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J Appl Physiol 106: 640-650, 2009. First published November 13, 2008; doi:10.1152/japplphysiol.90718.2008
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INNOVATIVE METHODOLOGY

INNOVATIVE METHODOLOGIES

Algorithm for transient response of whole body indirect calorimeter: deconvolution with a regularization parameter

Kumpei Tokuyama,1 Hitomi Ogata,1 Yasuko Katayose,2 and Makoto Satoh2

1Division of Sports Medicine and 2Division of Sleep Medicine, School of Comprehensive Human Science, University of Tsukuba, Ibaraki, Japan

Submitted 4 June 2008 ; accepted in final form 11 November 2008


   ABSTRACT
TOP
 ABSTRACT
METHODS
 RESULTS
 DISCUSSION
 GRANTS
 ACKNOWLEDGMENTS
 REFERENCES
 
A whole body indirect calorimeter provides accurate measurement of energy expenditure over long periods of time, but it has limitations to assess its dynamic changes. The present study aimed to improve algorithms to compute O2 consumption and CO2 production by adopting a stochastic deconvolution method, which controls the relative weight of fidelity to the data and smoothness of the estimates. The performance of the new algorithm was compared with that of other algorithms (moving average, trends identification, Kalman filter, and Kalman smoothing) against validation tests in which energy metabolism was evaluated every 1 min. First, an in silico simulation study, rectangular or sinusoidal inputs of gradually decreasing periods (64, 32, 16, and 8 min) were applied, and samples collected from the output were corrupted with superimposed noise. Second, CO2 was infused into a chamber in gradually decreasing intervals and the CO2 production rate was estimated by algorithms. In terms of recovery, mean square error, and correlation to the known input signal in the validation tests, deconvolution performed better than the other algorithms. Finally, as a case study, the time course of energy metabolism during sleep, the stages of which were assessed by a standard polysomnogram, was measured in a whole body indirect calorimeter. Analysis of covariance revealed an association of energy expenditure with sleep stage, and energy expenditure computed by deconvolution and Kalman smoothing was more closely associated with sleep stages than that based on trends identification and the Kalman filter. The new algorithm significantly improved the transient response of the whole body indirect calorimeter.

time resolution; Kalman-Bucy method

INDIRECT CALORIMETRY is the method by which energy expenditure and substrate oxidation rate are measured in vivo starting from gas exchange measurements. Its principles were described by Lavoisier in the 18th century, and the chamber for human indirect calorimeter was built by Pettenkoffer and Atwater et al. (16) before the turn of the 20th century. A detailed analysis of gaseous exchange in open-circuit, indirect calorimetry was completed by Brown et al. (5), and the relation between rate of production and concentration of any gas is given by the first-order, linear differential equation

Formula 1(1)
where F is flow rate in l/min, f is fractional concentration, R is rate of gas production in l/min, t is time in min, V is volume of chamber in liters, i is incoming, o is outgoing, and G is any gas [the symbols used throughout are consistent with those used by Brown et al. (5), and volume and flow rate were assumed to be corrected to STPD]. As room air is assumed to be well mixed before it is purged, fGo in the outgoing air represents the gas concentration in the chamber.

Although the equation provides accurate estimates of energy expenditure and the respiration quotient (RQ) for conventional 24-h measurements (13, 23, 31), simple application of the equation has limitations to assess dynamic changes in energy metabolism due to the amplitude of the signal in relation to the size of the room. The majority of existing indirect human calorimeters therefore bypass the low signal-to-noise ratio problem by making the measurement interval 15–60 min or longer. Introducing signal processing techniques, several human calorimeters stand out to achieve an excellent transient response. In Houston (17), Vanderbilt (27), Göteborg (12), and Pennington (19), a series of measurements is collected at any generic subinterval and the accuracy of the measurements is improved by averaging the gas concentration measurement, but the criteria for an acceptable error were ambiguous. In Rome (10), where a stochastic model was adopted to describe the dynamics of gas exchange in a respiratory chamber and of the measurement process, a different approach known as the Kalman-Bucy method was applied.

