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O₂, CO₂, and RQ in a respiratory chamber: accurate estimation based on a new mathematical model using the Kalman-Bucy method

¹Dipartimento di Informatica e Sistemistica "Antonio Ruberti," Facoltà di Ingegneria, Università di Roma "La Sapienza," 00184 Rome; and ²Istituto di Medicina Interna, Università Cattolica del Sacro Cuore, 00168 Rome, Italy

Submitted 28 July 2003 ; accepted in final form 31 October 2003

	ABSTRACT

TOP ABSTRACT METHODS RESULTS DISCUSSION APPENDIX: KB FILTER FORMULATION REFERENCES

 
A respiratory chamber is used for monitoring O₂ consumption (O₂), CO₂ production (CO₂), and respiratory quotient (RQ) in humans, enabling long term (24-h) observation under free-living conditions. Computation of O₂ and CO₂ is currently done by inversion of a mass balance equation, with no consideration of measurement errors and other uncertainties. To improve the accuracy of the results, a new mathematical model is suggested in the present study explicitly accounting for the presence of such uncertainties and error sources and enabling the use of optimal filtering methods. Experiments have been realized, injecting known gas quantities and estimating them using the proposed mathematical model and the Kalman-Bucy (KB) estimation method. The estimates obtained reproduce the known production rates much better than standard methods; in particular, the mean error when fitting the known production rates is 15.6 ± 0.9 vs. 186 ± 36 ml/min obtained using a conventional method. Experiments with 11 humans were carried out as well, where O₂ and CO₂ were estimated. The variance of the estimation errors, produced by the KB method, appears relatively small and rapidly convergent. Spectral analysis is performed to assess the residual noise content in the estimates, revealing large improvement: 2.9 ± 0.8 vs. 3,440 ± 824 (ml/min)² and 1.8 ± 0.5 vs. 2,057 ± 532 (ml/min)², respectively, for O₂ and CO₂ estimates. Consequently, the accuracy of the computed RQ is also highly improved (0.3 x 10^-4 vs. 800 x 10^-4). The presented study demonstrates the validity of the proposed model and the improvement in the results when using a KB estimation method to resolve it.

indirect calorimetry; respiratory gas exchange; energy expenditure; Kalman filter; stochastic model

To estimate O₂ and CO₂, conventional methods divide the observation period into small subintervals. For each such subinterval, the following three steps are executed (1, 2, 5, 10–13, 16, 18): measurement phase, in which a series of measurements is collected at any generic subinterval; prefiltering phase, in which the accuracy of the measurements is improved by applying suitable filtering methods (arithmetic mean, cubic splines); and finally, computation phase, in which the prefiltered data are related to the problem unknowns (O₂ and CO₂) by a mass balance equation that is simply inverted to produce estimates of the unknown values. Some differences do exist between the standard methods regarding mainly the formulation of the mass balance equation, rising from different simplifying hypotheses used.

Estimates performed as indicated above are affected by several error sources: 1) measurements are always corrupted by noise and, although prefiltering is applied, considerable residual error remains; 2) the used mass balance equation is only an approximate representation of reality due to the simplifying hypotheses used; 3) CO₂ and O₂ are obtained by simple inversion of the mass balance equation, without applying any mechanism to limit the propagation of the errors introduced by 1 and 2. The above error sources are always present in a real experimental setting and cannot be completely eliminated; therefore, it is of high importance to apply procedures and methods to minimize such negative effects.

The main goal of the present study is to propose adequate procedures to obtain more accurate and reliable estimates of the variables of interest (O₂, CO₂, and, moreover, RQ). Furthermore, it is of interest to validate the proposed method with respect to conventional ones. To do so, the following points were studied: 1) introduction of adequate stochastic model to describe the dynamics of gas exchange in a respiratory chamber and of the measurement processes; 2) analysis of the proposed model and identification of the statistical parameters of the stochastic processes involved; 3) use of the well-known Kalman-Bucy (KB) estimation method to realize online filtering and offline interpolation, thus producing accurate estimates of O₂ and CO₂; 4) experimental validation of the proposed model and of the estimation procedure by processing the data obtained by simulated (known injected gas rates) and real (humans) gas exchange in the respiratory chamber. Furthermore, the results obtained by using the proposed method are compared with the ones obtained by applying a conventional method (18).

