This variable results in the %V˙O2max,01 and %V˙O2max,02 being spread or reduced with the same distance (d), although the symmetry of the global shape did not change (s = b). The model, therefore, did not necessarily cross the abscissa axis at 0 and 100% of V˙O2max (Fig. 1A).
Substituting equations 4 and 7 into equation 6:
Substituting equations 5 and 10 into equation 9:
Substituting equation 8 into equation 11 and s = b (to keep the same global shape of the curve), e and g could be determined:
Symmetry and translation.
The variables of symmetry and translation were determined according a similar development but with different basic specifications. The variable of symmetry permitted movement of the peak of the curve while keeping the same %V˙O2max,01 and %V˙O2max,02 values (Fig. 1B). The variable of translation was used to translate the whole curve to the right or to the left toward the abscissa axis (Fig. 1C), which meant that %V˙O2max,01 and %V˙O2max,02 were shifted equally (t) in the same direction, whereas the symmetry of the global shape did not change (s = b).
Finally, when developing the global equation including by substitution these three variables (dilatation, symmetry, and translation), as summed up in Table 1, and regardless of the integrated chronological order, the final equation became:
where d, s, and t were the variables of dilatation, symmetry, and translation, respectively, and K the constant of intensity corresponded to (π / 100) (equation 2); development details are provided in the Appendix.
The basic symmetric curve (d = 0, t = 0, and s = 1), which has intersections with the abscissa axis at 0 and 100% V˙O2max, could therefore be modulated to fit the experimental data by independently changing the values of these three variables.
The independent variables of the model (dilatation, symmetry, and translation) were determined with an iterative procedure by minimizing the sum of the mean squares (SMS) of the differences between the estimated energy derived from lipid (Elipid) on the basis of the mathematical models and the measured values (MV) of Elipid.
All data are presented as means ± SD, and in the absence of normal distribution within sections of a data set, medians ± interquartile ranges (IQR) are also provided (Table 2). A Friedman repeated-measures ANOVA was used to compare SMS of the different fitting procedures to assess the accuracy of each method, and significant differences were isolated by using Turkey's post hoc test.
To compare the agreement between the different methods and their relative values of Fatmax and MFO, Pearson product-moment correlation coefficients were calculated and Bland-Altman plots (7) were used. The constructed graphs displayed scatter diagrams of the differences plotted against the mean of two measurements. The SD of the difference and the bias estimated from the mean difference (
) were calculated, and 95% limits of agreement were estimated by
± 1.96 SD.
Finally, Pearson product-moment correlations or Spearman rank-order correlation when the assumption of normality of distribution was violated were used to establish relationships between the three variables of SIN and the subjects' physical characteristics. For all statistical analyses, significance was accepted at P < 0.05.
The physical and performance characteristics of the subjects obtained during the graded exercise test to exhaustion are listed in Table 2. These include a mean V˙O2max of 47.9 ± 11.1 mL·kg−1·min−1 and HRmax of 181 ± 11 bpm.
Accuracy of the different methods.
Figure 2 provides various examples of subject's fat oxidation kinetics, expressed in %MFO and represented as a function of exercise intensity (%V˙O2max), obtained during graded test, with the different fitting curves corresponding to SIN, P3, and RER method (MRER).
No significant differences were found for the fitting accuracy, expressed in SMS between estimated Elipid and MV of Elipid, between SIN and P3 (943,598.1 ± 1,021,295.6 and 871,542.6 ± 940,802.1, P = 0.157, respectively), whereas MRER seemed to be less accurate than the two other methods (3,633,224.4 ± 6,123,245.6, P < 0.001 for both). The accuracy of MRER was dependant on the linear relationship between RER and submaximal exercise intensity up to an RER of 1.0 (r = 0.72, P < 0.001). When the RER value was, however, too high at rest, the linearity of the relationship between RER and exercise intensity was affected (r = −0.75, P < 0.001).
Agreement between methods.
