The theoretical model traditionally describes two metabolic transition points in the response of ventilation (VE) during incremental exercise, frequently called first (VT1) and second (VT2) ventilatory thresholds (Meyer et al., 2005; Skinner and Mclellan, 1980). When VE is plotted against VO₂ or workload, VT1 and VT2 are often determined in the first and second break point on the VE curve, respectively (Meyer et al., 2005; Skinner and Mclellan, 1980).

However, several researchers frequently make the assumption that there is only one break point (Crescêncio et al., 2003; Higa et al., 2007), not allowing to know if this point corresponds to VT1 or VT2. This supposition would impact on the training prescription because VT1 is used for aerobic training in elderly and diseased population (Meyer et al., 2005; Koufaki et al., 2000) while VT2 is employed for aerobic training in healthy subjects and elite athletes (Meyer et al., 2005; Lucía et al., 2000).

Divergences between the theoretical model and researcher’s assumptions make the conceptual and methodological understanding difficult. Orr et al., 1982 and Dennis et al., 1992 tried to solve this divergence through mathematical modeling which could bring valuable information about the phenomenon characteristics. For instance, Orr et al., 1982 found a better fitting on the E data with a bi-segmental linear regression model (_2SRM) than with a three-segmental linear regression model (3SRM). On the other hand, Dennis et al., 1992 reported better fitting with an exponential model than with _2SRM or 3SRM. The _2SRM as well as the exponential fitting produce only one metabolic transition point through intersection point between two segments or derivatives, respectively (Higa et al., 2007; Hughson et al., 1987; Orr et al., 1982; Morton, 1989; Santos e Giannella-Neto, 2004). Thus, these results are in contrast with the theoretical model in which two transitions points are expected.

Nevertheless, methodological issues may restrict the inferences from these studies (Myers and Ashley, 1997; Morton, 1993). Orr et al., 1982 did not compare _2SRM and 3SRM with the exponential model while Dennis et al., 1992 did not determine the intersection point(s) mathematically of _2SRM and of 3SRM fitting and did not use enough points on each curve due to the employment of 60 seconds averages on VE values. Then, the divergences between theory and experimental data concerning VE behavior could not be completely answered by results of these studies and a better mathematical description of VE data during incremental exercise remains to be determinate.

These divergences between theory and empirical data can be more confusing when the training status is taken into account. While aerobic training changes the VT1 and VT2 to the right and promotes lower VE values (Esteve-Lanao et al., 2007; Zapico et al., 2007), anaerobic training produces a longer bicarbonate buffering (distance between VT1 and VT2) (Röcker et al., 1994). Thus, it could be hypothesized that aerobically trained subjects should have a more smooth VE increase, mainly in the initial stages of an incremental test usually employed in practical situations. Consequently, the VE behavior should be exponential with only one ventilatory threshold. In addition, untrained subjects and anaerobically trained subjects should have an abrupt and less smooth VE increase which could allow the presence of two thresholds.

However, how the aerobic training status would change the feature of VE behavior during an incremental exercise and whether the occurrence of one or two metabolic transitions would depend upon the training status still needs to be experimentally answered. A more specific mathematical analysis of the VE curve could provide valuable practical information to elucidate this issue (Mader and Heck, 1986; Newell et al., 2006). Therefore, the objectives of this study were: 1) to describe and to compare VE behavior during incremental testing by mathematical modeling; 2) to compare the feature of VE curve fitted by the best mathematical model between untrained and aerobically trained subjects. To reach these goals, _2SRM, 3SRM and grow exponential model (GEM) as well as the derivatives of the best mathematical model were compared within and between groups.

Fifteen trained soccer players (TR) (26.8 ± 2.8 years; 75.6 ± 10.0 kg; 1.78 ± 0.11 m) from a Brazilian professional team and twenty healthy untrained men (UT) (28.9 ± 3.8 years; 80.6 ± 13.8 kg; 1.76 ± 0.07 m) volunteered to take part in the study. After giving information about the risks, an informed consent was obtained from each participant (study has approved by the Ethics Committee of School of Physical Education and Sport, São Paulo University).