	METHODS

TOP ABSTRACT METHODS RESULTS DISCUSSION APPENDIX: KB FILTER FORMULATION REFERENCES

 
The respiratory chamber. The respiratory chamber, built in the Metabolism Unit, Catholic University, School of Medicine, Rome, Italy, has a volume of 23.62 m³. It exchanges gas volumes with the external environment [input flow _i (l/min); output flow _o (l/min)] through two sole apertures by means of an adequate pump, mounted to the output aperture, creating a regular airflow. This _o is measured by a flowmeter generating an analog voltage output (0–10 V), directly proportional to the measured airflow (0–150 l/min). A certain portion (1 l/min) is directed to two gas analyzers (specified below), producing the fractions of oxygen and carbon dioxide in the sampled airflow.

The carbon dioxide concentration is measured by a 2% full scale (0–2%) infrared absorption analyzer (URAS 3G, Hartmann & Braun, Frankfurt, Germany), generating voltage output (2–10 V) directly proportional to the measured fraction (0–2%). The oxygen concentration is assessed by a 2% full scale (19–21%) paramagnetic analyzer (Magnos 4G, Hartmann & Braun), generating an analog voltage output (2–10 V) directly proportional to the measured fraction (19–21%). Both gas analyzers are operating with a precision of 0.02 volumetric percentages. The zero values of both analyzers were calibrated by allowing fresh air to flow through the sample and the reference lines simultaneously, whereas the span values were calibrated using commercially available gas mixtures (Rivoira, Torino, Italy). The composition of the gas mixture used to calibrate the O₂ analyzer was 19.48% O₂-balance N₂. The composition of the gas mixture used to calibrate the CO₂ analyzer was 1.5% CO₂-balance N₂. The calibration procedure is as specified by the manufacturers of the gas analyzer and was carried out at the beginning of each experimental session.

The three voltage signals (airflow, O₂ fraction, and CO₂ fraction) are sampled by a data-acquisition board (Keithley DAS-1601), mounted on a standard desktop. An adequate software realizes the necessary procedures to compute and memorize the sampled measurements at a preset sampling period. The memorized data is then used to compute the O₂ and the CO₂ of the subjects in the respiratory chamber, using either the proposed method or the method reported in Ref. 18, which is a representative example of most conventional methods presented in the literature.

A dynamic model for the gas exchange in the respiratory chamber. The scope of this section is to derive a general simple mathematical model to describe the following phenomenon: a controlled volume is subject to an _i, an _o, and within it a source produces or consumes a gas g (see Fig. 1). This model is later used to describe both CO₂ and O₂ by a subject in the respiratory chamber.

View larger version (9K): [in this window] [in a new window]   Fig. 1. Simplified presentation of gas exchange in the respiratory chamber. i(t), Flow rate of input fresh air at time t; o(t), flow rate of output stale air at time t; ug(t), gas production rate (positive or negative) at time t; V, volume.

The following notation is introduced: c^g(t), the mean volumetric spatial fraction of g at time t; _i(t), flow rate of input fresh air at t (in l/min); , volumetric fraction of g at the _i at t; , volumetric fraction of g at the _o at t; _o(t), flow rate of output stale air at t (in l/min); u^g(t), gas production rate (positive or negative) at t (in l/min); and [0,T], the time interval describing the duration of the experiment, where t represents a time instant within this interval.

With the above notation, the mass conservation principle allows the writing of the following fundamental equation

(1)

where V is volume, and the gas volumes and flows are expressed in STP conditions. The following simplifying hypotheses are now assumed.