According to the MV, fat oxidation rates increased with increasing of exercise intensity, up to a maximal of 0.39 ± 0.17 g·min−1 (range, 0.14-0.81 g·min−1) and occurred at an intensity of 45.2 ± 13.1% V˙O2max (range, 23%-70% V˙O2max). Mean values of Fatmax and several corresponding parameters such as fat oxidation rate or RER determined with the different methods are presented in Table 3. Values of each parameter were significantly correlated, and high correlation coefficients were obtained between methods, e.g., for Fatmax, r = 0.64 to 0.99 (P < 0.001), and MFO values, r = 0.97 to 1 (P < 0.001). Agreements between models were also confirmed by Bland-Altman plots. Biases and limits of agreement for values of Fatmax and MFO determined with the different techniques in comparison of MV are shown in Table 4. When values of Fatmax or MFO estimated by SIN were plotted against those determined with P3 (Fig. 3A), all data were close to the line of equality with correlation coefficients of 0.99 and 1 (P < 0.001), respectively, which was confirmed by biases close to zero (−0.26 ± 1.52 and 0.001 ± 0.005 for Fatmax and MFO, respectively) and narrow limits of agreement (from −3.234 to 2.708 and −0.008 to 0.010 for Fatmax and MFO, respectively; Fig. 3B).
Relationships between variables of SIN and physical characteristics.
Mean values of Fatmax and Fatmin determined with SIN model for the 32 subjects were 44.0 ± 10.1% V˙O2max (range, 29%-68% V˙O2max) and 90.1 ± 14.1% V˙O2max (range, 52%-99% V˙O2max), respectively, whereas MFO reached 0.37 ± 0.16 g·min−1 (range, 0.14-0.80 g·min−1). Correlation analyses were used to establish relationships between the three variables of the mathematical model (dilatation, symmetry, and translation) and the physical characteristics of the subjects. Although these variables were not correlated with the physical characteristics such as age, stature, body mass, or fat mass (P > 0.05), symmetry was positively correlated with Fatmax (r = 0.70, P < 0.001) and dilatation with Fatmax (r = 0.79, P < 0.001) and MFO (r = 0.60, P < 0.001). Fatmin was significantly correlated with the variables of dilatation (r = 0.67, P < 0.001) and translation (r = −0.76, P < 0.001). Translation was also linked with the index of physical activity estimated from questionnaire (r = −0.46, P < 0.05). MFO was correlated with maximal oxygen uptake (r = 0.44, P < 0.05), and V˙O2max with the index of physical activity (r = 0.49, P < 0.01). These correlations are presented in Figure 4.
The main objective of the present study was to develop a mathematical model (SIN) that includes three independent variables that accurately describe the different patterns of fat oxidation kinetics during an incremental exercise protocol and determines Fatmax, Fatmin, and MFO. In comparison with other methods currently used, the fitting curves obtained with SIN were as accurate as constructed P3 and were more accurate than MRER. SIN was effective because Fatmax (44 ± 10.1% V˙O2max) and MFO (0.37 ± 0.16 g·min−1) determined using this method were highly correlated with MV and those obtained with P3 or MRER and, in addition, allowed the calculation of Fatmin. Moreover, the three independent variables were directly related to the main expected modulations of the fat oxidation curve, and the variable of dilatation was found to be representative of a subjects' ability to oxidize lipid because it is significantly correlated with values of Fatmax, Fatmin, and MFO.
Graded exercise test.
In the present investigation, subjects reported to the laboratory either in the morning after a 12-h overnight fast, or in the afternoon 6 h after a standardized meal (∼300 kcal). Indeed, it has been previously shown that metabolic and hormonal responses during an exercise bout performed 3 h after a meal at moderate intensity, corresponding to the "crossover" point of substrate oxidation (i.e., the power output at which energy from CHO-derived fuel predominates over energy from lipids ), were closely similar to those targeted during the same submaximal exercise performed in a fasting state (11). The graded exercise protocol used was adapted from a previously validated one (1) in which the authors showed that when stage duration was reduced from 5 to 3 min or when increment size was reduced from 35 to 20 W; no significant differences were found in Fatmax, Fatmin, and fat oxidation rates in healthy subjects. However, when the moderately trained cyclists performed the graded exercise test with 5-min stages and 35-W increments, 7 of the 18 subjects did not have sufficient data points to construct the relationship between fat oxidation rate and exercise intensity (i.e., there were no or insufficient intensities below Fatmax) (1). Moreover, Stisen et al. (30) used a graded exercise protocol with 10- to 20-W increments to achieve an increment of around 10% V˙O2max in each step. Taking these results into account, and as the subjects of the present study were not cyclists, a graded exercise test with 3-min stages, 20-W increments, and starting at 40 W was used to ensure that a minimum number of values occurred before Fatmax. Moreover, the mean test duration (i.e., 8.9 ± 0.5 stages) was in line with previous study (30).
Accuracy of the methods.
