# A Mathematical Model to Describe Fat Oxidation Kinetics during Graded Exercise

Purpose: The purpose of this study was to develop a mathematical model (sine model, SIN) to describe fat oxidation kinetics as a function of the relative exercise intensity [% of maximal oxygen uptake (%V˙O_{2max})] during graded exercise and to determine the exercise intensity (Fat_{max}) that elicits maximal fat oxidation (MFO) and the intensity at which the fat oxidation becomes negligible (Fat_{min}). This model included three independent variables (*dilatation*, *symmetry*, and *translation*) that incorporated primary expected modulations of the curve because of training level or body composition.

Methods: Thirty-two healthy volunteers (17 women and 15 men) performed a graded exercise test on a cycle ergometer, with 3-min stages and 20-W increments. Substrate oxidation rates were determined using indirect calorimetry. SIN was compared with measured values (MV) and with other methods currently used [i.e., the RER method (MRER) and third polynomial curves (P3)].

Results: There was no significant difference in the fitting accuracy between SIN and P3 (*P* = 0.157), whereas MRER was less precise than SIN (*P* < 0.001). Fat_{max} (44 ± 10% V˙O_{2max}) and MFO (0.37 ± 0.16 g·min^{−1}) determined using SIN were significantly correlated with MV, P3, and MRER (*P* < 0.001). The variable of *dilatation* was correlated with Fat_{max}, Fat_{min}, and MFO (*r* = 0.79, *r* = 0.67, and *r* = 0.60, respectively, *P* < 0.001).

Conclusions: The SIN model presents the same precision as other methods currently used in the determination of Fat_{max} and MFO but in addition allows calculation of Fat_{min}. Moreover, the three independent variables are directly related to the main expected modulations of the fat oxidation curve. SIN, therefore, seems to be an appropriate tool in analyzing fat oxidation kinetics obtained during graded exercise.

^{1}Institute of Sport Sciences and Physical Education, University of Lausanne, Lausanne, SWITZERLAND; and ^{2}School of Medical Sciences, Faculty of Health Sciences, University of KwaZulu-Natal, SOUTH AFRICA

Address for correspondence: Xavier Chenevière, Institute of Sport Sciences and Physical Education, University of Lausanne, Bâtiment de Vidy, 1015 Lausanne, Switzerland; E-mail: xavier.cheneviere@unil.ch.

Submitted for publication May 2008.

Accepted for publication January 2009.

Obesity is a serious current health problem that is related to an increased risk of chronic diseases such as hypertension, coronary artery disease, and type 2 diabetes (^{13}). Interventions such as aerobic exercise, which have been found to impact positively on successful long-term reduced body weight maintenance in the previously overweight (^{29}), are therefore of considerable interest (^{34}). Although it is well documented that endurance training and regular physical activity increase daily energy expenditure and fat oxidation in healthy individuals and in the obese population (^{1,3,28,31,36}), the precise characteristics of an exercise protocol that optimizes weight management have not yet been defined (^{21,34}).

During aerobic exercise, carbohydrate (CHO) and fat are the two main sources of energy that sustain oxidative metabolism, and their relative utilization depends largely on the exercise intensity (^{8,26,37}). CHO oxidation increases with the workload, whereas absolute fat oxidation rates increase from low to moderate exercise intensities and decline at high intensities. The exercise intensity, at which the maximal fat oxidation (MFO) rate occurs, has been defined as Fat_{max} (^{1,21,30}), although Fat_{min} corresponds to the intensity at which the fat oxidation rate reached zero (i.e., respiratory exchange ratio (RER) ≥ 1) (^{1}).

In previous studies, fat oxidation rates have been measured at only two (^{15,34}), three (^{26,27,37}), four (^{6}), or six (^{19}) different exercise intensities, which made it difficult to identify with precision the exercise intensity that elicits peak fat oxidation (^{1,22}). A graded exercise protocol performed on a single day has therefore been proposed (^{1}) to determine fat oxidation rates over a wider range of intensities (^{1,2,4,22,30,38}).

