J Appl Physiol 96: 1045–1054, 2004.
First published November 14, 2003; 10.1152/japplphysiol.00788.2003.
V̇O2, V̇CO2, and RQ in a respiratory chamber: accurate estimation based on a
new mathematical model using the Kalman-Bucy method
L. Granato,1 A. Brandes,1 C. Bruni,1 A. V. Greco,2 and G. Mingrone2
1
Dipartimento 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
ACCURATE MONITORING OF GAS exchange [O2 consumption (V̇O2)
and CO2 production (V̇CO2)] of a subject in health and disease
is of large interest in medical research (4, 9, 17, 20–22). It
enables the assessment of important physiological indexes,
such as oxidation rate of energy substrate and energy expenditure (6, 7, 12). Although various instruments for indirect
calorimetry exist, the respiratory chamber is the only one
permitting continuous long-term monitoring (24 h) of patients,
thus offering a unique opportunity to study important aspects
of energy metabolism in humans practicing their daily life
activities (sleeping, eating, physical activity, etc.). The precision of the measurements of respiratory gas becomes of crucial
importance to obtain precise metabolic data, as for example,
computation of respiratory quotient (RQ) (V̇CO2-to-V̇O2 ratio),
which is very sensible to the accuracy of the V̇O2 estimates.
To estimate V̇O2 and V̇CO2, 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 (V̇O2
and V̇CO2) 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) V̇CO2 and V̇O2 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 (V̇O2, V̇CO2, 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 wellknown Kalman-Bucy (KB) estimation method to realize online filtering and offline interpolation, thus producing accurate
estimates of V̇O2 and V̇CO2; 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
Address for reprint requests and other correspondence: G. Mingrone, Istituto
di Medicina Interna, Universita Cattolica del Sacro Cuore, 00168 Rome, Italy
(E-mail: [email protected]).
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.
indirect calorimetry; respiratory gas exchange; energy expenditure;
Kalman filter; stochastic model
http://www.jap.org
8750-7587/04 $5.00 Copyright © 2004 the American Physiological Society
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Granato, L., A. Brandes, C. Bruni, A. V. Greco, and G.
Mingrone. V̇O2, V̇CO2, and RQ in a respiratory chamber: accurate
estimation based on a new mathematical model using the KalmanBucy method. J Appl Physiol 96: 1045–1054, 2004. First published
November 14, 2003; 10.1152/japplphysiol.00788.2003.—A respiratory chamber is used for monitoring O2 consumption (V̇O2), CO2
production (V̇CO2), and respiratory quotient (RQ) in humans, enabling
long term (24-h) observation under free-living conditions. Computation of V̇O2 and V̇CO2 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 V̇O2 and V̇CO2 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 V̇O2 and
V̇CO2 estimates. Consequently, the accuracy of the computed RQ is
also highly improved (0.3 ⫻ 10⫺4 vs. 800 ⫻ 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.
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V̇O2 AND V̇CO2 ESTIMATES VIA KALMAN-BUCY METHOD
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.
are compared with the ones obtained by applying a conventional method (18).
METHODS
J Appl Physiol • VOL
V䡠
d g
关c 共t兲兴 ⫽ ␸i共t兲 䡠 cgi 共t兲 ⫺ ␸o共t兲 䡠 cog共t兲 ⫹ ug共t兲
dt
(1)
where V is volume, and the gas volumes and flows are expressed in
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
STP
c og共t兲 ⫽ cg共t兲
(2)
2) The concentration of the g in the ␸i is constant, with a value
determined by its standard volumetric fraction in atmospheric air
c gi 共t兲 ⫽ c៮ g
(3)
3) The ␸i is equal to the ␸o and is perfectly measured by an
adequate sensor
␸ i共t兲 ⫽ ␸o共t兲 ⫽ ␸共t兲
(4)
Defining now the difference in the volumetric fractions of the gas
considered
c̃ g共t兲 ⫽ cog共t兲 ⫺ c៮ g
(5)
and substituting Eqs. 2–5 in Eq. 1 results in the following linear
differential equation
d g
␸共t兲 g
1
关c̃ 共t兲兴 ⫽ ⫺
䡠 c̃ 共t兲 ⫹ 䡠 ug共t兲
dt
V
V
(6)
Unfortunately, in real life, the hypotheses of Eqs. 2–4 are never
perfectly fulfilled, and, as a consequence, the right side of Eq. 6 is not
Table 1. Anthropometric characteristics of subjects
Subject No.
