A Single-Phase Modelling for the Oxygen Uptake Rate in Excess Post- 1 Exercise Oxygen Consumption 2

13 A two-parameters [a rate constant k and a factor f (0  f <1)] modelling, describing 14 satisfactorily the post-exercise oxygen uptake rate ( V O2 ) as a function of the recovery 15 time ( t ), is presented. f controls the rate equation d V O2 /d t , particularly at t = 0 where 16 (d V O2 /d t ) t =0   k (1  f ), a less abrupt decay than (d V O2 /d t ) t =0   k expected from an 17 exponential. Fitting the model to a set of experimental V O2 vs t data after a 3MT it was 18 found a set of values with f close to 0 and another with f >1/2, with a narrow distribution 19 of values for the half-recovery time  1/2 =(1/ k )ln[(2  f )/(1  f )] (  1/2  =0.641 min,  =0.062 20 min), very similar to that ( T ) found by fitting a model based on a logit transform 21 (  T  =0.672 min,  =0.081 min). The parameter f is a reliable index of the initial 22 acceleration of the oxygen uptake rate recovery (and likely of the heart rate recovery) 23 and, together with the half-recovery time  1/2 , may be a useful method in characterizing 24 and monitoring performs and exercise forms, very important in the physiology area.


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Three-minute all-out test (3MT) is a maximal effort evaluation protocol based 30 on the original model of critical power whose metabolic parameters of aerobic (EP) and 31 anaerobic (WEP) capacities are obtained through the mathematic analysis of a subject's 32 output power kinetics. The kinetics of output power of 3MT can determine important 33 parameters for training prescription such as some transition limits when it comes to 34 effort intensity 1 . EP is the threshold that measures the transition from intense to severe 35 exercise and uses mainly oxidative metabolism 1,2 , while WEP uses anaerobic sources 36 and is a finite amount of work carried until it runs out in severe intensity exercises 3 . The 37 post-exercise oxygen consumption (EPOC) after 3MT can provide important 38 information about the maximum effort performed by the individual. 39 Direct physiological parameters such as blood lactate concentration and oxygen 40 uptake measure the domains of intensity transition during exercise 4 . Researchers have 41 been using such physiological markers, collecting them right after exercise, to 42 determine the contribution of the aerobic and anaerobic energetic metabolisms 5 . EPOC 43 may physiologically represent the reestablishment of blood and muscular O2 to ATP and 44 PCr resynthesis, as well as removing lactate from high intensity efforts 6 . Blood lactate 45 concentration and EPOC can reliably estimate the anaerobic capacity in 3MT 7 . 46 The oxygen uptake rate (VO2) recovery is conventionally assumed to proceed in 47 5 The constraint equation (4) has been used to write Xinteg indistinctly in terms of k and f 116 or 1/2 and f in equation (7). If f = 0, equation (7) becomes Xinteg = 1/k = 1/2/ln2 just as in 117 case of an exponential decay, since the limit of (1/f)ln The total net volume of oxygen uptake up to complete recovery (t = ), VTO2net, 119 the so-called EPOC, is obtained by multiplying equation (7) by the quantity (VO2peak -120 VO2rest), or 121 To better illustrate the effect of the parameter f on the experimental VO2 vs t 123 curve, Figure 1 shows plots of equation (3) for several values of k (or f) for a fixed value 124 of 1/2 = ln2. The inclination (dX/dt)t=0 = k(1f) at t=0 is effectively less abrupt with 125 increasing f (insert in Figure 1), and particularly less abrupt than an exponential decay 126 where (dX/dt)t=0 = k. The increase of the parameter f retards effectively the decaying of 127 VO2 at the beginning of the recovery, however, the rate of decaying at t = 1/2, (dX/dt) 1/2 128 = (k/2)[1f/2], becomes more abrupt with increasing f, as shown by the tangent dash 129 lines plotted with slopes (dX/dt) 1/2 at t=1/2 in Figure 1 for several values of k (or f, since 130 1/2 = cte).
143 6 to keep the same notation for the fraction X = Xi /Xpeak = (VO2  VO2rest)/(VO2peak  VO2rest) 144 of the present work. So equation (9) could be written as 145 (10) 146 Equation (10), named here Stu model, could be fitted to the experimental VO2 vs t data 147 and the parameters , T, VO2peak and VO2rest determined. T represents the half-recovery 148 time, the time for X in equation (9) to decay to 1/2, the same meaning of the parameter 149 1/2 of the f1p model.