Many signals of interest for the quantitative understanding of the physiological system are not directly measured. Some examples include the secretion rate of a hormone, the production rate of a substrate, or the appearance rate of a drug in plasma after oral administration. In these cases, it is only possible to measure the causally related effects of these signals in the circulation (e.g., the time course of plasma concentration). Computation of the metabolic rate can also be viewed as reconstructing an unknown signal (metabolic rate) from the known causally related effects of the signal (gas concentration in the chamber). When dealing with linear time-invariant systems, like the chamber of whole body calorimeter, the computation of the input signal from the effects is called deconvolution (6). However, deconvolution is known to be ill conditioned, which means a small percent error in the measured effect (e.g., the measured gas concentration in the chamber) can produce a much greater percent error in the estimated cause (e.g., metabolic rate). In the literature, many methods have been developed to circumvent ill conditioning. By adopting a regularization parameter as a noise suppression method for the deconvolution, the present study proposes a new algorithm for whole body indirect calorimetry. In this stochastic approach, the relative weight given to data fit and solution regularity is governed by the so-called regularization parameter {gamma} (7). This approach has successfully been applied to estimate the rate of insulin secretion (26) and endogenous glucose production (28).

In the present study, the performance of algorithms was compared by validating against known input signals. First, in the in silico simulation study, the same synthetic data set corrupted with superimposed noise was used. Second, the CO2 infusion test, in gradually decreasing interval, was performed. Finally, as a case study, subtle changes in the sleeping metabolic rate estimated by different algorithms were compared.


   METHODS
TOP
 ABSTRACT
 METHODS
RESULTS
 DISCUSSION
 GRANTS
 ACKNOWLEDGMENTS
 REFERENCES
 
Specification of human calorimeter

The internal volume of the chamber converted to STPD is 16.626 m3, and the room air is mixed sufficiently by fans. The chamber is a "pull" calorimeter, i.e., the flow is controlled and measured at the outlet at a rate of 50–150 l/min. To measure the outgoing flow, the chamber is equipped with a mass flow controller (CMQ02, Yamatake, Tokyo). The concentrations of gases in outgoing air are measured with high precision by online process mass spectrometry (VG Prima {delta}B; Thermo Fisher Scientific). The mass spectrometer measures fractional concentrations of O2, CO2, 13CO2, N2, and argon. The concentrations of N2 and argon were included in the measurements to correct for pressure and sensitivity changes of the instrument via normalization. To calculate rates of gas production, the concentrations of 13CO2 and argon were added to those of CO2 and N2, respectively. Instrument measurements were supplied at every 1 min, and flow rates were converted to STPD. The precision of mass spectrometry, defined as SD for continuous measurement of calibration gas mixture (O2 15%-CO2 5%), was demonstrated to be 0.0016% for O2 and 0.0011% for CO2, respectively.

Respiration algorithms

Moving average method.   In Houston, the calorimeter was operated with closed-loop control of air inflow rate to maintain a constant room CO2 concentration (usually 0.45%). The tests began at a minimum air flow rate of 13 l/min until the CO2 concentration of the chamber approaches the set point, followed by an increase in flow rate. The changes in gas concentration in the chamber were calculated by reprocessing the data before and after 1, 2, or 3 min of the time of interest (17). The method relied on data at both endpoints of the interval, and the intermediate values between them were ignored. More efficient use of the data was adopted in Vanderbilt, where the sampling rate was optimized (every 5 s) and the accumulation rates of O2 and CO2 in the chamber were calculated over 1 min with a signal processing technique (moving average method). Additionally, the data were passed through a median filter: ymf(t) = median [y(t ± {tau})], where {tau} = 1 min (27).