1) The existence of an adequate ventilation system in the respiratory chamber allows us to assume that "a rapid mixing of respiratory gas with air occurs in the chamber" (explicit in Ref. 16 and implicit in all others); therefore

(2)

2) The concentration of the g in the _i is constant, with a value determined by its standard volumetric fraction in atmospheric air

(3)

3) The _i is equal to the _o and is perfectly measured by an adequate sensor

(4)

Defining now the difference in the volumetric fractions of the gas considered

(5)

and substituting Eqs. 2, 3, 4, 5 in Eq. 1 results in the following linear differential equation

(6)

Unfortunately, in real life, the hypotheses of Eqs. 2, 3, 4 are never perfectly fulfilled, and, as a consequence, the right side of Eq. 6 is not exactly equal to the left side. To account for the latter, it is possible to add a corrective term, w^m, denoted modeling error, resulting in

(7)

where w^m describes the cumulative effect of all approximations introduced by the simplifying hypotheses above (inhomogenity of gas mixture, variation of standard gas fractions, variation of airflows, etc.). The value of this error at each time instant t is not known; nevertheless, it can be well described by a stochastic variable, characterized by its statistical properties (expected value, variance, etc.). Equation 7 turns out to be a Stochastic Linear Differential Model of the phenomenon. A fundamental hypothesis that lies in the basis of this work is that the w^m can be accurately described as a zero mean white Gaussian stationary random process. Such a process is completely defined by its variance, denoted by .

Next, a dynamic description of the gas production/consumption rate u^g is proposed. A generic gas source having a constant production rate can be described by the simple model d/dt [u^g(t)] = 0. If, on the other hand, the production rate varies in an unknown manner around a nominal value, the following simple stochastic model can be used

(8)

where w^u is a zero-mean white Gaussian random process, describing the variations in the production rate, i.e., the error in assuming a constant production/consumption rate. This random process is completely defined by its variance .

The model in Eq. 8 is commonly referred to as random walk and, in this case, plays only a descriptive role and should not be interpreted as a model of the physiological phenomenon of the gas production. Because O₂ and CO₂ are highly related processes, it is reasonable to assume similar temporal behavior (dynamic model), which implies the use of the same . The results obtained, presented in the sequel, further validate the use of this simple stochastic model and the parameters here defined.

To complete the mathematical modeling of gas exchange in the respiratory chamber, the measurement process has to be formulated as well. The available measurements are the differences in the volumetric fractions of the gas considered, as defined in Eq. 5, and are affected by an additive measurement noise (present in any real experimental setting)

(9)

where is a white Gaussian process, fully identified by its expected value and variance . Equation 9 is widely referred to as the measurement equation.

Putting together Eqs. 7, 8, 9, the complete model describing a typical experimental setting of the respiratory chamber can be obtained in a compact matrix form

(10)

where

(11)

The model in Eq. 10 cannot be directly utilized due to the following reasons: 1) measurements are not available on any time instant t, but only at certain discrete time instants t_j = j·, j = 0, 1,..., N, where is a constant sampling period and N is the number of measurements, N = T/; 2) the model is to be resolved using computer software, which is by nature a discrete time digital process.

To account for the above, the following discrete time model is used in the sequel

(12)

where

(13)

Identification of model parameters. To complete the exact model formulation, specifying the values of the following parameters is needed: , denoting the process noise covariance matrix; , denoting the measurement noise covariance matrix. Note that, because the measurement is a scalar, then is also a scalar, and its covariance matrix reduces simply to the noise variance.

The has been experimentally obtained by performing a set of measurements of a known constant volumetric fractions (specifically zero) and calculating the variance of the obtained measurements.

As far as and are concerned, the authors have assessed their values by performing the following data fitting procedure.

1) Three experiments were carried out in which gas was injected into the respiratory chamber at accurately measured rates. These gas inputs simulated a step-type phenomenon, starting with zero rates for 20 min and followed by some constant value (200 ml/min in experiment A, 120 ml/min in experiments B and C) during the subsequent 20 min. Gas fractions and airflows were measured with a sampling period of 5 s.

2) The gas production has been estimated by applying the KB methods using different values for within the interval (10^-16... 10^-4) (min^-2), and for within the interval (10^-10... 10²) (l²/min⁴). The was fixed to the value derived experimentally as specified above.

3) The fitting error (mean of the absolute values of the instantaneous errors) when comparing the known gas inputs with their estimates was calculated for each variance pair ().

4) The selected values for and are the ones minimizing the fitting error.