After a 4-minute warm-up, the subjects underwent an incremental test on a treadmill with initial speed between 5.0-6.0 (UT) and 10.0 km.h^-1 (TR) and increases of 1 km·h^-1 every minute, until exhaustion (the slope was kept at 0%). Subjects wore a nose clip and mouthpiece connected to a heated wire flow meter with precision of ± 0.05 L and dead space lower than 100 ml (SensorMedics, CA, USA). The flow signal was digitally integrated to give tidal volume and inspired gases were continuously analyzed by rapidly responding O₂ (paramagnetic) and CO₂ (infrared) analyzers with response time lower than 130 ms (SensorMedics Vmax 229, SensorMedics, CA, USA). The analyzers were checked prior to each test using a calibration syringe containing precision O₂ and CO₂ gas mixes and a rapidly switching solenoid valve (SensorMedics, CA, USA). Expiratory flow, VE (BTPS) and fractional concentrations of expired O₂ and CO₂ were measured and digitally integrated on a breath by breath basis to provide estimates of O₂ uptake (VO₂) and expired CO₂ volume (VCO₂) at real time.

The VE, expired O₂ fraction (FeO₂) and expired CO₂ fraction (FeCO₂) data were smoothed to averages of 20 seconds and were plotted against the time (approximately 30 points over each curve). Three evaluators identified VT1 and VT2 by visual inspection following the hierarchic criteria’s: VT1: first linearity break point of VE curve and the break point of FeO₂; VT2: second linearity break point of VE curve and the break point of FeCO₂ (Meyer et al., 2005). The FeO₂ and FeCO₂ were utilized only when it was not possible to identify VT1 and VT2 utilizing VE curve. VT1 and VT2 identified by evaluators were useful as reference points for the beginning calculation of bi-segmental regression and three-segmental regression since these mathematical fittings can produce artificial break points in the curve, without physiological significance. Peak VO₂ (VO_2PEAK) was determined by averaging the three highest values found in last two speeds (Weston et al., 2002), and peak speed (Speed_PEAK) was identified as the highest speed reached.

The behavior of VE curve was fitted by 2SRM, 3SRM and GEM. The 2SRM was attained by linear regression with an initially unknown intercept calculated from every possible intersection between the three points immediately before to VT1 and three points immediately after to VT2. This criterion was employed to provide physiological limits in which the intercept(s) should be identified, avoiding intercept with artificial values, without physiological significance. The intercept that best shared the curve in two linear segments was assumed when the highest R² and the lowest residual sum of squares (RSS) were attained. The curve segments were predicted by the following equations:

The 3SRM fitting was calculated by linear regression with two initially unknown intercepts calculated from every possible intersection between the three points immediately before to VT1 and three points immediately after to VT2. The intercepts that best shared the curve in three linear segments were assumed when the highest R² value and the lowest RSS were attained. The curve segments were predicted by the same equation of 2SRM (equation 1), where x is the time, y is the predicted value of VE to each segment, a’ is the constant for the 1^st or 2^nd or 3^rd segment, and b’ is the slope of the 1^st or 2^nd or 3^rd segment, respectively. More than thirty possible combinations were tested from each linear regression model to each curve (Figure 2).

The exponential characteristic of VE was tested by the GEM, through the following equation:

The GEM was chosen considering important parameters such as the offset, delay and time constant of the data series, after comparison of the R², RSS and mean square error (MSE) with the continuous exponential function described by Hughson et al., 1987.

The parameters of each model were determined by least squares methods, which evaluate the sum of squares residuals for each combination. The sweep calculations were employed to obtain the minimized sum of squares residuals and to determine the parameters a’ and b’ in _2SRM and 3SRM. The Levenberg-Marquardt algorithm was used to determine the parameters in GEM.

The residuals, RSS and MSE were calculated for each mathematical model. To verify the robustness of mathematical models, the original residuals were plotted against the VE measured values and analyzed by graphic inspection, by Shapiro Wilk’s test and by Run test. Subsequently, the residuals were fitted by a linear regression fitting and the constant (α) and slope (β) obtained were used to give an approximation concerning the location and behavior of estimated VE data by mathematical models.

