} \tag{ 1 }$$

where A_c,aw is the airway cross-sectional area (cm²), which follows a power-law relation (∼1/z²) towards the lower airway generations (towards the mouth), A_c,A is the total cross-sectional area of the alveolar compartment (cm²), the function T describes the convective bulk flow and axial diffusion, and the function S represents the flux to and diffusion from the airway and alveolar regions per unit axial distance. The function T is given by

$$\begin{eqnarray}&&\begin{array}{l}T\left(\dot{V},{D}_{\mathrm{CO},\mathrm{air}}\right)=-\dot{V}\displaystyle \frac{{\rm{d}}{C}_{{\rm{CO}}}}{{\rm{d}}z}\\ \,+\,{D}_{\mathrm{CO},\mathrm{air}}\displaystyle \frac{{\rm{d}}}{{\rm{d}}z}\left[{A}_{{\rm{c}},\mathrm{aw}}(z)\displaystyle \frac{{\rm{d}}{C}_{{\rm{CO}}}}{{\rm{d}}z}\right],\end{array}\end{eqnarray} \tag{ 2 }$$

where $\dot{V}$ is the volumetric (inhalation and exhalation) flow rate (ml s⁻¹), D_CO,air the molecular diffusivity of CO in air (cm² s⁻¹). The function S has the form

$$\begin{eqnarray}&&\begin{array}{l}S(J{{\prime} }_{{\rm{awCO}}},D{{\prime} }_{{\rm{awCO}}},J{{\prime} }_{{\rm{ACO}}},D{{\prime} }_{{\rm{ACO}}})\\ \,=\,\left(J{{\prime} }_{{\rm{awCO}}}-D{{\prime} }_{{\rm{awCO}}}{C}_{{\rm{CO}}}\right)\left[1-\displaystyle \frac{{N}_{{\rm{alv}}}(z)}{{N}_{{\rm{\max }}}}\right]\\ \,\,+\,\left(J{{\prime} }_{{\rm{ACO}}}-D{{\prime} }_{{\rm{ACO}}}{C}_{{\rm{CO}}}\right)\left[\displaystyle \frac{{N}_{{\rm{alv}}}(z)}{{N}_{{\rm{t}}}}\right],\end{array}\end{eqnarray} \tag{ 3 }$$

where J'_awCO and J'_ACO are the maximum fluxes per unit axial distance (pl s⁻¹ cm⁻¹) and D'_awCO and D'_ACO the diffusing capacities per unit axial distance (pl s⁻¹ ppb⁻¹ cm⁻¹) in the conducting airways and the alveolar region, respectively, N_alv is the number of alveoli per unit axial distance, N_max the maximum number of alveoli at any axial position, and N_t is the total number of alveoli.

The mass balance equation, equation (1), is solved numerically using the method of lines, and provides the distribution of CO in the respiratory tract during inhalation, exhalation and breath-holding with spatial and temporal resolutions of 1 mm and 0.01 s, respectively. Single-exhalation profiles are extracted from the first simulated grid point (z = 0) representing the mouth. Details on the model parameters, the boundary conditions and the numerical solution can be found in [31].

The solution of the TMAD at the mouth grid point is fitted to the experimental eCO profiles using a weighted, nonlinear least-squares algorithm implemented in MATLAB. There are four open parameters representing the sinks and sources in the airway and the alveolar regions, namely J_awCO, D_awCO, J_ACO and D_ACO, which denote the total J'_awCO, D'_awCO, J'_ACO and D'_ACO in units of pl s⁻¹. and pl s⁻¹ ppb⁻¹, respectively. These four parameters affect the exhalation profile shape in different ways (no mutual dependencies) and are uniquely determined in the fitting process. Starting values for the open TMAD parameters have previously been estimated [31]. As mentioned above, other input data to the model were the actual IFRs and EFRs, inhaled/exhaled volumes and the inhaled CO concentration.

Due to the steep eCO increase in exhalation phase II, which represents the transition between the conducting airways and the alveolar region, this phase is particularly sensitive to discrepancies between the anatomical data assumed in the TMAD and the actual lung structure of the subject providing the sample. Therefore, a good fit cannot be expected in this region and more weight was put on the data points in phases I and III, which are the dominant regions for evaluation of the gas exchange in conducting airways and alveoli, respectively. The first 4% of the expirogram data points (corresponding to phase I) were weighted 20 time more and the last 74% (corresponding to phase III) 60 times more than the rest of the profile (phase II). Since the data acquisition rate and exhaled volume were kept constant, the relative amount of data points in each exhalation phase was the same regardless of exhalation flow rate and time. The TMAD fitting parameters, J_awCO, D_awCO, J_ACO and D_ACO, were free to vary in the range of 100–500 pl s⁻¹, 1.0–1.6 pl s⁻¹ ppb⁻¹, 5 × 10⁵–2 × 10⁸ pl s⁻¹, and 300–5 × 10⁵ pl s⁻¹ ppb⁻¹, respectively.