The KB estimation method. The problem of estimating the state x(j) using the available measurements y(j)in Eq. 12 turns out to be a linear Gaussian problem that can be solved by applying the KB estimation method. This is a well-known estimation procedure, which processes the measured data using rational criteria and accounts for the available a priori information regarding both the deterministic and the stochastic nature of the observed phenomena. The estimated state at time j based on the set of measurements y(0)... y(k) is denoted (j|k). The larger the set of measurements, the more accurate becomes the estimate.

The KB procedure has an important theoretical property, being the Best Linear Unbiased Estimator, in the sense of minimizing the expected value of the squared estimation error. The estimation error is defined as the difference between the estimated and the true value, i.e., ê(j|k) = x(j) - (j|k). The demonstration of this property is not straightforward; the interested reader is, therefore, referred to various texts, e.g., Refs. 3, 8, 19.

In general, the estimation problem can be resolved for the following three possibilities: 1) j > k, prediction, where the time index of the estimate j is in the future of the available measurements; 2) j = k, filtering, where the time index of the estimate is equal to that of the available measurements [this case is used for online (real-time) implementation]; and 3) j < k, interpolation, where the time index of the estimate j is in the past of the available measurements, thus enabling the use of more information with respect to the other cases.

The KB method computes iteratively both the state estimate, (j|k), and the covariance matrix of the estimation error

(14)

Application of the KB methods requires the specification of the covariance matrices describing the random processes used, i.e., _w = cov (w) and = cov (). Furthermore, it is necessary to specify initial conditions for both the estimated state, (j = 0), and the _ê, _ê(j = 0), the choice of which is not critical to the performance of the KB method because it is proved that, as the index j grows, the estimate becomes independent of this selection (8). According to the information available at the initial conditions of the experiments, the following values are used in this study: 1) initial state, for the O₂ estimates and for the CO₂ estimates , where the first component is the initial differential volumetric fraction (unitless) and the second is the initial gas production rate (in l/min); 2) initial , where 0.1 is the variance associated with the gas fraction estimation error, thus dimensionless, and 1 is associated with the u^g [therefore, in (l/min)²].

Various formulations of the above algorithms are known in literature. The APPENDIX reports the formulation realized by the authors (3). In this study, the authors present the implementation of both interpolator (offline) and filter (online) versions of the KB method.

Experimental validation. Experimental validation is performed by two sets of experiments. The first set is realized by simulating gas production within the respiratory chamber, at accurately measured rates. Gas mixtures of known composition (20% CO₂-1% O₂-balance N₂) were injected into the respiratory chamber at an accurately measured flow rate. The goals of this procedure were to confirm and validate the ability of the proposed method to reproduce a good estimate of the known gas production. Three such experiments were carried out, starting with zero CO₂ injection for 20 min followed by 20 min of a known level of CO₂ gas (200 ml/min in experiment A; 120 ml/min in experiment B; 120 ml/min in experiment C), thus creating a step-type phenomenon, to be estimated by the proposed method. Flow and concentration measurements were performed at the frequency of 0.2 Hz.

The second set involves humans occupying the respiratory chamber. Eleven subjects (4 men and 7 women) were studied. None had diabetes mellitus or any endocrine disease. Their anthropometric characteristics are reported in Table 1. The nature and purpose of the investigations were explained to all subjects before they agreed to participate in the study, which followed the protocol guidelines of the Institutional Review Board. At the time of the examination, all of the subjects studied were on an "ad libitum" diet with the following average composition: 60% carbohydrates, 30% fat, and 10% proteins (at least 1 g protein/kg body wt). This dietary regimen was maintained for 1 wk before the study. The day preceding the experimental session, the patients entered the Energy Metabolism Research Unit in the fasted state. Each subject occupied the respiratory chamber for 24 h, practicing normal activity and also a physical exercise during 30 min, as specified below. Before the experiment, the subjects were allowed to practice walking on the motorized treadmill, until they were able to walk without holding on to the railings, to become familiar with the testing equipment. The physical exercise includes walking for 30 min at 10% grade and with a constant speed of 3 km/h. Flow and concentration measurements were performed at the frequency of 1 Hz and averaged over a 5-min period.

