### Model for the MC and BbB variabilities and averages definition

The Figure 3 shows a hybrid calorimeter model with the MC and the BbB techniques where the VO_{2} and VCO_{2} variabilities are defined at the output as *SD*
_{
VO2
} and *SD*
_{
VCO2
} and *vVO2*[*n*] and *vVCO2*[*n*], respectively. Likewise, the averages are defined as V\dot{O}2{\left[n\right]}_{\mathit{MC}} and *VĊO* 2[*n*]_{
MC
} for the MC technique and V\dot{O}2\left[n\right] and *VĊO* 2[*n*] for the BbB method.

This model for the MC and BbB techniques considers a discrete gas exchange at the input of the mouth (*VO2*[*n*] and *VCO2*[*n*]) in which the variabilities are implicitly included before they are separately measured. The argument n = 1,2,3… stands for a discrete time series that represents the breath by breath gas exchange during an IC study. The model in Figure 5 explains how the discrete gas exchange is formed at the alveolar level.

### Model for the alveolar discrete gas exchange

The alveolar discrete gas exchange is modelled in Figure 5. The assumption is that the continuous gas exchange at the alveoli (*VO2*(*t*) and *VCO2*(*t*)) is sampled by the lung’s mechanical ventilation. Then a discrete gas exchange *VO2*[*n*] and *VCO2*[*n*] is generated when the breath by breath instant flow *f*(*t*) works as a sampling function as in Equations (1) and (2).

\begin{array}{c}\mathit{VO}2\left(t\right)\times f\left(t\right)={V}_{T}\times \left(\mathit{FI}{O}_{2}\left(t\right)-\mathit{FE}{O}_{2}\left(t\right)\right)\\ \phantom{\rule{2em}{0ex}}\phantom{\rule{2em}{0ex}}\phantom{\rule{2em}{0ex}}\times f\left(t\right)=\mathit{VO}2\left[n\right]\end{array}

(1)

\begin{array}{c}\mathit{VCO}2\left(t\right)\times f\left(t\right)={V}_{T}\times \mathit{FEC}{O}_{2}\left(t\right)\times f\left(t\right)\\ \phantom{\rule{1em}{0ex}}=\mathit{VCO}2\left[n\right]\end{array}

(2)

Where: *f* (*t*) is the instant expired flow (L/sec). *V*
_{
T
} is the tidal volume (ml) without BTPS (body, temperature, pressure, saturated) to STPD (standard, temperature, pressure, dry) volumetric corrections in order to preserve the simplicity of the model, *FIO*
_{2}(*t*) – *FEO*
_{2}(*t*) is the inspired-expired oxygen fraction difference and the *FECO*
_{2}(*t*) is the expired CO_{2} gas fraction. All gas fractions are in atmospheric percentages (%).

The products *f*(*t*) × *VO2*(*t*) and *f*(*t*) × *VCO2*(*t*) generate the continuous sampling for the O_{2} and CO_{2} uptake during each expired breath with time duration *D*
_{
1
}, *D*
_{
2
}, …*D*
_{
n
}. Hence, individual and different breath-by-breath sample volumes are produced as *VO2*[*Dn*] and *VCO2*[*Dn*]. These volumes are computed as in Equations (3) and (4). The instantaneous products are done between signals analog to digital converter (A/D) at the rate of 10 milliseconds per sample in order to avoid numerical integration errors and to be according with the sampling Nyquist theorem when it is assumed signals with bandwidths below 100 Hz (Proakis and Manolakis 1998).

\begin{array}{l}\mathit{VO}2\left[{D}_{n}\right]=\frac{1}{{V}_{T}}{\int}_{0}^{{\mathit{D}}_{n}}{V}_{T}\times \left(\mathit{FI}{O}_{2}\left(t\right)-\mathit{FE}{O}_{2}\left(t\right)\right)\times f\left(t\right)\\ \phantom{\rule{2em}{0ex}}\phantom{\rule{2em}{0ex}}\phantom{\rule{2em}{0ex}}\phantom{\rule{1em}{0ex}}\times g\left(t-{D}_{n}\right)\mathit{dt}\end{array}

(3)

\mathit{VCO}2\left[{D}_{n}\right]=\frac{1}{{V}_{T}}{\displaystyle {\int}_{0}^{{\mathit{D}}_{n}}{V}_{T}\times \mathit{FEC}{O}_{2}\left(t\right)\times f\left(t\right)\times g\left(t-{D}_{n}\right)\mathit{dt}}

(4)

The *g* (*t* – *D*
_{
n
}) are continuous gate functions with the same time duration D_{1}, D_{2},..D_{n} that allow synchronization to integrate the products between the instant flow and the instant gas fractions as it is seen in Figures 6 and 7. Normalized products *f*(*t*) × *VO2*(*t*) and *f*(*t*) × *VCO2*(*t*) are needed in order to match with the *f*(*t*) peak amplitude the *VO2*(*t*) and *VCO2*(*t*) values so that Equations (3) and (4) should be divided by *1*/*V*
_{
T
}. Figure 6 shows a real example how gas fractions signals and the expired instant flow signal are synchronized to compute each *VO2*[*Dn*] and *VCO2*[*Dn*].

