Dynamic characteristics of oxygen consumption
 Lin Ye^{1},
 Ahmadreza Argha^{2},
 Hairong Yu^{1},
 Branko G. Celler^{2},
 Hung T. Nguyen^{1} and
 Steven Su^{1}Email author
Received: 19 February 2018
Accepted: 16 April 2018
Published: 23 April 2018
Abstract
Background
Previous studies have indicated that oxygen uptake (\(VO_2\)) is one of the most accurate indices for assessing the cardiorespiratory response to exercise. In most existing studies, the response of \(VO_2\) is often roughly modelled as a firstorder system due to the inadequate stimulation and low signal to noise ratio. To overcome this difficulty, this paper proposes a novel nonparametric kernelbased method for the dynamic modelling of \(VO_2\) response to provide a more robust estimation.
Methods
Twenty healthy nonathlete participants conducted treadmill exercises with monotonous stimulation (e.g., single step function as input). During the exercise, \(VO_2\) was measured and recorded by a popular portable gas analyser (\(K4b^2\), COSMED). Based on the recorded data, a kernelbased estimation method was proposed to perform the nonparametric modelling of \(VO_2\). For the proposed method, a properly selected kernel can represent the prior modelling information to reduce the dependence of comprehensive stimulations. Furthermore, due to the special elastic net formed by \(\mathcal {L}_1\) norm and kernelised \(\mathcal {L}_2\) norm, the estimations are smooth and concise. Additionally, the finite impulse response based nonparametric model which estimated by the proposed method can optimally select the order and fit better in terms of goodnessoffit comparing to classical methods.
Results
Several kernels were introduced for the kernelbased \(VO_2\) modelling method. The results clearly indicated that the stable spline (SS) kernel has the best performance for \(VO_2\) modelling. Particularly, based on the experimental data from 20 participants, the estimated response from the proposed method with SS kernel was significantly better than the results from the benchmark method [i.e., prediction error method (PEM)] (\(76.0\pm 5.72\) vs \(71.4\pm 7.24\%\)).
Conclusions
The proposed nonparametric modelling method is an effective method for the estimation of the impulse response of VO_{2}—Speed system. Furthermore, the identified average nonparametric model method can dynamically predict \(VO_2\) response with acceptable accuracy during treadmill exercise.
Keywords
Background
Oxygen uptake (\(VO_2\)) onkinetics is an important physiological parameter for the determination of functional health status and muscle energetics during physical exercise [1]. In addition, the \(VO_2\) kinetics provides a useful assessment of the body’s ability to support a change in metabolic demand and an insight into the circulatory and metabolic response to exercise. Several studies confirmed that oxygen consumption is mainly controlled by intramuscular factor related metabolic system [2, 3]. Different from heart rate, the oxygen uptake cannot be affected by mood, stress, etc., and is generally considered as the most accurate measurement of the fitness for cardiorespiratory system [4, 5]. The main goal of this paper is to establish a nonparametric model to describe the onkinetics of the oxygen uptake in response to the speed of treadmill exercise.
Previous researches conducted on the oxygen uptake modelling can be divided into two categories: (i) static status modelling and (ii) dynamic status modelling. For the static status modelling, an early stage study in [6] proposes a linear static model to approximately estimate oxygen uptake for a given range of walking speed. Simple nonlinear static models are also discussed in [7–9] for the compensation of nonlinearities. On the other hand, the transient response of oxygen uptake has captured the interests of many researches. For example, the authors of [10, 11] have developed a firstorder system to approximate the process based on step response. Later, the work in [12] has developed a nonlinear dynamic model for oxygen uptake modelling during treadmill exercise with pseudo random binary signal (PRBS) as the input. However, it is relatively difficult for the exercisers to follow the PRBS signal during the treadmill exercise generally.
In real life, the standard deviation of noise in \(VO_2\) measurements is quite large due to the limitations of portable gas analyser. For the modelling of a process with large noise, as determining the order is difficult, a nonparametric model such as impulse response (IR) model is a good choice. However, conventional system identification methods for impulse response estimation normally requires relatively complex input such as PRBS [13] to significantly stimulate the system. In previous studies, the response of oxygen uptake can only be roughly modelled as a firstorder system due to the lack of suitable modelling techniques. Recently, a new kernel based estimation method has been developed for nonparametric model estimation [14, 15]. To avoid illconditioned solutions due to the existence of large noises, a regularised term is incorporated into the cost function [16], which can limit the onestep variation of the estimated parameters. This new kernel based method projects the parameters of IR into a reproducing kernel Hilbert space (RKHS) which can reduce high frequency components in IR model. Furthermore, by using this method, more accurate results can also be obtained enabling us to employ simple inputs such as step input.
In this paper, in order to implement nonparametric modelling of \(VO_2\) response to dynamic exercises, the kernel based estimation method has been adopted and modified. An \(\mathcal {L}_1\) regularisation term has been added into the cost function to penalise the least significant term of IR which can result in reducing the order of the impulse response model. Particularly, we have demonstrated that this method is still valid when the input of the system is a single step response for this specific VO_{2}—Speed system. For this research, several popular kernels were tested, such as stable spline (SS) kernel, diagonal kernel (DI) and diagonal/correlated (DC) kernel. Furthermore, we showed through several simulation examples that SS and DC kernels can achieve higher accuracy compared to DI kernel for this problem. Eventually, the proposed method was experimentally validated by using the \(VO_2\) data collected from 20 participants. The results were compared with the estimated model based on Akaike’s Information Criterion (AIC) selected autoregressive with exogenous terms (ARX) model with predicted error method for parameter estimation.
The main contributions of this work can be summarised as follows. Firstly, a new nonparametric modelling approach has been developed based on the kernelbased impulse response estimation approach, which can efficiently reduce the order of the IR model by incorporating an \(\mathcal {L}_1\) penalty term. Secondly, for the developed IR model identification, appropriate kernels selection has been investigated using extensive simulations, and the stable spline kernel (SS) was recommended as the best candidate. Thirdly, it was demonstrated by both experiment and simulation that the proposed method is efficient for the modelling of IR of cardiorespiratory response to dynamic exercise, which often confronts a highly noisy measurement under the stimulation of a simple input signal. Finally, an averaged impulse response model has been established, which is able to quantitatively describe the oxygen update onkinetics for treadmill exercise.
This paper is organised as follows. In the "New modelling method for \(VO_2\) during exercise" section, the nonparametric method for \(VO_2\) modelling is proposed and kernel selection is also discussed. In the "Simulations" section, the simulation is carried out for the validation of the proposed method. In the "Experiments" section, the experimental results are presented. The "Conclusions" section concludes the paper.
New modelling method for \(VO_2\) during exercise
Kernel based estimation method of finite impulse response
In this section, a new kernel based nonparametric estimation method is exploited to model the oxygen uptake during treadmill exercise. For this nonparametric estimation method, it is not necessary to predefine the order of the model in advance. Furthermore, it will be shown that the proposed method can provide stable and smooth estimation comparing to other estimation methods, cf [15].
Kernel selection