After the best mathematical model had been found, the first derivative was calculated to the points corresponding to workloads below and above ventilatory threshold with the assumption that lower workload derivatives (LWD) should indicate moderate intensities and upper workload derivatives (UWD) should indicate heavy intensities. The first derivative obtained in moderate and heavy intensities of data fitted by the best mathematical fitting were employed to investigate the relationship between VE behavior during the incremental test and aerobic fitness. The Speed_PEAK (km·h^-1), VO_2PEAK (ml·kg^-1·min^-1), VT1 and VT2 (ml·kg^-1·min^-1, km·h^-1 and %Speed_PEAK) were accepted as aerobic fitness markers.

The data distribution was verified by Shapiro Wilk’s test and the data were reported as mean and standard deviation (SD) to make future comparisons easier, even when they did not show normal distribution. Factorial ANOVA (3 x 2) was carried out to detect interaction effects between the factors (mathematical fitting versus groups). F test was utilized to determine the differences between RSS and MSE for comparison intra-group and in the pooled data. The tests of Mann-Whitney, Wilcoxon and Student were carried out when necessary, for comparisons among derivatives, α and β parameters, Speed_PEAK, VO_2PEAK, VT1 and VT2 intra- group or between UT and TR. Pearson correlation coefficient determined the degree of association between α and β parameters of residuals, and between the derivatives of the best mathematical model and the aerobic markers. The analysis was carried out in SPSS (15.0) and in Origin (6.0) softwares and the significance was accepted at p < 0.05.

VO_2PEAK, VT1 and VT2 (expressed as ml·kg^-1·min^-1) were, respectively, 39.7 ± 6.3, 19.1 ± 5.1 and 31.5 ± 4.4 ml·kg^-1·min^-1 in UT and 54.7 ± 3.2, 38.6 ± 3.8 and 46.1 ± .41 ml·kg^-1·min^-1 in TR. These variables showed significant differences between UT and TR groups. Other aerobic parameters are listed in the Table 1.

The distribution of residuals of 2SRM, 3SRM and GEM showed a Gaussian behavior in almost every subject and the fittings attained significant R² (R² = 0.97 to 0.99 to more than 90% of values). Some subjects did not show Gaussian behavior with 2SRM (n = 5), 3SRM (n = 2) and GEM (n = 5) when analyzed by Shapiro Wilk’s test. These values were reduced for 2SRM (n = 1), but not for 3SRM (n = 3) and GEM (n = 6), when analyzed by Run test. No interaction effect was observed between the mathematical fittings (2SRM, 3SRM and GEM) and groups (UT and TR).

The RSS and MSE for GEM in pooled data were higher than 2SRM and 3SRM (p < 0.01). In intra-group comparisons, RSS and MSE for GEM were higher than 2SRM and 3SRM in the UT, but no difference was observed among mathematical fittings in the TR (Table 2).

The parameter β obtained on the residuals was not different from zero for 2SRM, 3SRM and GEM fittings in UT, while the constant α was negative and different from zero (a= -2.01 ± 3.20 l·min^-1) for GEM. Together, the parameters α and β of GEM showed a modest tendency of estimated VE values to be located below measured E values, throughout the whole curve (Table 2). In TR group, α and β did differ from zero, with negative values for α and positive values for β. Together, the parameters α and β of three mathematical models showed a modest tendency for underestimation of VE values during moderate intensities, and this tendency was attenuated according to increasing of VE (Figure 4 and Table 2).

The 2SRM was chosen as the best model once it is more parsimonious due to the employment of fewer estimative parameters than 3SRM and GEM models. So, the first derivative was calculated for workloads below and above intersection point of two regression lines. In this case, we accepted the intersection point as ventilatory threshold (VT_2SRM) (Higa et al., 2007; Orr et al., 1982), with LWD and UWD indicating moderate and heavy intensities, respectively.

In the pooled data, the LWD showed significant correlation with VT_2SRM and Speed_PEAK, while the UWD only showed significant correlation with VT_2SRM. When analyses were carried out separately, there was a significant correlation between LWD and VT_2SRM and Speed_PEAK in UT. However, in TR group a significant correlation was observed between UWD and VT_2SRM, Speed_PEAK and VO_2PEAK (Table 3). The intra-group comparisons showed that LWD and UWD differed significantly in UT (0.120 ± 0.062 l·min^-1·sec^-1 versus 0.228 ± 0.058 l·min^-1·sec^-1) as well as in TR (0.117 ± 0.025 l·min^-1·sec^-1 versus 0.251 ± 0.055 l·min^-1·sec^-1). The LWD and UWD did not differ between UT and TR groups.