The fitting curves constructed with SIN and P3 appeared to correlate strongly, and no significant differences were found between these two methods. Although the accuracy was not significantly different (P = 0.157), P3, however, presented a lower mean SMS. This could be explained by the fact that P3 did not necessarily cross the abscissa axis twice and could therefore better accommodate the experimental data. On the other hand, the absence of this constraint implied that P3 could not determine Fatmin in every case, which could be a disadvantage (discussed in "Agreement between methods"). Fitting curves constructed with MRER, however, seemed to be less accurate than those that resulted from the two other methods, and SMS was significantly higher than those obtained with SIN and P3. However, the RER technique is based on the theoretical linear relationship between RER and submaximal exercise intensity. In the present study, a positive correlation was found between this linearity and the accuracy of MRER (r = 0.72, P < 0.001), which was not the case with SIN and P3. This implies that the more linear the relationship, the more accurate the fitting curve determined with MRER. As Goedecke et al. (16) observed with trained athletes, a large variability occurred in resting RER, and nine subjects presented high values (i.e., 0.88-0.96). For these individuals, the RER decreased to "normal value" when the exercise protocol started and then increased proportionally with exercise intensity. Because all stages, including the rest period, were therefore taken into consideration, the linear rise between RER and exercise intensity was affected (r = −0.75, P < 0.001), which could explain why the fat oxidation curve constructed with MRER did not accurately fit the experimental data in some cases. Although the variability in resting RER remained unclear, the main finding was that it did not affect the accuracy of SIN and P3.
Agreement between methods.
To analyze agreement between the different methods, MV of Fatmax and MFO were compared with data determined with SIN, P3, and MRER using the mean of Bland-Altman plots. Small positive biases between MV and SIN or P3 indicated that these two methods tend to underestimate the MV to a small degree (Table 4). This observation was confirmed by lower mean values of Fatmax and MFO determined with these models (Table 3). MRER, however, had wider limits of agreement and tended to overestimate Fatmax, although it underestimated MFO. In the present study, the RER technique therefore seemed to be less accurate and efficient than SIN and P3.
In this analysis, MV was considered as reference method. Although this technique functions best when the set of measures forms a clear parabolic curve with one distinct peak, it can also have shortcomings. For example, when two similar MFO rates occur at two different exercise intensities, in the case of SIN or P3 models, Fatmax determination would not be influenced by the order of the two peaks. The use of a mathematical model is therefore more consistent than MV when analyzing data that do not align in a perfect curve.
Among the different procedures currently used, P3 is the only method that also models the kinetics. It is therefore interesting to compare this technique with values obtained with SIN. As previously analyzed, SIN was as accurate as P3 in fitting experimental data of fat oxidation rates obtained during a graded exercise protocol. When comparing values of Fatmax and MFO determined with these two methods, results were also very similar. In Figure 3A, most of the points lie along the line of equality, and correlation coefficients are equal to 0.99 and 1 (P < 0.001), respectively, which confirm the nearly perfect agreement between measurements by SIN and P3. Negligible biases and narrow limits of agreement represented in Figure 3B confirm that these two models are consistent with each other. Therefore, SIN could be considered as accurate and as efficient as P3 in describing fat oxidation kinetics and in determining parameters such as MFO or Fatmax. On the other hand, SIN has the additional advantage of also being able to determine Fatmin in every case, which is not possible with P3. In fact, according to Brooks and Mercier (8), prior endurance training results in muscular biochemical adaptations characterized by an increase in lipid oxidation and a decrease of the sympathetic nervous system (SNS) activity in response to given submaximal exercise intensities. After aerobic training, the crossover point would shift to the right. Consequently, Fatmax and Fatmin should also occur at a higher intensity. Fatmin could therefore be an interesting additional parameter when analyzing the effect of a specific training program on fat oxidation kinetics. Another limitation of third polynomial equations is that parameters are dependant and do not correspond to particular elements. A mathematical model including independent variables that are directly related to the main expected modulations of the curve could therefore be of considerable interest. Although studies have shown that several factors such as training level (2,9,25,30), gender (9,18,32,33,38), mode of exercise (4), or body composition (23) could influence fat oxidation kinetics, Fatmax, or MFO, some uncertainties still remain. For example, although it has been shown that fat oxidation rates increased with endurance training when differences in substrate utilization were investigated between trained and untrained subjects (2,9,25,30), nobody has studied whether the curve depicting fat oxidation as a function of the relative exercise intensity is projected upward (i.e., increase of MFO), rightward (i.e., Fatmax and Fatmin occur at higher intensities), or both after training. A better quantification of the above-mentioned factors that affect fat oxidation kinetics could assist in improving exercise interventions and result in more effective treatment of conditions in which fat oxidation patterns are disturbed (38). The SIN model has therefore been developed with three independent variables, which are directly related to the main expected modulations of the curve (Fig. 1), to accommodate all fat oxidation kinetics, while accurately determining Fatmax, Fatmin, and MFO.