Trends identification method.   In Göteborg, another algorithm for noise suppression was developed using trends identification. At every minute, the program looks back at the preceding 30 min and partitions the interval (30 min) into two steady state conditions (12). Using the exact solution of the equations for steady state, each gas concentration was fitted by a least square method to two connected exponential segments of variable length for the preceding 30-min period. Independently of the location of the joint between the two segments, the gas concentration and its time derivative were evaluated at –15 min and gas production rates were computed from Eq. 1.

Kalman-Bucy method.   In Rome, the Kalman-Bucy method was applied for whole body indirect calorimetry (10). This data processing algorithm combines available measurement data plus prior knowledge about the system and measuring devices to produce an estimate of the desired variables in such a manner that the error is statistically minimized. Recursively, the algorithm projects the current state estimate ahead of time, which is subsequently corrected by measurement. The Kalman-Bucy method realizes "filtering" or "smoothing"; the former estimates the gas production rate at each time instant based on past measurements, while the latter is based on past and future measurements. It is necessary to specify initial conditions for both the estimated state and the covariance of estimation error, the choice of which is not critical to the performance of the Kalman-Bucy method, because as the recursive process continues, the estimate becomes independent of this selection. In the present study, the initial condition of O2 consumption and CO2 production was set at 0.3 and 0.2 l/min, respectively. An introduction to the Kalman-Bucy method can be found in the literature (15).

Deconvolution.   Viewing a chamber for indirect calorimetry as a linear time-invariant system, it is possible to relate mass of any gas in the chamber at t and its rate of appearance at time {tau} by a convolution summation (6; Fig. 1).

Formula 2(2)
where the function h(t) describes the input-output behavior of the chamber and is called the impulse response of the system. If we split Ra in two components, gas in the incoming air [Fi({tau})fGi({tau})] and metabolism of the subject [RG({tau})], Eq. 2 becomes

Formula 3(3)
The left-hand side and the first term of the right-hand side of the equation 3 can be estimated from chamber volume (V), gas concentration (fGi, fGo), and flow rate (Fi). Rearranging the equation, the inverse problem to solve is

Formula 4(4)
Adopting matrix notation, it is expressed as

Formula 5(5)
where c is a vector of dimension n containing changes of gas in the chamber sampled at times t1 < t2,..., tn (t1 > 0); u is a vector of dimension n containing the metabolic rate sampled at times {tau}1 < {tau}2,..., {tau}n ({tau}1 = 0; {tau}i = ti-1 for i = 2,..., n) and assumed to be piecewise constant; and H is a n x n lower triangular matrix of impulse response, whose entries are given analytically by

Formula 6(6)
where {lambda} is F/V. Each nonzero element describes the output of the model at time ti when all initial conditions are zero and the input is a pulse of a unit amplitude between tj-1 and tj. Since direct solution of the deconvolution problem can be ill conditioned, the Phillips-Tikhonov regularization approach was adopted in the present study (7). The regularized estimate u is defined as

Formula 7(7)
where y denotes the n-dimension vector of the noisy data and its measurement error was assumed uncorrelated (E[e] = 0, Cov[e] = B). Q is the square Toeplitz matrix (n x n), whose first column is [1, –2, 1, 0,..., 0]T. The first term on the right-hand side in Eq. 7 measures the fidelity to the data, whereas the second term is introduced to penalize the roughness of the estimate. The relative weight given to data fit and solution regularity is governed by the regularization parameter {gamma}. Large values of {gamma} will return very smooth solutions, while small values of {gamma} will lead to ill-conditioned estimates. The closed-form solution of Eq. 7 is

Formula 8(8)


Figure 1
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  		Fig. 1. Input estimation by deconvolution. Input signals at time {tau} (left) applied to a linear time-invariant dynamic system result in series of impulse responses (center), and output at time t is summation of each response (right). Assuming that the output (mass of gas in the chamber) is measurable and system dynamics is known, input signal can be obtained as a piecewise constant function by deconvolution. An input that begins at {tau} = 3 (left) and its response (center) is highlighted. Response depends on t – {tau}, and impulse response to a unit signal is described in Eq. 6, which reaches to (1-e–{lambda})/{lambda} at (t – {tau}) = 1 followed by a decrease as {(1-e–{lambda})/{lambda}}e–{lambda}({tau} – {tau}).