The main findings of this study were: 1) no interaction between the type of curve fitting and aerobic training status; 2) highest RSS and MSE of GEM in pooled data and in UT group; 3) change in the feature of E curve between untrained and aerobically trained men. The differences showed in RSS and MSE between GEM and segmental regression models in pooled data and the UT group and the absence of difference between _2SRM and 3SRM in pooled data as well as in UT and TR groups make us accept the _2SRM as the best fitting. This choice is based on the parsimony principle since this model is simpler than other models and it utilizes fewer estimative parameters.

Orr et al., 1982 and Dennis et al., 1992 were the only studies to employ mathematical fittings on VE data which could identify one or two metabolic transition points. Our results partially corroborate these findings since they suggested a VE behavior with only one transition point during incremental exercise. However, contradicting the Dennis et al., 1992 findings, a better exponential fitting was not observed here. One possible explanation for this discrepancy is the exponential equation type adopted here.

The GEM equation was chosen after previous comparisons with the continuous exponential described by Hughson et al., 1987. As it was not observed difference between RSS and MSE, the GEM was chosen because it considers important parameters such as offset, delay and grow time constant of the data series, adding only one more parameter than the other exponential.

However, methodological differences between the studies must be a probable justification for the disagreement between them. Dennis et al., 1992 applied a protocol of 15 w·min^-1 increments utilizing the average VE over each minute, which allowed around 15 points on each curve. Our experimental protocol with 1 km·h^-1 increments allowed around 9 stages. As an average VE over each 20 seconds was employed, our experimental design allowed the analysis with around 27 points over each curve, which was almost twice that utilized by Dennis et al., 1992. As a number greater than 20 points is necessary for a successful segmental linear fitting (Orr et al., 1982) it is possible that the lower number of points in Dennis’ study as well as others (Campbell et al., 1989; Orr et al., 1982), allowed a better continuous exponential fitting since this model maximizes the correlation coefficient between x and y values by a linear coefficient (y= a + b exp^(cx)). Then, a better continuous exponential fitting should be observed whenever long time averages (≥ 1 minute) rather than short time averages (≤ 30 seconds) are employed for data analysis.

It is likely that the difference observed between the linear regression fitting and continuous exponential fitting is also linked to ergometer employed because a better linear regression model was found in treadmill (Myers et al., 1994) while a better continuous exponential model was found in cycle ergometer (Campbell et al., 1989; Dennis et al., 1992; Hughson et al., 1987). An incremental treadmill test with lowest initial velocity implies a change in energy expenditure of the transition phase between walking and running (Mercier et al., 1994). Due to lower mechanical efficiency during the transition between walking and running (around 7 km·h^-1) there is a higher increase in VO₂ and VE than during velocities below (< 7 km·h^-1) or above this phase (>7 km·h^-1) (Mercier et al., 1994). This methodological detail can be responsible for a worse fitting with GEM in UT than in TR, similarly to Myers et al., 1994 results.

On the other hand, the exponential model seems to be more sensitive on VO₂ values above 28 ml·kg^-1·min^-1, approximately (Morton, 1993; Beaver et al., 1988). In previous works which found a better continuous exponential (Dennis et al., 1992; Hughson et al., 1987) the VO₂ value in the third minute (or third stage) was around 27 ml.kg^-1.min^{- 1} showing that the main part of the curve was fitted with values above this. Differently, other works that have found a better _2SRM employed an experimental protocol that allowed VO₂ values to be quite below 28 ml.kg^-1.min^-1 in the third minute, and a greater sampling frequency (Myers et al., 1994; Orr et al., 1982). In the present study, VO₂ values around 22 ml.kg^-1.min^-1 were found in UT group and 35 ml·kg^-1·min^-1 in TR group for the third minute (p < 0.001). So, a greater part of the curve built with values less than 28 ml·kg^-1·min^-1 in UT than TR can be the reason for the difference between the GEM and 3SRM in UT group. The modest tendency to underestimate VE values observed mainly in the beginning of the curve (α = -2.01 ± 3.20 l·min^-1) of UT corroborates this supposition. It is also likely that the utilization of VO₂ in x axis (rather than velocity or time) affects the quality of fitting because VO₂ has a curvilinear pattern in incremental tests (Kelly and Lindsey, 2004).