Age, yr
Gender
Height, m
Weight, kg
BMI, kg/m2
1
2
3
4
5
6
7
8
9
10
11
46
38
51
30
26
43
42
55
52
59
47
F
M
F
F
M
F
F
F
M
F
M
1.52
1.78
1.61
1.75
1.62
1.55
1.56
1.50
1.76
1.63
1.83
95
145
123
119
116
96
112
90
132
63
130
41.1
45.8
47.5
38.9
44.2
40.0
46.0
40.0
42.6
23.7
38.8
F, female; M, male; BMI, body mass index.
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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 V̇O2 and the V̇CO2 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 V̇CO2 and
V̇O2 by a subject in the respiratory chamber.
The following notation is introduced: cg(t), the mean volumetric
spatial fraction of g at time t; ␸i(t), flow rate of input fresh air at t (in
l/min); cgi (t), volumetric fraction of g at the ␸i at t; cgo(t), volumetric
fraction of g at the ␸o at t; ␸o(t), flow rate of output stale air at t (in
l/min); ug(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
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V̇O2 AND V̇CO2 ESTIMATES VIA KALMAN-BUCY METHOD
Fig. 2. Mean fitting error vs. different values
of the variance pair (modeling error variance
2
␴m
, variance related with random walk
model ␴2u).
d g
␸共t兲 g
1
关c̃ 共t兲兴 ⫽ ⫺
䡠 c̃ 共t兲 ⫹ 䡠 ug共t兲 ⫹ wm共t兲
dt
V
V
(7)
where wm 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 wm can be accurately described as a zero mean
white Gaussian stationary random process. Such a process is com2
pletely defined by its variance, denoted by ␴m
.
Next, a dynamic description of the gas production/consumption
rate ug is proposed. A generic gas source having a constant production
rate can be described by the simple model d/dt [ug(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
d g
关u 共t兲兴 ⫽ wu共t兲
dt
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)
y共t兲 ⫽ c̃ g共t兲 ⫹ ␯共t兲
where ␯ is a white Gaussian process, fully identified by its expected
value and variance ␴2␯. Equation 9 is widely referred to as the
measurement equation.
Putting together Eqs. 7–9, the complete model describing a typical
experimental setting of the respiratory chamber can be obtained in a
compact matrix form
再
d
关x共t兲兴 ⫽ A共t兲 䡠 x共t兲 ⫹ w共t兲
dt
y共t兲 ⫽ C 䡠 x共t兲 ⫹ ␯共t兲
where
x共t兲 ⫽
冋 册
J Appl Physiol • VOL
冋
册
冋 册
␸共t兲 1
c̃ g共t兲
wm共t兲
⫺
,
A共t兲
⫽
, C ⫽ 关1
V V , w共t兲 ⫽
g
u 共t兲
wu共t兲
0
0
(8)
where wu 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 ␴2u.
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 V̇O2 and V̇CO2 are highly related processes, it is reasonable to
assume similar temporal behavior (dynamic model), which implies the
use of the same ␴2u. The results obtained, presented in the sequel,
further validate the use of this simple stochastic model and the
parameters here defined.
(9)
(10)
0兴
(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
Table 2. Variance of noise processes
Parameter
Value
Units
Evaluation Method
2
␴m
␴2u
␴2␯
10⫺12
2.5 ⫻ 10⫺3
10⫺8
min⫺2
l2/min4
Optimal fitting of known inputs
Optimal fitting of known inputs
Measurements of null gas fractions
2
␴m
, modeling error variance; ŷ2u, variance related with random walk model;
␴2␯, variance of measurement noise.
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exactly equal to the left side. To account for the latter, it is possible to
add a corrective term, wm, denoted modeling error, resulting in
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V̇O2 AND V̇CO2 ESTIMATES VIA KALMAN-BUCY METHOD
Table 3. Results of the experiments with
simulated gas production
Experiment A
Experiment B
Experiment C
Mean
KB
Interpolator
KB Filter
Standard
Method
(18)
19⫾1.0
18⫾1.0
10⫾0.7
15.6⫾0.9
42⫾2.7
28⫾2.0
21⫾1.6
30.3⫾2.1
298⫾63.2
127⫾19.8
132⫾24.0
186⫾35.6
Values are means of the absolute values of the fitting error (with respect to
the known gas production rate) ⫾ SE and are reported in ml/min. KB,
Kalman-Bucy.