DI kernel:where \(c>0\), \(1 {>} \lambda {>} 0\).$$\begin{aligned} P(i,j)=\left\{ \begin{aligned}&c\lambda ^{i},\quad i=j\\&0 \end{aligned}\right. , \end{aligned}$$(18)

SS kernel:where \(c >0,\lambda >0\).$$\begin{aligned} P(i,j)=\left\{ \begin{aligned}&c\frac{\lambda ^{2i}}{2}\left( \lambda ^i\frac{\lambda ^{j}}{3}\right) ,\quad i\ge j\\&c\frac{\lambda ^{2j}}{2}\left( \lambda ^j\frac{\lambda ^{i}}{3}\right) ,\quad j\ge i \end{aligned}\right. , \end{aligned}$$(19)

DC kernel:where \(c> 0\), \(1>\lambda > 0\), \(\rho \le 1\) and \(\rho \ne 0\).$$\begin{aligned} P(i,j)=c\rho ^{ij}\lambda ^{i+j/2}, \end{aligned}$$(20)
Simulations
Mostly, the relationship between the oxygen uptake and the jogging speed was considered as a firstorder system. To the authors’ best knowledge, due to the individual differences of the body condition of human beings, it is likely that the transfer function model of the \(VO_2\) for each person is different in terms of gain value and order. For some people, the relationship between the oxygen uptake and the joggling speed may not be described by a firstorder transfer function. Furthermore, it is generally hard to correctly identify the exact order of system through a single input response, especially under large observation noise. The major difference between a firstorder system and a highorder system in step response is in their slope and damping. Therefore, it is likely that a second or higher order system is identified as a firstorder system through single step response. Therefore, during this “Simulations” section, both firstorder systems and secondorder systems were considered.