The 2SRM and 3SRM fittings also differed between our study and other studies (Dennis et al., 1992). Some studies employed the segmental regression fittings with intersection points previously determined by visual inspection, making the calculation of an R² possible for each segment (Campbell et al., 1989; Dennis et al., 1992; Hughson et al., 1987). In the present study the intersection points were determined mathematically after testing more than thirty possible combinations, allowing only one R² for the whole data series. These methodological differences may have influenced the quality of fittings because, while the studies which utilized intersection points previously determined by visual inspection found a better exponential fitting (Campbell et al., 1989; Dennis et al., 1992; Hughson et al., 1987), the studies which employed intersection points mathematically determined found a better segmental regression fitting (Myers et al., 1994).

It is not possible to justify statistically or mathematically why 3SRM was not better than 2RSM. Our results are close to other works (Dennis et al., 1992; Orr et al., 1982) and show that adding one more segment does not improve the fit of the data series. As this result was observed in UT group as well as TR group, it can be suggested that a better mathematical fitting is not related to aerobic training status. Therefore, at least in part, it is suggested that aerobic training does not change the identification possibility of one or two metabolic transition points when identified by VE data.

On the other hand, the absence of difference between 2SRM and 3SRM could also be due to type of incremental protocol employed here. The time constant of VE is around 59 s in moderate intensities (~40% VO_2PEAK) (Keslacy et al., 2005) and it increases according to exercise intensity (Casaburi et al., 1989). Thus, it is likely that the incremental protocol has not allowed a complete VE response during intermediate and heavy intensities (≥ 50% VO_2PEAK), leading a delayed response of VE in these stages. Consequently, the first linearity break point of VE could be carried towards the second linearity break point, providing only one break point and making the 3SRM unsuccessful. Nevertheless, as the time constant of VE depends on workload increase-duration relationship (Debigar et al., 2000), it is unlikely that the VE behavior observed in our data would be different from the majority of studies that identified the ventilatory threshold by VE curve in incremental treadmill tests once workload increase-duration relationship is very close between them (Bunc et al., 2001; Esteve-Lanao et al., 2007). Thus, if the workload increase is fixed, a behavior of VE curve different from that observed here could occur only when longer stages (≥ 2 minutes) are employed.

Mathematical models are frequently employed to aid the explanation about physiological mechanisms. Taking into account both previous results and the results obtained here, it would be suggested the presence of only one metabolic transition point in VE curve (Dennis et al., 1992; Orr et al., 1982). However, if it is a continuous process rather than a threshold model it can only be a methodological question.

Admitting a very close relationship between lactate production and ventilatory responses, the presence of only one metabolic transition point could be linked to a modification point on the lactate/pyruvate ratio around 65% VO_2PEAK (i.e., 4-6 mmol·l^-1 of lactate) due to saturation on pyruvate dehydrogenase (PDH) activity and an elevation on lactate dehydrogenase (LDH) activity around twice or three times greater than the former (Spriet et al., 2000). Specifically, the saturation on PDH activity can be a result of modification of the redox state (NADH/NAD ratio), substrate availability and product accumulation relationship, and mitochondrial O₂ supply (Hollidge-Hovart et al., 1999; Parolin et al., 2000).

On the other hand, the control of VE during exercise is suggested as a multifactorial process (Ward, 2007). Potassium (K⁺), catecholamine and ammonia (NH₃^-) exert humoral control on the ventilation by carotid body chemoreceptors, promoting an elevation in ventilatory stimulus as it increases (Hopkins et al., 2003; Paterson, 1997; Tong et al., 2003). Evidences show the presence of only one inflection point in K⁺, catecholamine and NH₃^- in workloads around 40% and 70% of VO_2PEAK (or 60% and 80% maximal power output) during incremental exercise (Busse et al., 1992; Fraser et al., 2002; Mcmorris et al., 2000; Yuan et al., 2002). Whether the increasing pattern of K⁺, catecholamine and NH₃^- in incremental exercise is linked to one metabolic transition point on VE is a matter that future studies should analyze.