### BbB discrete time series analysis

The *VO2*[*n*] *and VCO2*[*n*] discrete time series in Figure 7 are generated when the computations of each *VO2*[*Dn*] and *VCO2*[*Dn*] are carried out over the continuous signal outputs corresponding to the flow, O_{2} and CO_{2} sensors. The constant time delay of 800 msec in Figure 6 is for the synchronization between the instantaneous flow *f*(*t*) and the time gas fraction signals (*FEO2*(*t*) *and FECO2*(*t*)). This time lag depends on the sensors’ time response and the delay produced by the tubing length which utilizes a flow of 150 ml/min to sample the sensors. The hybrid calorimeter with the open pneumatic circuit is sketched in Figure 9. Each *VO2*[*Dn*] and *VCO2*[*Dn*] value is placed in a time series using the sequence *δ*[*n*] with mathematical proprieties that allow the generation of the BbB discrete time series according to Equations (5) and (6).

\mathit{VO}2\left[n\right]={\displaystyle {\sum}_{{D}_{n=1}}^{N}\mathit{VO}2\left[{D}_{n}\right]\times \delta \left[n-{D}_{n}\right]}

(5)

\mathit{VCO}2\left[n\right]={\displaystyle {\sum}_{{D}_{n=1}}^{N}\mathit{VCO}2\left[{D}_{n}\right]\times \delta \left[n-{D}_{n}\right]}

(6)

Then, the sequence *δ* [*n* – *D*
_{
n
}] is used to geometrically place each value of *VO2*[*Dn*] and *VCO2*[*Dn*] as a series of coefficients at the end of each *f*(*t*) as it is seen in Figure 7. Here, the meaning of *Dn* is extended as a dumb variable (*Dn* = 1,2,..n) just to be interpreted as an index to generate the BbB discrete time series *VO2*[*n*] and *VCO2*[*n*]. The Figure 7 shows an example of a discrete time series from which the *νVO2*[*n*] and *νVCO2*[*n*] variabilities are computed. The average values (V\dot{O}2\left[n\right] and *VĊO* 2[*n*]) are calculated from the discrete gas exchange as in Equations (7) and (8).

\mathit{vVO}2\left[n\right]=\mathit{VO}2\left[n\right]-\dot{V}{O}_{2}\left[n\right]

(7)

\mathit{vVCO}2\left[n\right]=\mathit{VCO}2\left[n\right]-\dot{V}C{O}_{2}\left[n\right]

(8)

Where:

\dot{V}{O}_{2}\left[n\right]=\frac{1}{N}{\displaystyle {\sum}_{n=1}^{N}\mathit{VO}2\left[n\right]}

(9)

\dot{V}C{O}_{2}\left[n\right]=\frac{1}{N}{\displaystyle {\sum}_{n=1}^{N}\mathit{VCO}2\left[n\right]}

(10)

The above averages are computed with approximately N = 225 breaths, which are equivalent to a data acquisition window of 15 minutes.

An example of the power spectrum analysis of the *νVO2*[*n*] and *νVCO2*[*n*] is shown in the Figure 8. A linear data interpolation function was used to reformat the discrete time series *VO2*[*n*] and *VCO2*[*n*]. Then, one sample per second was used to resample the reformatted discrete time series in order to obtain a frequency domain analysis in the range of 0.0 to 0.5 Hz. The processing window was selected to capture at least 15 minutes of data so that a Welch power spectrum estimator allowed a maximum resolution of 0.005 Hertz using 50% of data overlapping. The frequency band analysis was defined in three main regions: low frequencies (LF = 0–0.04 Hz), medium frequencies (MF = 0.05–0.15 Hz) and high frequencies (HF = 0.16–0.5 Hz). These band divisions are similar to the heart rate variability analysis with special emphasis in the LF and MF bands since the HF band is assumed to be related with instant flow’s frequency (respiratory frequency) activity as it is seen in the example of the Figure 8.

### Averages and variabilities in the MC technique

The measurement of the averages and variabilities using the MC technique requires modeling the effect of the mixing chamber upon the discrete gas exchange *VO2*[*n*] and *VCO2*[*n*], having the model in Figure 3 in mind. The MC averages should be computed as in Equations (11) and (12) using a digital moving average which depends on the chamber volume and the number of breaths that the chamber storages as a manner of pipe-line, prior to obtaining one sample average every 20 seconds. In our case, the hybrid calorimeter has a chamber with a volume of 1.8 Liters so that the number of breaths storage in the MC, when the patient’s respiratory frequency is approximately 15 breaths/min, is approximately M = 4 in Equations (11) and (12). And the V\dot{O}2{\left[n\right]}_{\mathit{MC}} and *VĊO* 2[*n*]_{
MC
} are computed using the criteria of 30 averages to smooth enough the gas exchange. Thus, each one of the 30 averages is formed with M breaths to obtain one average sample during a total of 15 minutes per each IC study.