 
The regularization parameter {gamma} was tuned as maximum likelihood estimate, assuming the accuracy of the gas concentration measurement was known (7, 28)

Formula 9(9)
where SSU is sum of the squared estimate and

Formula 10(10)

Validation study

Simulation study.   The model assumed the chamber as a single compartment (V = 16.626 m3), and completely mixed air in the chamber was constantly removed (FO = 62.0 l/min). The gas concentration of incoming air (20.93% O2-0.03% CO2) was assumed to be constant. Changes in the gas concentration in the chamber were generated from periodic input signals for the O2 and CO2 production rates by simulation software SAAMII (University of Washington). To simulate experimental conditions, the generated data for O2 and CO2 concentrations were corrupted by artificially generated noises. Random noises added to the measurement of O2 and CO2 concentrations were uncorrelated, and the SD was 0.0016% for O2 and 0.0011% for CO2, respectively. As input signals, rectangular (Fig. 2) and sinusoidal signals (Fig. 3) were used in the present study. Estimates of the O2 consumption and CO2 production rate from the noisy data by moving average, trends identification, Kalman-Bucy (filtering and smoothing), and deconvolution method were compared with the known input signal.


Figure 2
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  		Fig. 2. Simulation with rectangular input signal. The chamber of whole body calorimeter was modeled as a single compartment (V = 16.626 m3), from which air was pulled out at a rate of 62 l/min. Rectangular inputs for the system, 0.4 l/min for O2 consumption and CO2 production, were applied in gradually decreasing intervals. At 45 min of the simulation, input of a 64-min period (32 min on and 32 min off) was initiated, followed by those of 32-, 16-, and 8-min periods. Against input signal shown as straight line, oxygen consumption estimated by moving average (A), trends identification (B), Kalman filter (C), Kalman smoothing (D), and deconvolution (E) is shown as circles. Regularization parameter for deconvolution ({gamma}) was 189.

 

Figure 3
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  		Fig. 3. Simulation with sinusoidal input signal. The chamber of whole body calorimeter was modeled as a single compartment (V = 16.626 m3), from which air was pulled out at a rate of 62 l/min. After 45 min of constant signal, sinusoidal input, 0.4 + 0.2sin[t(2{pi}/period)] l/min for O2 consumption and CO2 production, of a period 64 min (45–109 min), 32 min (109–173 min), 16 min (173–237 min), and 8 min (237–301 min) was applied. Against input signal shown as a line, oxygen consumption estimated by moving average (A), trends identification (B), Kalman filter (C), Kalman smoothing (D), and deconvolution (E) is shown as circles. Regularization parameter for deconvolution ({gamma}) was 401.

 
CO2 infusion test.   The CO2 infusion test was performed by infusing a mixture of CO2 and N2 (20 and 80%, respectively) at a rate of 1.888 l/min (i.e., 0.3776 l CO2/min) in gradually decreasing intervals using mass flow controller (MQV0020; Yamatake, Tokyo). The gas concentration of outgoing air was measured at every 1 min, and the CO2 infusion rate was estimated by trends identification, Kalman filter, Kalman smoothing, and deconvolution. The moving average method was not applied for the present data, since the gas concentration measurement at every 5 s was not available in our system.

Case study.   During sleep in the whole body indirect calorimeter, the energy metabolism of a male subject (176.7 cm, 78.2 kg) was measured at a 1-min interval. Air in the chamber is pumped out at a rate of 70 l/min. The temperature and relative humidity of incoming fresh air were controlled at 25.0 ± 0.5°C and 55.0 ± 3.0%. The gas concentration of outgoing air was measured at every 1 min, and O2 consumption and CO2 production rate were estimated by trends identification, Kalman filter, Kalman smoothing, and deconvolution. The moving average method was not applied for the present data, since the gas concentration measurement at every 5 s was not available in our system. Energy expenditure was calculated from the rates of O2 consumption and CO2 production, according to Weir's equation (30).