Independent variables of the SIN model.
Basic values of 0 for dilatation, 1 for symmetry, and 0 for translation determine a symmetric curve that has intersections with the abscissa axis at (0,0) and (100,0) (i.e., 0 and 100% V˙O2max). A variable of symmetry between 0 and 1 therefore indicates a leftward asymmetry, whereas a value >1, a rightward asymmetry. Together with the translation variable, this could be, for example, linked to the effect of training level on fat oxidation kinetics because trained people have been reported to reach Fatmax at higher intensities (2) and to oxidize more fat than untrained subjects during intense exercise (10). In the present study, the correlations found between symmetry and Fatmax and between translation and Fatmin or the physical activity level therefore seem to indicate that trained people may be able to use energy derived from lipids at higher intensity than sedentary individuals and that their fat oxidation curve, or the peak of the curve, tend to be shifted to the right. These observations confirm previous findings of higher Fatmin in trained people than those with lower V˙O2max values (1,2,38). The variable of dilatation seems to be representative of a subjects' ability to oxidize lipid because it is significantly correlated with values of Fatmax, Fatmin, and MFO. These relationships suggest that when Fatmax occurs at a higher intensity, the curve becomes more dilated, and the Fatmax zone (1) tends to be larger.
Practical illustration of the SIN model.
To illustrate the practical relevance of the model, examples of the fat oxidation kinetics of two different individuals obtained during the graded exercise test and constructed with SIN are presented in Figure 5. The curves present the absolute and relative fat oxidation kinetics of two men matched according to their physical characteristics (i.e., age, mass, height, body mass index, fat-free mass) but with different training levels (including V˙O2max, Fatmax, and MFO). Figure 5B, which presents the relative fat oxidation kinetics, clearly quantifies the differences in these variables between the trained and untrained subjects. The endurance individual's fat oxidation kinetics is more dilated than that of the untrained (dilation values of 0.18 vs −0.52, respectively). His curve also has a rightward asymmetry (symmetry value of 1.52), as opposed to that of the untrained subject (symmetry value of 0.60), and is more translated to the right (translation value of 0.21 vs 0.25, respectively). The three independent variables therefore precisely characterize and quantify the shape of fat oxidation kinetics of the trained and untrained subjects. However, the large degree of variability between individuals in Fatmax (range, 29%-68% V˙O2max), MFO (0.14-0.80 g·min−1), or V˙O2max (25.4-65.2 mL·min−1·kg−1) and the absence of underweight or obese subjects made it difficult to confirm significant relationships with training level or body composition. However, this was not the goal of this study. Additional research involving more homogenous groups within the sample is therefore required to investigate the sensitivity of the SIN model in identifying differences in the fat oxidation kinetics responses to these factors. Moreover, although women seem to have a greater reliance on fat oxidation than men during exercise (9), both genders were included in the present protocol to test the accuracy and efficiency of the SIN model to fit all different shapes of fat oxidation kinetics that may occur.
In summary, the SIN model provides a mathematical description of fat oxidation kinetics during graded exercise, which presents the same precision as other methods currently used in determination of Fatmax and MFO but in addition allows calculation of Fatmin. In addition, the variables of dilatation, symmetry, and translation account for the main expected modulations of the curve and can be adjusted separately to accurately accommodate the data. The degree of dilatation also seems to be a sensitive marker of the ability to oxidize fat. The SIN model developed in the present study therefore seems to be a valuable tool and, with its independent variables, could be of considerable interest when investigating the impact of a specific factor (e.g., training program or diet) on fat oxidation kinetics.
No funding received for this work.
The results of the present study do not constitute endorsement by ACSM.
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Development of the global equation, including by substitution the three variables (dilatation, symmetry, and translation), whose parameters are summed up in Table 1.
Including the variable of dilatation:
Including the variable of translation:
Including the variable of symmetry:
Because the standard basic curve used is symmetric and crosses the abscissa axis in (0,0) and (100,0), therefore, a = 1 (no dilatation or retraction), b = 1 (symmetric curve), and c = 0 (no translation).
Finally, with x = %V˙O2max:
with the constant of intensity K = π / 100.
Keywords:© 2009 American College of Sports Medicine
FATmax; MAXIMAL FAT OXIDATION; INDIRECT CALORIMETRY; EXERCISE INTENSITY