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The rate equation that governs the decaying law of equation (9) is given by 151 given. The test only was considered successful after 3 minutes of non-stoppable 171 running. The mechanical output power generated by each subject during the 3MT test 7 was recorded and analyzed against time to find the aerobic capacity (EP) and anaerobic 173 work capacity (WEP) values of the power output graph 2 . 174 The oxygen uptake and carbon dioxide production were measured by using a gas 175 analyzer (Cosmed Italy K4b²) that was calibrated after each session. The gas analyzer was 176 integrated to an online system of breath-by-breath data caption. The oxygen uptake rate (VO2) 177 was measured as a function of the recovery time (t) up to about 10 min after the start of 178 the 3MT recovery. The models were fitted to the experimental VO2 vs t data. 179 The fitting processes were carried out using a routine of least-squares method 180 based on the Levenberg-Marquardt algorithm. When was the case, the similarity 181 between the parameters given by different models was statistically tested for equal 182 means using One-Way ANOVA. 183  Table 1 The standard biphasic two-exponential model, which can be cast as VO2 = VO2rest 205 + A1exp(t/1) + A2exp(t/2), where A1 and A2 are constants and 1 and 2 are the "fast" 206 and the "slow" decaying constant times, respectively, was also fitted to the present 207 experimental data. In order to minimize the parameters to be fitted, VO2rest was fixed as 208 the mean value of VO2 in the last minute of recovery and the parameters A1, 1, A2, and 209 2 were obtained by fitting. Table 1 (two-exponential model) shows the fitted 210 parameters with the standard errors. 211

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The mean values for 1/2 (1) and 1/2 (2) were 1/2 (1)  = 0.476 min, with standard deviation zero (subjects 1 to 5), while  2 were found minor than those of the standard model for 269 the group of individuals with f >1/2 (subjects 6 to 10). The same general aspects of 270 quality apply to the R-square values from fitting both models (Figure 4(a)). This 271 suggests that the standard two-exponential model could be not quite appropriated to fit 272 experimental data that exhibit a not so high decaying of the rate VO2 at the beginning of 273 recovery, where the f1p model previews a less abrupt decaying [(dX/dt)t=0 = k (1f)] 274 when f > 0. This seems to be the case for the recovery following peak exercise, for 275 which a first order equation is far to confirm to be the optimal model to establish the 276 most appropriate exercise protocol 13 . 277 Apart from the fact that the standard two-exponential model to have more 278 parameters to be fitted with respect to the single-phase models (f1p or Stu), Figure 5 shows that the distribution of the "fast" half-recovery time 1/2 (1) was even larger 280 (1/2 (1)  = 0.476 min,  = 0.082 min) than 1/2 of the f1p model, and the distribution of 281 the "slow" half-recovery time 1/2 (2) was too large (1/2 (2)  = 3.94 min,  = 1.91 min), 282 likely meaningless, even by fixing the rest parameter VO2rest in the fitting process. 283

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The present f-single-phase modelling is a simple and powerful method able to fit 285 satisfactorily with only two parameters, a rate constant (k) and a factor f (0  f < 1), 286 experimental data of oxygen uptake rate in recovery process. The parameter f controls 287 the rate equation dVO2/dt, particularly at t = 0 where (dVO2/dt)t=0  k(1f), a decay 288 effectively less abrupt than (dVO2/dt)t=0  k, expected from an exponential.