SS kernel: \(c=1\), \(\lambda =0.98\)

DC kernel: \(c=1\), \(\lambda =0.9\), \(\rho =0.8\)

DI kernel: \(c=1\), \(\lambda =0.9\)

regulariser: \(\gamma =8\), \(\alpha =10\)
Fit ratio comparison of firstorder system simulation
Method  Mean  Standard deviation  Best 

PEM  0.7525  0.0444  0.8475 
Kernel (SS)  0.8705  0.0264  0.9170 
Kernel (DC)  0.8738  0.0254  0.9141 
Kernel (DI)  0.8678  0.0254  0.9114 
Fit ratio comparison of secondorder system simulation
Method  Mean  Standard deviation  Best 

PEM (firstorder system)  0.7087  0.0554  0.8157 
PEM (secondorder system)  0.8133  0.0484  0.8955 
SS kernel  0.8694  0.0246  0.9075 
DC kernel  0.8758  0.0248  0.9184 
DI kernel  0.8615  0.0234  0.9073 
Experiments
Age and BMI of participants
Subject  Age (year)  Height (m)  Mass (kg)  BMI (kg/m^{2}) 

Average  38.02  1.77  86.10  27.16 
Standard deviation  5.28  0.06  14.05  3.61 
All data was acquired by a gas analyser \(K4b^2\) (COSMED), which is a portable system for pulmonary gas exchange measurement with true breathbybreath analysis. The UTS Human Research Ethics Committee (UTS HREC 2009000227) approved this study and an informed consent was obtained from every participant before the commencement of data collection.
Prior to the experiments, all participants were ask to observe the following requirements: including the nutritional intake, physical activity and environment conditions. The participants were instructed to consume a standardised light meal at least 2 h before the experiment. Meanwhile, they were asked not to engage in any other exercises for one day before the experiment. The temperature and humidity of the laboratory were set to 20–25 °C and around \(50\%\) relative humidity respectively.
Goodness of fit
Subject  Nonparametric (%)  AIC + PEM (%)  AIC 

Participant 1  83.3  83.3  11 
Participant 2  77.0  67.4  7 
Participant 3  61.5  54.7  12 
Participant 4  75.3  65.4  7 
Participant 5  74.7  61.0  4 
Participant 6  81.1  73.7  8 
Participant 7  81.7  80.1  13 
Participant 8  82.2  80.6  11 
Participant 9  78.3  74.5  12 
Participant 10  72.6  63.6  10 
Participant 11  74.3  65.1  9 
Participant 12  72.9  71.0  13 
Participant 13  71.1  67.0  17 
Participant 14  77.8  74.3  15 
Participant 15  76.6  74.1  17 
Participant 16  76.5  71.6  11 
Participant 17  78.0  76.0  11 
Participant 18  78.4  72.1  11 
Participant 19  76.3  69.4  10 
Participant 20  76.4  70.8  13 
Average  76.0  71.4  – 
Standard deviation  5.72  7.24  – 
Conclusions
This paper reports our proposed method for nonparametric modelling of \(VO_2\) response to treadmill exercise using a kernel based modelling approach. Several kernel functions have been exploited and tested using different numerical simulations. The stable spline kernel was chosen as it can achieve expected results. With stable spline kernel, the proposed estimation method were tested experimentally using 20 participants. The obtained results showed that the goodness of fit of the proposed method can significantly exceed the prediction error method. We conclude that the kernel based nonparametric modelling method is an effective method for the estimation of the impulse response of the VO_{2}—Speed system. We also believe that the identified FIR model can provide accurate dynamic prediction of \(VO_2\) response during treadmill exercise.
Declarations
Authors' contributions
LY carried out the detailed theoretical derivation and drafted the manuscript. AA checked the theoretical derivation and revised the manuscript. HY collected the data and carried out the experimental analysis. BGC and HTN supervised the project. SWS supervised the theoretical development and revised the manuscript. All authors read and approved the final manuscript.
Acknowledgements
The work is supported by “THE AUSTRALIA CHINA JOINT INSTITUTE FOR HEALTH TECHNOLOGY AND INNOVATION” .
Competing interests
The authors declare that they have no competing interests.
Availability of data and materials
The datasets generated or analysed during the current study are not publicly available due to the privacy of the participants but are available from the corresponding author on reasonable request.
Consent for publication
The consent form for publication is signed.
Ethics approval and consent to participate
The UTS Human Research Ethics Committee (UTS HREC 2009000227) approved this study and an informed consent was obtained from every participant before the commencement of data collection.
Funding
Not applicable.
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Authors’ Affiliations
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