To assess sleep stages, a standard polysomnogram recording (22) was recorded on Alice5 1848 (Respironics, Murrysville, PA). The records were coded, and 30-s epochs were used to score sleep stages, according to conventional criteria (1). If the sleep stage was constant during a 1-min interval of energy expenditure measurement, the data were used for statistical analysis. To isolate the an effect of sleep stage on energy metabolism, the effect of sleep time on energy metabolism was removed using analysis of covariance (ANCOVA). The values of energy expenditure of each sleep stage were then analyzed by one-way ANOVA followed by Bonferroni's tests.

The protocol was approved by the local ethical committee of the University of Tsukuba, and was conducted in accordance with the Helsinki Declaration.

Calculations

Recovery in the simulation and CO2 infusion test was defined as the area under the curve of estimate divided by that of the known input signal. The program for the trends identification algorithm, Kalman-Bucy method, and deconvolution was written using C++ on Linux operating system. In particular, source programs (lubksb, ludcmp, svbksb, svdfit, and svdcmp) supplied by Press et al. (21) were adapted to solve linear algebraic equations.


   RESULTS
TOP
 ABSTRACT
 METHODS
 RESULTS
DISCUSSION
 GRANTS
 ACKNOWLEDGMENTS
 REFERENCES
 
Simulated problem

The time course of estimates of O2 consumption were compared with the known input signal of rectangular (Fig. 2) and sinusoidal signals (Fig. 3). Comparison between the estimates and known input signal revealed several features of algorithms. First, the Kalman filter consistently returned estimates several minutes behind the known signal. Second, trends identification returned delayed estimates against a signal of a 16-min period. Third, estimates often preceded the signal when the metabolic rate suddenly increased. For example, before the first increase in the rectangular signal at 45 min, estimates by deconvolution and Kalman smoothing began to increase. Although the time course was not shown, the same characteristic features of algorithms were also observed in estimates of CO2 production rate.

The performance of algorithms against the input signal in a gradually decreasing period of 64 min (45–109 min of the simulation), 32 min (109–173 min), 16 min (173–237 min), and 8 min (237–301 min) was summarized as recovery (Table 1), mean square error (MSE; Table 2), and Pearson's correlation coefficient (Table 3). In terms of recovery, the Kalman filter, Kalman smoothing, and deconvolution were satisfactory. Recovery of the moving average for the overall period of a simulation was also satisfactory, but it ranged from 90.7 to 107.6% depending type of input signals. It is worth mentioning that recoveries of trends identification dropped, when a rectangular signal of a 16- and 8-min period was applied. In terms of MSEs for the overall period of the simulation, deconvolution performed better than the trends identification, Kalman filter, and Kalman smoothing. The MSEs of trends identification were the smallest against the rectangular signal of a long period (64 and 32 min), but it was not the case for the rectangular signal of a short period (16 and 8 min) and all sinusoidal signals. The MSEs of the moving average were roughly comparable with those by the other algorithms. However, compared with the other algorithms, estimates of the moving average were under-damped and limitation of this algorithm became evident when estimates of the RQ were compared among the algorithms. During 256 min of simulation against the sinusoidal signal, the RQ by moving average (means ± SD; 1.10 ± 0.57) was highly variable, while that by trends identification (0.98 ± 0.01), Kalman filter (1.00 ± 0.01), Kalman smoothing (1.00 ± 0.00), and deconvolution (1.00 ± 0.02) was satisfactory. Similarly, during the 128 min when the rectangular input was "on," the RQ by moving average was highly variable (1.29 ± 1.02), compared with those by trends identification (0.98 ± 0.08), Kalman filter (0.99 ± 0.07), Kalman smoothing (1.00 ± 0.01), and deconvolution (1.01 ± 0.05). Pearson's correlation coefficients of estimates by moving average and deconvolution with known input were higher than those by trends identification, Kalman filter, and Kalman smoothing. Unexpectedly trends identification (for rectangular signal of 16 min and sinusoidal signal of 16 min and shorter) and Kalman filter (for input signals of 32 min or shorter) returned estimates, the correlation coefficients of which with the known input were negative.