For healthy, non-smoking subjects, the airway CO contribution is usually very small and the airway TMAD parameters (J_awCO and D_awCO) cannot be accurately determined from normal breathing profiles. During a BH maneuver, however, there is more time for airway tissue CO to diffuse into the gas stream, which increases the sensitivity to the airway parameters. Thus, in this work, J_awCO and D_awCO were first determined from an expirogram recorded after 10 s BH, and then fixed in the fits to the normal breathing exhalation profiles. The alveolar and airway (tissue) CO concentrations predicted for equilibrium conditions, C_ACO and C_awCO, were obtained from the ratios J_ACO/D_ACO and J_awCO/D_awCO, respectively. The computational time for fitting a typical normal breathing eCO profile recorded at an EFR of 250 ml s⁻¹ and an exhalation volume of 1250 ml was around 45 s on a standard office PC. For a 10 s BH profile, the time was 4–5 min As expected, the computational time increases with decreasing IFR and EFR, and increasing BH time.

The effect of CO exposure on the eCO profiles and TMAD parameters was investigated in a human exposure study including intermittent exercise. The subjects stayed in a controlled environment exposure chamber (18 m³) [36] for 2 h, breathing a mixture of 10 ± 2 ppm CO in air derived from a 300 ppm CO gas standard (AGA Gas AB) by dilution with air. The expected increase in blood carboxyhemoglobin (COHb) concentration due to the CO exposure was calculated using the differential Coburn–Forster–Kane (CFK) equation [37] and healthy non-smoker blood and lung properties. An increase in COHb level of around 60% compared to normal (up to 1.3% saturation from an initial value of 0.8%) was predicted for the 2 h exposure assuming an alveolar ventilation rate of 30 l min⁻¹. The COHb level can also be calculated from the measured alveolar CO concentration (C_ACO) using an empirical relationship given by [38]

$$\begin{eqnarray}&&{\rm{COHb}}=0.63+0.16\,{C}_{{\rm{ACO}}},\end{eqnarray} \tag{ 4 }$$

where C_ACO is in ppm, and COHb is obtained in units of % saturation.

Two healthy, male non-smokers (subjects 1 and 2, aged 42 and 37 years, respectively) participated in the pilot-study. No diary restrictions were imposed on the subjects, but they were asked not to exercise for 3 h prior to the test. All eCO measurements were conducted between 1 pm and 3 pm on weekdays in late October in Umeå, Sweden. Two different breathing maneuvers were considered. The first maneuver, here referred to as 'normal breathing', comprised 5 s inhalation of ambient air at 250 ml s⁻¹ IFR, followed by 5 s exhalation at 250 ml s⁻¹ EFR. This resulted in inhalation/exhalation volumes of about 1250 ml, depending on how well the subjects followed the audiovisual indicators. The second maneuver consisted of 10 s inhalation at 120 ml s⁻¹ IFR, followed by 10 s breath-holding and 10 s of exhalation at 120 ml s⁻¹ EFR. These two breathing maneuvers were also performed 7 min after the end of the 2 h exposure to 10 ppm CO. During the exposure in the chamber, the subjects alternated between 15 min of moderate cycling and 15 min of rest. The sequence of breath cycles was recorded by performing twenty successive normal breathing maneuvers. All breath samples were given through the mouth, while subjects were sitting upright. The study protocol was approved by the Regional Ethical Review Board at Umeå University (2017/306-31 and 2018-35-23 M).

Figure 4 presents measured, normal breathing CO exhalation profiles (red markers, every 2nd data point shown) from subjects 1 and 2, together with weighted TMAD curve fits (blue lines) and fit residuals (lower panels). The corresponding breath sampling data for these expirograms is shown in figure 2. Time zero indicates the start of the exhalation. The prevailing ambient air CO concentration was continuously sampled during inhalation. In figure 4(a), the three exhalation phases are indicated with Roman numerals. Phase I represents the air from anatomical dead-space and the conducting airways, phase III represents the air from the alveoli, and phase II denotes the transition between phases I and III. The airway TMAD parameters (J_awCO and D_awCO) used in the fits to the normal breathing eCO profiles were fixed to those determined from the BH profiles shown in figure 5, which depicts single-exhalation profiles recorded after 10 s BH (red markers, every 2nd data point shown) with curve fits (blue lines) and the fit residuals (lower panels). Independent of the starting values and allowed optimization ranges, the TMAD parameters converged to the values summarized in table 1.

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Table 1.  Gas exchange parameters extracted from the TMAD fits to the experimental data in figures 4 and 5, and corresponding respiratory data. ETCO—end-tidal CO concentration, V—average of inhaled/exhaled volume, ETCO₂—end-tidal CO₂ concentration.

								Respiratory dataa
Figures	JawCO	DawCO	JACO	DACO	ETCOa	CACO	CawCO	$\dot{{V}_{{\rm{I}}}}$	$\dot{{V}_{{\rm{E}}}}$	V	ETCO2	Camb
	pl s−1	pl s−1 ppb−1	pl s−1	pl s−1 ppb−1	ppb	ppb	ppb	ml s−1	ml s−1	ml	%	ppb
4(a)	192	1.6	2.21 × 107	10 461	1969	2113	120	242	246	1201	6.4	120
4(b)	291	1.6	1.23 × 107	10 867	1050	1136	182	255	257	1289	5.8	154
5(a)	192	1.6	1.35 × 107	6361	2062	2118	120	115	119	1121	7.7	90
5(b)	291	1.6	8.37 × 106	6434	1248	1301	182	130	128	1177	6.1	127

^aDirectly measured; all other parameters are derived from the model fits.