As previously stated, conventional methods all compute respiratory volumes by inverting a mass balance equation, similar to the one in Eq. 6. In this study, the estimates performed using the proposed method are compared with the ones obtained by applying the method in Ref. 18, based on the three phases described in the Introduction. Gas fraction measurements are collected over a certain time period (5 min with humans and 1 min for the experiments with simulated gas production) and prefiltered using arithmetical mean. The u^g are then computed by solving the equations specified there (Eqs. 3, 4, 5, 6 in Ref. 18). Note that, when applying this method, prefiltering by averaging is needed, whereas, when applying the KB methods, no such prefiltering is required, and estimates are available with the same temporal resolution as the measurements.

Analysis of the results. Regarding the experiments with simulated gas production rates, it is possible to compare the estimates with this known input and to compute the instantaneous fitting error. The authors have chosen the mean of the absolute value of this fitting error as an index for the quality of the estimates.

Concerning humans, the true u^g are unavailable for comparison. An alternative comparison approach makes use of the spectra of the estimates. The spectrum of a time series provides information with regard to its temporal behavior. Respiratory gas exchange, as a natural biological phenomenon, has a limited bandwidth; therefore, its spectrum should be concentrated on the lower frequencies and should tend to zero on high frequencies. The existence of high-frequency components in a computed spectrum denotes the presence of estimation errors. To quantify these noisy components, the area contained below the spectrum was calculated, starting from the frequency of 0.05 (min^-1) (i.e., oscillation periods shorter than 20 min). The resulting area (signal power) is proportional to the power of the noisy content and has the dimension of the signal squared, dimensionless for the RQ case and squared milliliters per minute for the O₂ and CO₂.

Statistics. For the experiments with simulated gas production rate, the results are reported as means of the absolute values of the fitting error ± SE, whereas, for experiments with humans, the results are reported as means ± SE, computed over the 11 subjects.

	RESULTS

TOP ABSTRACT METHODS RESULTS DISCUSSION APPENDIX: KB FILTER FORMULATION REFERENCES

 
Model parameters identification. Statistical properties of are assessed by measurements of null gas fractions in the respiratory chamber, resulting in a zero-mean noise {E[(t)] = 0} and (unitless). As for _w, Fig. 2 describes the fitting error obtained when the computed estimate is compared to the known gas production rate. Those values are plotted vs. different values of the pair () used in the KB method. The minimizing values are and . Table 2 summarizes the values of all of the parameters used in the sequel.

View larger version (33K): [in this window] [in a new window]   Fig. 2. Mean fitting error vs. different values of the variance pair (modeling error variance , variance related with random walk model ).

Experimental validation. Results in terms of mean absolute fitting error with regard to the known inputs are presented in Table 3. Figure 3 shows the results of experiment A. The three subfigures plot the performance of the methods applied: KB interpolator, KB filter, and the conventional method (18). The known value of the gas production rate is plotted by a thin line. Note that KB estimates are available at the measurement sampling period (i.e., every 5 s), whereas the conventional method provides estimates every 60 s, due to the prefiltering performed.

View larger version (19K): [in this window] [in a new window]   Fig. 3. Estimates of the known input in experiment A. The Kalman-Bucy (KB) interpolator estimates (top), the KB filter estimates (middle), and the estimates computed by a conventional method (bottom) are indicated by thick lines. The reference input is indicated by thin lines.

As for experiments with humans, typical results of obtained estimates are portrayed in Fig. 4 (O₂ and CO₂) and in Fig. 5 (RQ). Figure 6 shows the mean power spectral density (computed over the 11 subjects) of the O₂ and CO₂ estimates. Note that the spectra are almost identical, but at high frequencies the spectra of the KB estimates are clearly lower than the spectrum of the estimate by the standard method and are converging to zero. Figure 7 plots the results obtained by spectral analysis of the RQ estimates. Note that the units of the signal power are the square of the signal units over frequency units. Because the RQ is dimensionless, the resulting spectrum is quoted in minutes, whereas, for the gas rates, the spectra are in units of milliliters squared per minute. The time scale in Figs. 4, 5, 6, 7 refers to the experiment duration, where the initial time (instant 0) coincides with 0800.

