Submitted on June 16, 2005
Accepted on October 19, 2005
On the modeling of breath-by-breath oxygen uptake kinetics at the onset of high-intensity exercises: Simulated Annealing versus GRG2 method
Olivier BERNARD1*, Olivier ALATA2, and Marc FRANCAUX3
1 Laboratoire des Adaptations Physiologiques aux Activites Physiques (LAPHAP, EA 3813), Universite de Poitiers, Faculte des Sciences du Sport, Poitiers, France
2 Laboratoire Signal Image Communications (SIC, FRE-CNRS 2731), Universite de Poitiers, UFR Sciences Fondamentales et Appliquees, Poitiers, France
3 Institut d'Education Physique et de Readaptation, Universite Catholique de Louvain, Louvain-la-Neuve, Belgium
* To whom correspondence should be addressed. E-mail: olivier.bernard{at}univ-poitiers.fr.
Modeling in the time domain, the non-steady-state VO2 on-kinetics of high-intensity exercises with empirical models is commonly performed with Gradient-Descent based methods. However, these procedures may impair the confidence of the parameter estimation when the modeling functions are not continuously differentiable and when the estimation corresponds to an ill-posed problem. To cope with these problems, an implementation of Simulated Annealing (SA) methods was compared to GRG2 algorithm (a gradient-descent method known for its robustness). Forty simulated VO2 on-responses were generated to mimic real time course for transitions from light- to high-intensity exercises, with a Signal-to-Noise (SNR) ratio equal to 20 dB. They were modeled twice with a discontinuous double-exponential function using both estimation methods. GRG2 significantly biased two estimated kinetic parameters of the first exponential (the time delay td1 and the time constant 
1) and impaired the precision (i.e. standard deviation) of the baseline A0, td1 and 1 compared to SA. SA significantly improved the precision of the three parameters of the second exponential (the asymptotic increment A2, the time delay td2 and the time constant 2). Nevertheless, td2 was significantly biased by both procedures and the large confidence intervals of the whole second component parameters limit their interpretation. To compare both algorithms on experimental data, twenty six subjects each performed two transitions from 80 W to 80%VO2max on a cycle ergometer and VO2 was measured breath by breath. More than 88% of the kinetic parameter estimations done with SA algorithm produced the lowest residual sum of squares between the experimental data points and the model. Repeatability coefficients were better with GRG2 for A1 while better with SA for A2 and 2. Our results demonstrate that the implementation of SA improves significantly the estimation of most of these kinetic parameters but a large inaccuracy remains in estimating the parameter values of the second exponential.