Various methods of using graphical depiction of fat oxidation rate values as a function of exercise intensity to determine Fat_{max} have also been described. These include determination of the maximal value of measured fat oxidation rates reached during each stage of the graded exercise test and identification of the corresponding intensity (^{1,2,4,38}), and construction of a standard (^{22}) or a third polynomial fitting curve (P3) (^{30}). Pérez-Martin et al. (^{23}) calculated the MFO point by differentiating the simplified equation, fat oxidation rate = 1.7 (1 − RER) V˙O_{2}, according to the nonprotein respiratory quotient technique (^{24}).

It is well documented that the pattern of fat oxidation kinetics, including MFO and Fat_{max}, presented as a function of exercise intensity, may be influenced by gender, training level, body composition, or exercise mode (^{3,4,20,38}). Regardless of the method used, exercise that elicits MFO does not seem to exceed a moderate intensity. In a recent cross-sectional study conducted on a wide range of healthy men and women performing a graded exercise, Venables et al. (^{38}) reported an average Fat_{max} at 48% (range, 25%-77%) of maximal oxygen uptake (V˙O_{2max}), with a significant difference between males and females (45% and 52%, respectively), whereas others (^{1,2,4}) measured MFO at an exercise intensity between 62% and 64% V˙O_{2max} in moderately trained subjects. Obese subjects have been found to present with lower rates of fat oxidation at given submaximal workloads and Fat_{max} at a lower exercise intensity (^{23}). It has also been observed that between 50% and 70% V˙O_{2max}, fat oxidation rates are significantly higher during running than cycling, with MFO occurring at the same relative intensity in both exercise modalities (^{4}). Compiling the above-mentioned elements, various fat oxidation kinetics may occur. However, despite identification of these potential regulatory factors, a large degree of interindividual variability in substrate utilization at rest and during exercise, which remains unexplained, exists (^{3,16,17,35,38}), even within a homogenous group of trained subjects (^{2}). To our knowledge, no study has been carried out to date to describe the pattern of fat kinetics curve during graded exercise quantitatively.

The main purpose of the present study was therefore to develop a mathematical model that (i) accurately describes specific fat oxidation kinetics during graded exercise; (ii) determines Fat_{max}, Fat_{min}, and MFO; and (iii) incorporates independent variables that account for expected modulations of the pattern of fat oxidation because of factors such as training level, gender, or body composition. A second aim was to test the validity of the findings and their correlation with those obtained from other currently used methods.

## METHODS

### Subjects

Thirty-two healthy, sedentary to moderately trained volunteers (17 women and 15 men; age range, 21-59 yr) were recruited to participate in this study. All subjects considered for the study were nonsmokers, disease-free, not taking any medication, and were screened for the absence of electrocardiographic abnormalities. Each volunteer completed a self-reported measurement of habitual physical activity questionnaire divided into three sections: physical activity at work, sport during leisure time, and physical activity during leisure excluding sport (^{5}). The participants were considered sedentary if the sum of the three derived indices (i.e., work, sports, and leisure) was below an arbitrary score of 8, and their physical activity was <1 h·wk^{−1}. All test procedures, risks and benefits associated with the experiment were fully explained, and subjects were asked to provide written consent for participation. The protocol and the consent form were approved by the local ethics committee.

### General Design

Each subject performed a graded exercise test to exhaustion on a cycle ergometer. Averaged fat and CHO oxidation obtained during the last minute of each submaximal work stage were determined by indirect calorimetry and plotted as a function of exercise intensity. The data over a wide range of intensities were used to develop and to test a mathematical model to describe the fat oxidation kinetics during graded exercise and to determine Fat_{max}, MFO, and Fat_{min}.