The findings of present study have practical implications. Usually, VT1 is employed for aerobic training prescription in elderly and diseased population (Koufaki et al., 2000; Meyer et al., 2005) while VT2 is used for aerobic training prescription in healthy subjects and elite athletes (Lucía et al., 2000; Meyer et al., 2005), existing a broad difference of intensity between them. As it was experimentally observed only one ventilatory threshold, it is not possible to know if VT_2SRM would correspond at training intensity for elderly and diseased population or for healthy subjects and elite athletes since VT_2SRM was significantly higher than VT1 (p < 0.01) and significantly lower than VT2 (p < 0.01) in pooled data as well as in UT and TR. Consequently, VT_2SRM could be just employed as an upper intensity limit for aerobic training prescription in elderly and disease population and a lower intensity limit for aerobic training prescription in healthy subjects and elite athletes. While intensities around ~80% VT_2SRM could be employed for elderly and diseased population, intensities around ~120% VT_2SRM could be employed for healthy subjects and elite athletes. However, further studies should be realized to confirm this proposal.

In the present study the relationship between parameters obtained by the best fitting and aerobic fitness level also was investigated. As proposed in pattern increasing on lactate curve (Mader and Heck, 1986), it was hypothesized that the aerobic training would allow a more smooth VE increase during incremental exercise, providing a different behavior in VE curve between untrained and aerobically trained subjects. As there was no difference in VE curve fitted between UT and TR groups, it could be suggested that the aerobic training status does not have effects on the mathematical description of VE behavior during incremental exercise. However, as the lower and upper derivatives of _2SRM were associated to markers of aerobic fitness, it could be suggested that these parameters of best mathematical fitting can be used to discriminate the aerobic training status.

In the pooled data, significant and negative correlations were observed among lower and upper derivatives of _2SRM and the markers of aerobic capacity (VT_2SRM) and aerobic power (Speed_PEAK). The results show that in a heterogenic group an improvement in the aerobic fitness also suggests an improvement in gas exchange leading the aerobically trained subjects to need a smaller elevation in the ventilatory responses over each increase of the mechanic power output during incremental exercise (Bickham et al., 2004; Ward, 2007). When analysis was carried out separately, differences were observed. In UT group the correlation coefficients showed that the subjects who had higher aerobic capacity and aerobic power also had a slower VE increase during moderate workloads (i.e. < VT_2SRM). In TR group, those who had higher aerobic capacity and aerobic power showed a slower VE increase in heavy workloads (i.e. > VT_2SRM).

It is not possible to establish any cause and effect relationship between slower VE increase and aerobic fitness markers in the present study. In spite of this, some suppositions should not be discarded: 1) for heterogenic groups the aerobic training seems to improve the duration of physical work in moderate workloads as well as heavy workloads, moving the whole VE-workload curve to the right with a smaller slope; 2) in untrained subjects the adaptations to aerobic training would mainly affect the work capacity in moderate workloads; 3) in endurance homogeny groups the time of permanence in heavy intensities can better distinguish the aerobic fitness level than VT_2SRM, once these subjects already show high values of ventilatory threshold. Actually, these results would corroborate other studies (Bickham et al., 2004; Ward, 2007), which can be a consequence of interval training carried out by soccer players (Bunc and Psotta, 2001).

The physiological mechanisms responsible for a slower VE increase in moderate (UT group) and heavy (TR group) intensities cannot be explained here. However, the following aspects linked to ventilation control should be investigated in future studies to aid in clarifying the possible mechanisms: 1) the association between the modifications on the bicarbonate buffering efficiency, PFK and PDH activity, NH₃^- and catecholamine levels, and the VE behavior in moderate intensities (i. e. < VT_2SRM) after aerobic training; 2) the association between the modifications on the tolerance to metabolic acidity, LDH isozyme expression (LDH-M isozyme and LDH-H isozyme), K⁺ concentration, fiber recruiting pattern, and the VE behavior in heavy intensities (i.e > VT_2SRM) after aerobic training.

The behavior of VE during incremental treadmill test tends to show only one metabolic transition point. Although some possible physiological mechanisms can justify this increasing on VE data, the presence of a threshold process rather than a continuous process seems to be more of a methodological than physiological matter. When the feature of VE curve is analyzed according to the training status it is possible to verify that untrained subjects show a slower VE increase at moderate rather than heavy intensities and that aerobically trained subjects show a slower VE increase at heavy rather than moderate intensities.