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  		Table 1. Recovery for in silico simulation

 

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  		Table 2. Mean square error for in silico simulation

 

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  		Table 3. Correlation coefficient of estimates with known signals for in silico simulation

 
CO2 infusion test

A mixture of CO2 and N2 was infused at a constant rate in gradually decreasing intervals, and the CO2 production rates estimated by algorithms were compared with the CO2 infusion rate (Fig. 4). Even for an infusion of a 16-min period (8 min "on" and 8 min "off") and for shorter periodic infusions, deconvolution returned estimates synchronized with the input signal. Compared with estimates by deconvolution, those by Kalman smoothing were slightly over damped. The Kalman filter consistently returned estimates several minutes behind the known input signal. The estimates of the CO2 production rate by trends identification fit nicely to the infusion rate when the standard gas was infused with a 64-min period (32 min "on" and 32 min "off"). However, against a repeated infusion of a 16-min period, the recovery by this algorithm dropped and the delayed time course of the estimates was evident. Similar to simulation studies with the rectangular signal, trends identification (16-min period) and the Kalman filter (16 min or shorter) returned estimates, the correlation coefficients of which with the CO2 infusion rate were negative. Collectively in terms of overall recovery, MSE, and correlation with the known input signal, deconvolution performed better than the other algorithms (Table 4).


Figure 4
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  		Fig. 4. CO2 infusion test. A mixture of CO2 and N2 (20 and 80%, respectively) was infused at a rate of 1.888 l/min (STPD) in gradually decreasing intervals. At 45 min of the test, infusion of a 64-min period (32 min on and 32 min off) was initiated, followed by those of 32-, 16-, and 8-min periods. Against CO2 infusion rate shown as a line, estimate of CO2 by trends identification (A), Kalman filter (B), Kalman smoothing (C), and deconvolution (D) is shown as circles. Regularization parameter for deconvolution ({gamma}) was 209.

 

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  		Table 4. Recovery, mean square error, and correlation coefficient for CO2 infusion test

 
Case study

The time course of energy expenditure during sleep was shown with changes in sleep stage in Fig. 5. Compared with deconvolution, Kalman smoothing returned similar but slightly over-damped estimates and the Kalman filter returned a delayed response. As shown in Fig. 5, inset, trends identification returned a time course with sudden changes (from 464 to 465 min and from 502 to 503 min of the sleep). Figure 5, inset, also shows that apparent energy expenditure based on deconvolution and Kalman smoothing increased before awakening, when data after awakening were included for calculation of O2 consumption and CO2 production.


Figure 5
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  		Fig. 5. Energy expenditure during sleep. Energy expenditure during sleep (0–503 min) was estimated by trends identification (green line), Kalman filter (red line), Kalman smoothing (blue line), and deconvolution (black line). To compute O2 consumption and CO2 production, data collected until awakening were used for the Kalman filter, Kalman smoothing, and deconvolution, while trends identification required data until 15 min after awakening. From O2 consumption and CO2 production, energy expenditure was calculated according to Weir's equation (21). Sleep stages identified by a standard polysomnogram were shown in a lower panel. Insert of the figure shows energy expenditure during the final 40 min of sleep. When computed using data including 30 min after awakening (0–533 min), Kalman smoothing (blue dotted line) and deconvolution (black dotted line) returned enhanced metabolic rate before awakening.