### Experimental Design

Before the experiment, subjects were familiarized with equipment and procedures. Volunteers were requested to avoid strenuous exercise and drinking alcohol and caffeine for the 24-h preceding the test, but no dietary restriction was imposed during the days before the day of the test. Subjects reported to the laboratory either in the morning after a 12-h overnight fast (*n* = 22) or in the afternoon 6 h after a standardized meal made of 19% protein, 77% CHO, and 4% fat providing an energy yield of 4.3 kcal·kg^{−1} (*n* = 10). Body mass and stature were measured, and body composition (body fat mass and % body fat) was estimated from skinfold thickness measurements at four sites according to the method of Durnin and Womersley (^{12}). After a 3-min rest period, volunteers began with a 5-min warm-up at 40 W on a cycle ergometer (Ebike Basic BPlus; General Electric, Niskayuna, NY) after which the workload was, in keeping with the method described by Achten et al. (^{1}), increased by 20 W every 3 min until RER reached 1.0. Thereafter, the work rate was increased by 20 W every minute until exhaustion to obtain a measure of V˙O_{2max} within a short time. For five subjects who fell into the sedentary category according to their physical activity level, the exercise protocol was similar, but started at 20 W, and the workload was increased by 10 W to ensure that a minimum number of values occurred before RER = 1.0. Oxygen uptake (V˙O_{2}) was considered to be maximal when at least two of the following three criteria were met: 1) a leveling off of V˙O_{2} (defined as an increase of no more than 2 mL·kg^{−1}·min^{−1}) during the latter stages of the exercise test, 2) a heart rate (HR) >90% of the age-predicted maximum (220 − age), and 3) a RER > 1.1. V˙O_{2max} was calculated as the average V˙O_{2} during the last 20 s of the last stage of the test.

During the test, HR was recorded continuously from an ECG (GE Cardiosoft Corina; GE Medical Systems, Freiburg, Germany). Breath-by-breath measurements were performed throughout the exercise using an Oxycon Pro gas analysis system (Jaeger, Würzburg, Germany). Before each test, the gas analyzers were calibrated with gases of known concentration (16.0% O_{2}; 5.02% CO_{2}), and the volume was automatically calibrated at flow rates of 0.2 and 2 L·s^{−1}.

### Indirect Calorimetry and Calculation of Substrate Oxidation Rates

During the test, average values for V˙O_{2} and CO_{2} output (V˙CO_{2}) were calculated during the last minute of every stage, during which the RER ≤ 1.0. Fat and CHO oxidation rates and energy expenditure were calculated using stoichiometric equations (^{14}) and appropriate energy equivalents, with the assumption that urinary nitrogen excretion rate was negligible.

where *V˙O2* and *V˙CO2* are in liters per minute.

### Fat Oxidation Kinetics: Current Methods Used to Determine Fat_{max} or MFO

In previous studies focusing on lipid kinetics during graded exercise, fat oxidation rate values have been depicted graphically as a function of exercise intensity (^{1,2,4,22,23,30,38}), and the methods that have been used to calculate or to determine Fat_{max} or MFO include the following:

#### Measured values.

Determination of the maximal value of measured fat oxidation rates reached during every stage of the graded exercise test and identification of the corresponding intensity (^{1,2,4,38}).

#### RER method.

The calculation of lipid oxidation rates is estimated from gas exchange measurement according to the nonprotein RER technique (^{24}): fat oxidation rate (g·min^{−1}) = 1.6946 V˙O_{2} − 1.7012 V˙CO_{2}, which is simplified as fat = 1.7 (1 − RER) V˙O_{2} (^{23}). This equation is composed of two linear relationships: the decrease of (1 − RER) and the increase of V˙O_{2} as a function of exercise intensity. Fat oxidation rates can therefore be calculated by differentiating this equation, and MFO is the point at which the value of the differentiated equation is equal to zero.

#### P3.

Graphical depiction of fat oxidation values as a function of exercise intensity consists of constructing a standard (^{22}) or a P3 with the intersection at (0,0) (^{30}).

### Fat Oxidation Kinetics: Development of a New Mathematical Model

#### Sine model.

The sine model (SIN) was developed to model and describe mathematically fat oxidation kinetics obtained during graded exercise and to determine Fat_{max}, MFO, and Fat_{min}. The equation included three independent variables that accounted for theoretical and main expected modulations of fat oxidation curve because of factors such as gender, training level, or body composition: *dilatation*, *symmetry*, and *translation*. *Dilatation* refers to the degree of dilatation or retraction of the curve (Fig. 1A), *symmetry* variable is used to break the symmetry of the standard basic sine curve (Fig. 1B), and *translation* refers to translation of the whole curve toward the abscissa axis (Fig. 1C).

The basic equation used to develop the mathematical model was a simple sine function, which is *π*-periodic and has an intersection in (0,0).