View larger version (25K): [in this window] [in a new window]   Fig. 4. O2 consumption (O2) and CO2 production (CO2) of subject 11. Proposed method, solid line; standard method, dotted line. The peak corresponds with the physical exercise.

View larger version (31K): [in this window] [in a new window]   Fig. 5. Estimated respiratory quotient (RQ) of subject 11 throughout the 24-h experiment. Proposed method, solid line; standard method, dotted line.

View larger version (19K): [in this window] [in a new window]   Fig. 6. Spectral analysis of the O2 and CO2 estimates. Mean spectra are shown of KB interpolator, KB filter, and standard method, computed over the 11 subjects. The spectra of the KB methods are almost identical.

View larger version (17K): [in this window] [in a new window]   Fig. 7. Spectral analysis of the RQ estimates. Mean spectra over the 11 subjects are shown.

Table 4 presents the high-frequency signal power (i.e., oscillatory components with periods shorter than 20 min) computed as the area of the spectra at high frequencies for the estimates performed by the three methods. These data are proportional to the residual noise content in the estimates. Figure 8 shows the temporal evolution of the _ê as produced iteratively by the KB filter algorithm.

View larger version (14K): [in this window] [in a new window]   Fig. 8. Temporal behavior of the estimation error covariance matrix, computed by the KB filter.

	DISCUSSION

TOP ABSTRACT METHODS RESULTS DISCUSSION APPENDIX: KB FILTER FORMULATION REFERENCES

 
Methods presented in the literature to compute respiratory gas production/consumption of a subject in a respiratory chamber are all based on similar mass balance equations, relating the available observations (measurements of volumetric gas fractions, _o) with O₂ and CO₂ (1, 2, 5, 10, 11–13, 16, 18). Some differences do exist between the standard methods, mainly regarding the formulation of the mass balance equation, rising from the different simplifying hypotheses used. Some authors (2, 18) address the problem of noise attenuation by applying certain prefiltering methods but do not explicitly consider the effects of measurement noise and other uncertainties. Heymsfield et al. (10) have formulated for the first time the problem using the terminology of dynamic systems theory, introducing the concepts of stochastic modeling, explicitly considering measurement noise and process noise, fundamental in real applications. However, no solution to the estimation problem was proposed based on those concepts.

In this paper, the authors propose a general stochastic mathematical model describing the dynamics of gas exchange within a controlled volume using an input-state-output approach. The used model is linear, and the stochastic variables describing the uncertainties are considered as zero mean white Gaussian processes, therefore formulating the problem as a Linear Quadratic Gaussian (LQG) problem. Different from Ref. 10, the u^g (O₂ and CO₂) are treated as part of the system state, and, therefore, their estimation using the classic KB methods is made possible. As it is well known, the KB estimation method is optimal for such problems (LQG) in the sense of minimizing the expected value of the squared estimation error. The accuracy improvement is verified by the experimental validation performed in this study.

Optimal fitting of the known simulated gas production rates is achieved by using the model parameters reported in Table 2. The low value of points out that the modeling errors are negligible, thus implying that the stochastic modeling (Eq. 7) is very adequate for this class of problems (dynamics of gas exchange in a controlled volume) and that the assumed hypotheses are well satisfied. Regarding , its value describes the uncertainty in variations in the temporal evolution of the u^g and should be, therefore, large enough to account for abrupt changes, e.g., the ones occurring at the beginning and ending of the physical exercise. One possible modification might be the use of a time-dependent ), in particular, considering large at the instants of beginning and ending of the physical exercise. Such modification requires the use of a priori information regarding these instants.

The experiments with simulated gas production rates are an important procedure to validate both the proposed model and the estimation method, because they offer a unique opportunity to compare the estimates of a variable with its true values. Clearly the results obtained by applying the proposed method are much better than those obtained by using the standard (see Table 3). Note in Fig. 3 the large variations in the standard method, due, in all probability, to the propagation of the measurement and model errors into the computation of the gas production. Contrarily, the KB estimates are much smoother, where KB interpolator performs better than the KB filter, as expected, and removes the delay in detecting the step phenomenon.