The repeatability of the breath sampling procedure and robustness of the TMAD fitting routine were studied by looking at the intra-individual variations of the model parameters under normal breathing conditions.

As presented in figure 6, sequences of twenty consecutive breath-cycles were measured for both subjects (left panels) and analyzed using TMAD fits. The right panels in figure 6 show the TMAD fits (blue lines) to five of the measured eCO profiles (red markers). The TMAD parameter mean values and coefficients of variation (CV) extracted from fits to all twenty profiles are given in table 2. The mean IFR and EFR were 242 ml s⁻¹ and 243 ml s⁻¹ for subject 1, and 263 ml s⁻¹ and 258 ml s⁻¹ for subject 2, respectively. The mean ambient air CO concentrations were 115 ppb and 149 ppb, respectively. The intra-individual CV of J_ACO and D_ACO were less than 6% for subject 1 and less than 5% for subject 2. Inter-individually, the mean value of J_ACO for subject 1 is almost twice that of subject 2, whereas there is less than 9% difference in the mean D_ACO.

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Table 2.  Mean values and coefficients of variation (CV (%) = Standard deviation/Mean ×100) of the TMAD and respiratory parameters derived from the 20 breath-cycles shown in figure 6 for subjects 1 (S1) and 2 (S2). ETCO—end-tidal CO concentration, V—average of inhaled/exhaled volume.

							Respiratory dataa
	JawCO	DawCO	JACO	DACO	ETCOa	CACO	$\dot{{V}_{{\rm{I}}}}$	$\dot{{V}_{{\rm{E}}}}$	V	ETCO2	Camb
	pl s−1	pl s−1 ppb−1	pl s−1	pl s−1 ppb−1	ppb	ppb	ml s	ml s−1	ml	%	ppb
Subject 1
S1	192	1.6	2.04 × 107	9652	1943	2120	242	243	1196	6.5	115
CV	—	—	5.1	5.8	2.1	2.2	2	2	3.4	1.5	5
Subject 2
S2	291	1.6	1.20 × 107	10 515	1063	1139	263	258	1301	5.8	149
CV	—	—	4.7	4.9	1.5	1.5	3	4	3.7	1.1	6

^aDirectly measured; all other parameters are derived from the model fits.

Exposure to exogenous CO gives rise to an increase in blood COHb, which, in turn, results in elevated eCO concentrations during elimination [28, 37]. We hypothesize that, for healthy non-smokers, exposure will mainly affect the maximum CO flux, but not the lung diffusion properties. Figure 7 shows measured single-exhalation eCO profiles (markers) for the two subjects before and 7 min after the 2 h exposure to 10 ± 2 ppm CO, together with least-squares fits (lines) to the experimental data. In each case, the airway TMAD parameters (J_awCO and D_awCO) have first been determined from fits to the corresponding BH curves (not shown). The extracted parameters are presented in table 3.

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Compared to the control values, J_ACO was substantially increased after exposure (by factors of 2.7 and 4.9 for subjects 1 and 2, respectively), which suggests elevated COHb levels. Indeed, the COHb values calculated from C_ACO using equation (4) were increased accordingly and similar to those predicted with the CFK model (section 2.5) considering ventilation rates of 26 l min⁻¹ and 36 l min⁻¹ for subjects 1 and 2, respectively. For both study participants, a slightly elevated equilibrium airway (tissue) CO concentration was found after exposure. The alveolar diffusing capacities, on the other hand, did not change significantly for subject 1 (6% increase), and not at all for subject 2.

The potential of the novel methodology to assess health conditions connected to respiratory diseases was investigated by simulating eCO profiles that could be expected from subjects with airway inflammation (increased J_awCO) and an obstructive lung disease (decreased D_ACO). We hypothesize that a curve fit to highly precise and time-resolved TDLAS data, as presented in this study, can be used to extract parameters that reflect changes in the exhalation profile shape associated with the diseases.

Figure 8(a) presents eCO profiles (10 s BH, 240 ml s⁻¹ IFR/EFR) for a healthy subject (blue solid lines) and for a subject with three-fold increased maximum airway flux (red dashed line). Figure 8(b) shows a comparison between CO expirograms (normal breathing, 240 ml s⁻¹ IFR/EFR) for a healthy subject (blue solid line) and for a subject with an alveolar CO diffusing capacity reduced to 40% of normal with (red dashed line) and without (green dashed-dotted line) simultaneously increased COHb level (controlled by means of J_ACO). COHb is believed to be elevated in COPD [39], but this may not always be the case. The initial TMAD parameters representing a healthy subject were taken from the average data of subject 1 given in table 2. Ambient air CO concentrations and mean inhaled/exhaled volumes were assumed to be 100 ppb and 1200 ml, respectively. For all cases, the TMAD parameters used for the simulated CO expirograms are provided in table 4.

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