V\dot{O}2{\left[n\right]}_{\mathit{MC}}=\frac{1}{30}{{\displaystyle {\sum}_{j=1}^{30}\left\{\frac{1}{M=4}{\displaystyle {\sum}_{m=1}^{M}V{O}_{2}\left[n-m\right]}\right\}}}_{j@20sec}

(11)

V\dot{O}2{\left[n\right]}_{\mathit{MC}}=\frac{1}{30}{{\displaystyle {\sum}_{j=1}^{30}\left\{\frac{1}{M=4}{\displaystyle {\sum}_{m=1}^{M}V{O}_{2}\left[n-m\right]}\right\}}}_{j@20\mathit{sec}}

(12)

The measurement of the variabilities in the MC technique was computed as in Equations (13) and (14). Even, these equations allow the calculation of the VCs according to the clinical practice guidelines as it is shown in (15).

S{D}_{\mathit{VO}2}=\sqrt{\frac{1}{30-1}{\displaystyle {\sum}_{i=1}^{30}{\left(\dot{V}O2{\left[n\right]}_{\mathit{MC}}-{\left(\frac{1}{M}{\displaystyle {\sum}_{m=1}^{M}\mathit{VO}2\left[n-m\right]}\right)}_{i}\right)}^{2}}}

(13)

S{D}_{\mathit{VCO}2}=\sqrt{\frac{1}{30-1}{\displaystyle {\sum}_{i-1}^{30}{\left(V\dot{C}O2{\left[n\right]}_{\mathit{MC}}-{\left(\frac{1}{M}{\displaystyle {\sum}_{m=1}^{M}\mathit{VCO}2\left[n-m\right]}\right)}_{i}\right)}^{2}}}

(14)

\begin{array}{ccc}\hfill \mathit{VC}=\frac{S{D}_{\mathit{VO}2}}{V\dot{O}2{\left[n\right]}_{\mathit{MC}}}\hfill & \hfill \mathit{or}\hfill & \hfill \mathit{VC}=\frac{S{D}_{\mathit{VCO}2}}{V\dot{C}O2{\left[n\right]}_{\mathit{MC}}}\hfill \end{array}

(15)

### Hybrid indirect calorimeter hardware

A specific open-circuit hybrid indirect calorimeter (MGM-3) was designed and manufactured for the purpose of this work which was based in the design of Westenskow et al. (1984). The MC and the BbB techniques were fused in the MGM-3 as it can be seen in Figure 9. The patient’s half mask works either by passing the expired gas through the 1.8 L mixing chamber to implement the MC technique or by directly connecting the expired gas to a hot-wire flowmeter (TSI Inc, USA) to implement the BbB technique.

The MGM-3 calibration and quality control unit was a microprocessor based design and was calibrated every 5 minutes using a reference gas cylinder with a certified mixture of 21% O_{2}, 10% CO_{2}, complemented with N_{2}. Additionally, two more gas certified mixture cylinders (15% O_{2}, 4% CO_{2} and 18% O_{2}, 3% CO_{2}, Praxair) were used to adjust the transducer offsets and gains for the case of doing IC studies in ambulatory patients. The MC technique was implemented by displaying values of V_{E} (expired volume minute in L · min^{-1}), RF (respiratory frequency in breaths · min^{-1}), VO_{2} (ml · min^{-1}), VCO_{2} (ml · min^{-1})_{,} V_{T} (tidal volume in ml · breath^{-1}) and RQ (respiratory quotient VO_{2}/VCO_{2}) every 20 seconds. These readings were automatically corrected and displayed at STPD conditions after measuring volumes and fractions at BTPS conditions (2400 meters above the sea level at Mexico City, 590 ± 3 mmHg, and expired gases’ temperature). The VO_{2} was computed using the Haldane correction.

### Experimental design and data processing

A population of 15 young normal volunteer subjects without a history of any chronic disease was studied. The ages ranged from 18 to 30 years with a body mass index (BMI) average of 24.2 ± 3.8 Kg · m^{-2}. All subjects gave signed informed consent to be studied in the morning after 8 hours of fasting. Subjects were asked to perform the active clino-orthostatic maneuver (COM). First, a 5 minute period of relaxing was used before he/she lied down on a couch and was then submitted to the COM while connected to the MGM-3 calorimeter. Two 30 minutes periods were used to implement the measurement protocol: 15 minutes for the MC technique and 15 min for the BbB technique in each COM position. All of the measurements were made in the same room maintaining constant temperature and data collection by the same expert team in all cases.

Comparative statistical paired data analysis was applied intra-groups. The MC averages V\dot{O}2{\left[n\right]}_{\mathit{MC}} and *VĊO* 2[*n*]_{
MC
} were compared against the BbB averages V\dot{O}2\left[n\right] and \dot{V}\mathit{CO}2\left[n\right]. Similarly, variances averages (*SD*
_{
VO 2})^{2} and (*SD*
_{
VCO 2})^{2} were compared against total spectral energy averages. The statistics analysis was parametric since the variables were considered to be Gaussians, once they were tested by the Kolmogorov–Smirnov test. Then, two-tailed paired t-tests (Welch version) were used as appropriate for unequal variances. In all cases, the null hypothesis was rejected when p ≤ 0.1 since this experiment was considered to be a pilot study.