 
ANOCVA revealed an association of energy expenditure with sleep stage. Energy expenditures estimated by deconvolution and Kalman smoothing were more closely associated with sleep stage than those based on the Kalman filter and trends identification (Table 5).


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  		Table 5. Relation between energy expenditure and stage of sleep

 

   DISCUSSION
TOP
 ABSTRACT
 METHODS
 RESULTS
 DISCUSSION
GRANTS
 ACKNOWLEDGMENTS
 REFERENCES
 
The whole body indirect calorimeter has been a major tool of nutrition research for more than a century and has been labeled the "golden standard" for energy expenditure measurements. Improvements in noise reduction algorithms made the whole body calorimeter useful not only for traditional 24-h energy expenditure measurements but also for experiments requiring rapid response (10, 12, 17, 19, 27). Although several noise reduction algorithms have been proposed, there is a paucity for rigorous comparison among the algorithms under various conditions. As the first step to compare the performance of algorithms, the present study performed in silico simulation using rectangular and sinusoidal input signals. The rectangular input signal modeled intermittent CO2 infusion used in previous studies (12, 19) to validate whole body indirect calorimeter. Since energy metabolism gradually changes at the onset and end of an exercise (2), the sinusoidal input signal was also included in the simulation study.

The moving average method is straightforward, and its performance was comparable with that of the other algorithms when evaluated from overall recovery and MSE. However, the time course of estimates by this method is rather erratic or under-damped. This prevented us from relying on this algorithm for estimating the RQ value in a short interval.

The trends identification algorithm initially developed for "push" calorimeter at Göteborg (12) was subsequently installed for a "pull" calorimeter at Pennington (19). This algorithm for noise suppression allows one to compute energy metabolism during the stable stage of respiration and the transition between states. The obvious limitation of this algorithm is that, if there are more than two states within the 30 min, the states are smoothed. Consistent with the intermittent CO2 infusion test of previous studies (12, 19) and the present study, this algorithm returned a poor estimate in the simulation study, when the input signal changed frequently. Furthermore, simulation also revealed a characteristic delay of the estimate for the signal of a 16-min period, which has been demonstrated but not mentioned in the CO2 infusion test of previous studies (12, 19). This delay of the estimate by trend identification resulted in negative or no correlations between the estimate and input signal when the period of the input signal was 16 min (8 min "on" and 8 min "off") or shorter. In addition, recovery of this algorithm decreased against a signal of a 16-min period during the simulation with the rectangular input and CO2 infusion test. Since the sinusoidal signal continuously changes its magnitude, it was not surprising that trends identification did not perform well for this signal. Thus simulation study highlighted the limited ability of the trends identification algorithm against the input signals of a short period, and this was confirmed by the CO2 infusion test.

The Kalman filter, which depends on only past measurements, consistently returned estimates several minutes behind the known signal. When the delay became significant in relation to the period of the input signal (16 min or shorter), correlation between estimates by the Kalman filter and the input signal became negative. Kalman smoothing, which depends on past and future measurements, performed better than the Kalman filter in terms of recovery and MSE. However, estimates by Kalman smoothing were slightly over damped compared with those by deconvoution. Correlations between estimates by Kalman smoothing and the input signal were lower for an input signal of 16 min or a shorter period, compared with those by moving average and deconvolution.

The overall performance of deconvolution was best among the algorithms, although trends identification performed well in a limited condition (rectangular input of 32 min or longer period). The limitation of deconvolution as an algorithm for the transient response of whole body indirect calorimeter should be pointed out. Compared with input signal, the estimate by this algorithm is also over damped, and visual inspection of the data in Figs. 2–4 clearly showed that this approach did not track well with the input signal of an 8-min period. If expired air collected through the mouthpiece or face mask is directly fed into a gas analyzer, breath-by-breath analysis of gas exchange is possible using commercially available equipment (3). Thus there remains room for improvement in the time resolution of the whole body indirect calorimeter, which allows long and continuous measurement of energy metabolism, including during a period of meal and sleep without wearing any attachment for collecting expired air.