再
where
H共 j兲 ⫽
冋
x共 j ⫹ 1兲 ⫽ H共 j兲 䡠 x共 j兲 ⫹ ⌬ 䡠 w共 j兲
y共 j兲 ⫽ C 䡠 x共 j兲 ⫹ ␯共 j兲
m共 j兲
0
n
1
册
m共 j兲 ⫽ 1 ⫺
⌬ 䡠 ␸共 j兲
V
(12)
n ⫽ ⌬/V (13)
Identification of model parameters. To complete the exact model
formulation, specifying the values of the following parameters is
wm
␴m2 0
needed: 1) ⌿w ⫽ cov 共w兲 ⫽ cov
⫽
, denoting the
wu
0 ␴u2
2
process noise covariance matrix; 2) ⌿␯ ⫽ cov (␯) ⫽ ␴␯, 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 ␴2␯ 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.
冉 冊 冉
冊
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.
J Appl Physiol • VOL
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only at certain discrete time instants tj ⫽ j䡠⌬, 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
2
As far as ␴m
and ␴2u 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
2
methods using different values for ␴m
within the interval (10⫺16 . . .
2
⫺4
⫺2
10 ) (min ), and for ␴u within the interval (10⫺10 . . . 102)
(l2/min4). The ␴2␯ 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
2
estimates was calculated for each variance pair (␴m
, ␴2u).
2
4) The selected values for ␴m
and ␴2u 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 x̂( jⱍk). 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( jⱍk) ⫽ x( j) ⫺ x̂( jⱍk). 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,
V̇O2 AND V̇CO2 ESTIMATES VIA KALMAN-BUCY METHOD
1049
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, x̂( jⱍk),
and the covariance matrix of the estimation error
⌿ ê 共 j兩k兲 ⫽ cov 关ê共 j兩k兲兴
(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, x̂( 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 V̇O2 estimates
0
0
x̂共 j ⫽ 0兲 ⫽
and for the V̇CO2 estimates x̂共 j ⫽ 0兲 ⫽
,
0.3
0.2
where the first component is the initial differential volumetric fraction
(unitless) and the second is the initial gas production rate (in l/min);
0.1 0
2) initial ⌿ê, ⌿ê共 j ⫽ 0兲 ⫽
, where 0.1 is the variance
0 1
associated with the gas fraction estimation error, thus dimensionless,
and 1 is associated with the ug [therefore, in (l/min)2].
冉 冊
冉 冊
冉
冊
J Appl Physiol • VOL
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
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Fig. 4. O2 consumption (V̇O2) and CO2 production (V̇CO2) of subject 11. Proposed method, solid line; standard method, dotted line.
The peak corresponds with the physical exercise.
1050
V̇O2 AND V̇CO2 ESTIMATES VIA KALMAN-BUCY METHOD
Fig. 5. Estimated respiratory quotient (RQ)
of subject 11 throughout the 24-h experiment. Proposed method, solid line; standard
method, dotted line.
J Appl Physiol • VOL
trum 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 V̇O2
and V̇CO2.
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
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 ␴2␯ ⫽ 10⫺18 (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
2
plotted vs. different values of the pair (␴m
, ␴2u) used in the KB
2
method. The minimizing values are ␴m ⫽ 10⫺12 min⫺2 and ␴2u
⫽ 2.5 ⫻ 10⫺3 l2/min4. Table 2 summarizes the values of all of
the parameters used in the sequel.
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
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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 ug are then
computed by solving the equations specified there (Eqs. 3–6 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 ug 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 spec-
V̇O2 AND V̇CO2 ESTIMATES VIA KALMAN-BUCY METHOD
DISCUSSION
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 V̇O2 and V̇CO2 (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
Fig. 6. Spectral analysis of the V̇O2 and V̇CO2 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.
J Appl Physiol • VOL
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sampling period (i.e., every 5 s), whereas the conventional
method provides estimates every 60 s, due to the prefiltering
performed.