Moreover, the KB methods are not constrained regarding the time resolution, as it is possible to apply the KB method for any sampling period . In fact, at the experiments with simulated gas production, the KB methods process the measurements at 5-s intervals, with no averaging (prefiltering) involved, whereas the conventional method (18) demands averaging of the measurements over some significant period (60 s, i.e., 12 samples). The KB methods produce estimates that are more accurate and with higher temporal resolution (5 s in the KB estimates vs. 60 s in the standard method).

Regarding experiments with humans, the measurement data were available already averaged over a 5-min period (prefiltered). In this case, the obtained results demonstrate that, by using the KB method, it is possible to further improve the estimates and to attenuate residual noise content that is not removed by the prefiltering process. Note in Figs. 4 and 5 that the standard method produces highly scattered estimates, making it hard to deduce the physiological condition of patients, whereas the ones obtained using the proposed methods are much smoother. Moreover, as it is well known, an acceptable range for RQ values is from 0.7 to 1.0. Clearly the estimates by the standard method fail to comply with this range, whereas the KB estimates generally meet this constraint.

The spectral analysis demonstrates that, at low frequencies, all spectra (by the different methods) are practically identical. Nevertheless, Fig. 6 clearly shows that the spectra of the standard method estimates do not go to zero on high frequencies (thus denoting high-noise content), whereas the spectra of the KB estimates do converge to zero on those frequencies. A similar and even more distinguished phenomenon is revealed by the spectra of the RQ in Fig. 7. Table 4 quantifies the noise contents, clearly visible in Figs. 6 and 7. The noise content in the estimates obtained by the standard method is two to three orders of magnitude larger than that obtained by the KB methods, thus confirming the superiority of the proposed method in the case of human patients as well. Although variations between different subjects are sometimes large, the superiority of the KB methods over the conventional method is, again, very clear.

Figure 8 plots the temporal evolution of the diagonal elements of the _ê (Eq. A6). These values are, in fact, online estimates of the accuracy of the estimation procedure. Note that this variance converges rapidly to a steady-state value, thus confirming the filter stability and its robustness with regard to the initial condition values. This fact indicates quasi-stationary behavior of the filter, resulting from the fact that the temporal variations in the matrix H(j) (Eq. 13) are negligible. In such case, it might be possible to use the steady-state version of the KB algorithm (the Wiener filter) and reduce the computational load, although in this study it was preferred to use the more general method.

In conclusion, a general mathematical model is presented, accurately describing gas exchange in a respiratory chamber. This model serves to estimate O₂ and CO₂ by applying the KB methods, resulting in estimates highly superior to the ones obtained by conventional method in all cases studied. Computation of RQ, based on the obtained estimates, complies better with biological understanding of such phenomenon.

	APPENDIX: KB FILTER FORMULATION

TOP ABSTRACT METHODS RESULTS DISCUSSION APPENDIX: KB FILTER FORMULATION REFERENCES

 
Following the presentation in Ref. 3, the general LQG estimation problem can be solved by three sequential phases: 1) single-step prediction, 2) filtering, and 3) interpolation.

Single-step prediction.

(A1)

(A2)

(A3)

(A4)

With the use of initial conditions for the state (j = 0) and the initial error covariance matrix, _ê(j = 0), as provided in METHODS.

Filtering.

(A5)

(A6)

Smoothing.

(A7)

(A8)

(A9)

The smoother problem is solved backward (from j = k to j = 1), making use of the solution to the filtering problem, which, in turn, is based on the solution of the single-step prediction problem.

	FOOTNOTES

 

Address for reprint requests and other correspondence: G. Mingrone, Istituto di Medicina Interna, Universita Cattolica del Sacro Cuore, 00168 Rome, Italy (E-mail: gmingrone{at}rm.unicatt.it).

The costs of publication of this article were defrayed in part by the payment of page charges. The article must therefore be hereby marked "advertisement" in accordance with 18 U.S.C. Section 1734 solely to indicate this fact.

	REFERENCES

TOP ABSTRACT METHODS RESULTS DISCUSSION APPENDIX: KB FILTER FORMULATION REFERENCES

 

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