It is worth mentioning that the generalization of results from the present simulation requires cautious interpretation. Gas concentration in simulation of the present study was corrupted by artificially generated noise, the SD of which was set at 0.0016% for O2 and 0.0011% for CO2, reflecting specification of our gas analyzer. When simulations were repeated by setting the SD of noise at 0.010% for O2 and 0.004% for CO2, corresponding to an accuracy of conventional gas analysis system with paramagnetic O2 and infrared CO2 analyzer (27), the performance of deconvolution was better than that of the moving average, trends identification, and Kalman filter method. The overall performance of deconvolution and Kalman smoothing was comparable, but the transient responses of deconvolution seemed to be better than those of Kalman smoothing. Compared with Kalman smoothing, correlation of estimates by deconvolution with the known input signal was higher for CO2 production for the rectangular signal, and for O2 consumption and CO2 production for the sinusoidal input signals of a 16-min period (see Supplemental Tables S1 and S2; supplemental data for this article are available online at the J Appl Physiol website). Thus the deconvolution was effective in improving the transient response of whole body indirect calorimeter, without jeopardizing recovery and MSE.

As a case study, changes in sleeping metabolic rate estimated by different algorithms were evaluated in relation to sleep stage. Rapid-eye-movement (REM) sleep is accompanied by an increase in cerebral blood flow and oxygen and glucose uptake (14). Some but not all previous studies (4, 11, 20, 29, 32) performing indirect calorimetry using face mask, hood, or small chamber called sleep calorimeter (V = 1.022m3) suggested higher metabolic rate during REM and drowsiness/light sleep (sleep stages 1 and 2) compared with that during non-REM and deep sleep (sleep stages 3 and 4), respectively. Given that there is a relation between sleep stage and energy metabolism, it is expected that delayed estimates by the Kalman filter and trends identification would fail to identify the relation between sleep stage and energy metabolism. Indeed, estimates by the deconvolution and Kalman smoothing suggested more significant relations between sleep stage and energy expenditure. As discussed above, the overall performance of deconvolution and Kalman smoothing was rated better than other algorithms, these two algorithms require cautious execution. Deconvolution and Kalman smoothing returned an enhanced metabolic rate before awakening if gas concentration measurements after awakening were included in the calculation (Fig. 5, inset). Although there is a general agreement among the literature (29, 32) that energy expenditure starts to rise before awakening, it is conceivable that previous reports might contain artifacts due to information overflow from measurements after awakening. The Kalman filter, which depends only on past measurement, may have a role when overflow of information from the future measurement could be critical.

Bioactive substances known to enhance energy metabolism such as caffeine and catechin polyphenols and genetic polymorphism affecting energy metabolism draw recent attention to nutritional studies using indirect calorimetry. However, the effects of these substances and physiological condition are usually subtle (8). Furthermore, it is preferred to capture changes in the metabolic rate with higher resolution in time, which allows one to compare it with timing of experimental protocols and changes in physiological parameters such as heart rate and sympathetic nervous system activity. The noise suppression algorithm of whole body indirect calorimeter, particularly deconvolution with regularization parameter proposed in the present study, plays a role in experiments requiring analysis of rapid response.


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This research was supported in part by the 21st Century Center of Excellence Program (2002–2006 Project: Promotion of Health and Sport Scientific Research), Ministry of Education, Culture, Sports, Science and Technology.


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We thank Shigeru Nakajima (Fuji Medical Science) for technical assistance and Daniel Merriman (Thermo Fisher Scientific) for helpful discussion and comments.


   FOOTNOTES
 

Address for reprint requests and other correspondence: K. Tokuyama, Division of Sports Medicine, School of Comprehensive Human Science, Univ. of Tsukuba, Ibaraki, Japan 305-8574. (e-mail: tokuyama{at}taiiku.tsukuba.ac.jp)

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.


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