As for experiments with humans, typical results of obtained
estimates are portrayed in Fig. 4 (V̇O2 and V̇CO2) and in Fig. 5
(RQ). Figure 6 shows the mean power spectral density (computed over the 11 subjects) of the V̇O2 and V̇CO2 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. 4–7 refers to the experiment duration, where the initial
time (instant 0) coincides with 0800.
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.
1051
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V̇O2 AND V̇CO2 ESTIMATES VIA KALMAN-BUCY METHOD
Linear Quadratic Gaussian (LQG) problem. Different from
Ref. 10, the ug (V̇O2 and V̇CO2) 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.
2
The low value of ␴m
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 ␴2u, its value describes
the uncertainty in variations in the temporal evolution of the ug
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
Table 4. Spectral content at noise (high) frequencies
Units
RQ
V̇O2 (ml/min)2
V̇CO2 (ml/min)2
KB Interpolator
KB Filter
Standard Method
(3.2⫾2.0)⫻10⫺5
2.9⫾0.8
1.8⫾0.5
(6.1⫾2.2)⫻10⫺4
90.8⫾22.5
57.6⫾15.1
(8.1⫾4.0)⫻10⫺2
3,440⫾824
2,057⫾532
Values are means ⫾ SE for 11 subjects. RQ, respiratory quotient; V̇O2, O2
consumption; V̇CO2, CO2 production.
J Appl Physiol • VOL
use of a time-dependent ␴2u ⫽ ␴2u(t), in particular, considering
large ␴2u 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).
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Fig. 7. Spectral analysis of the RQ estimates. Mean spectra over the 11 subjects are shown.
V̇O2 AND V̇CO2 ESTIMATES VIA KALMAN-BUCY METHOD
1053
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
J Appl Physiol • VOL
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 quasistationary 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.
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Fig. 8. Temporal behavior of the estimation error covariance matrix, computed by the KB filter.
1054
V̇O2 AND V̇CO2 ESTIMATES VIA KALMAN-BUCY METHOD
In conclusion, a general mathematical model is presented,
accurately describing gas exchange in a respiratory chamber.
This model serves to estimate V̇O2 and V̇CO2 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
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.
F共 j兲 ⫽ H共 j兲 䡠 ⌿ ê 䡠 共 j兩 j ⫺ 1兲 䡠 C T 䡠 关C 䡠 ⌿ê 䡠 共 j兩 j ⫺ 1兲 䡠 CT ⫹ ⌿␯兴⫺1
(A1)
G共 j兲 ⫽ H共 j兲 ⫺ F共 j兲 䡠 C
(A2)
2
⫹ F共 j兲 䡠 ⌿␯ 䡠 FT共 j兲
x̂共 j ⫹ 1兩 j兲 ⫽ G共 j兲 䡠 x̂共 j兩 j ⫺ 1兲 ⫹ F共 j兲 䡠 y共 j兲
(A3)
(A4)
With the use of initial conditions for the state x̂( j ⫽ 0) and the initial
error covariance matrix, ⌿ê( j ⫽ 0), as provided in METHODS.
Filtering.
x̂共 j兩 j兲 ⫽ H ⫺1 共 j兲 䡠 x̂共 j ⫹ 1兩 j兲
⫺1
⌿ ê 䡠 共 j兩 j兲 ⫽ H 共 j兲 䡠 ⌿ ê 䡠 共 j ⫹ 1兩 j兲 䡠 H
⫺1 T
(A5)
共 j兲
⫺ ⌬ 䡠 H⫺1共 j兲 䡠 ⌿w 䡠 H⫺1 共 j兲
T
2
(A6)
Smoothing.
M共 j兲 ⫽ H ⫺1 共 j兲 ⫺ ⌬ 2 䡠 H ⫺1 共 j兲 䡠 ⌿ w 䡠 ⌿ ê⫺1 共 j ⫹ 1兩 j兲
⫺1
N共 j兲 ⫽ H 共 j兲 ⫺ M共 j兲
(A7)
(A8)
x̂共 j兩k兲 ⫽ M共 j兲 䡠 x̂共 j ⫹ 1兩k兲 ⫹ N共 j兲 䡠 x̂共 j ⫹ 1兩 j兲
j⬍k
(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.
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