As the fat oxidation kinetics represented in the present study in percentage of MFO (%MFO) during graded exercise is depicted graphically as a function of percentage of maximal exercise intensity (%V˙O_{2max}), where the rate of fat oxidation increases from low to moderate exercise intensities and decreases again at high intensities, the standard basic curve used should only cross the abscissa axis at (0,0) and (100,0) (i.e., 0 and 100% of V˙O_{2max}). A constant of intensity *K* was therefore included in the basic sine equation.

Further in the development of the equation, *K**x*, which corresponds to *K* · %V˙O_{2max}, is only represented by %V˙O_{2max}.

The sine equation, in accordance with the three identified variables (*dilatation*, *symmetry*, and *translation*), could be expressed by the following equation:

According to this equation, two points (%V˙O_{2max,01} and %V˙O_{2max,02}), where the contribution of fat oxidation to energy expenditure reaches zero (fat oxidation = 0% MFO), could be identified:

Use of equation 3 would, however, imply that all the three parameters (*a*, *b*, and *c*) are dependant and move together when the mathematical model is adapted to fit the data, making it difficult to interpret the results or to establish correlation with factors such as training level or gender. The equation was therefore amended to permit independent modulation of the variables of *dilatation*, *symmetry*, and *translation*, which could function in isolation on the curve. When developing these different variables, parameters *a*, *b*, and *c* were replaced in the basic equation 3 by *e*, *s*, and *g*, respectively, to avoid confusion.

#### Dilatation.

This variable results in the %V˙O_{2max,01} and %V˙O_{2max,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˙O_{2max} (Fig. 1A).

For %V˙O_{2max,01}:

with

Substituting equations 4 and 7 into equation 6:

For%V˙O_{2max,02}:

with

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:

and

#### 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˙O_{2max,01} and %V˙O_{2max,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˙O_{2max,01} and %V˙O_{2max,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˙O_{2max}, could therefore be modulated to fit the experimental data by independently changing the values of these three variables.

### Data Analysis

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 (*E*_{lipid}) on the basis of the mathematical models and the measured values (MV) of *E*_{lipid}.

### Statistical Analysis

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 Fat_{max} 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.

## RESULTS

#### Subject characteristics.

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˙O_{2max} of 47.9 ± 11.1 mL·kg^{−1}·min^{−1} and HR_{max} 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˙O_{2max}), 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 *E*_{lipid} and MV of *E*_{lipid}, 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˙O_{2max} (range, 23%-70% V˙O_{2max}). Mean values of Fat_{max} 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 Fat_{max}, *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 Fat_{max} and MFO determined with the different techniques in comparison of MV are shown in Table 4. When values of Fat_{max} 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 Fat_{max} and MFO, respectively) and narrow limits of agreement (from −3.234 to 2.708 and −0.008 to 0.010 for Fat_{max} and MFO, respectively; Fig. 3B).

#### Relationships between variables of SIN and physical characteristics.

Mean values of Fat_{max} and Fat_{min} determined with SIN model for the 32 subjects were 44.0 ± 10.1% V˙O_{2max} (range, 29%-68% V˙O_{2max}) and 90.1 ± 14.1% V˙O_{2max} (range, 52%-99% V˙O_{2max}), 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 Fat_{max} (*r* = 0.70, *P* < 0.001) and *dilatation* with Fat_{max} (*r* = 0.79, *P* < 0.001) and MFO (*r* = 0.60, *P* < 0.001). Fat_{min} 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˙O_{2max} with the index of physical activity (*r* = 0.49, *P* < 0.01). These correlations are presented in Figure 4.

## DISCUSSION

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 Fat_{max}, Fat_{min}, 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 Fat_{max} (44 ± 10.1% V˙O_{2max}) 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 Fat_{min}. 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 Fat_{max}, Fat_{min}, 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 [^{8}]), 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 Fat_{max}, Fat_{min}, 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 Fat_{max}) (^{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˙O_{2max} 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 Fat_{max}. 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 Fat_{min} 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 Fat_{max} 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 Fat_{max} and MFO determined with these models (Table 3). MRER, however, had wider limits of agreement and tended to overestimate Fat_{max}, 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, Fat_{max} 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 Fat_{max} 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 Fat_{max}. On the other hand, SIN has the additional advantage of also being able to determine Fat_{min} 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, Fat_{max} and Fat_{min} should also occur at a higher intensity. Fat_{min} 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, Fat_{max}, 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., Fat_{max} and Fat_{min} 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 Fat_{max}, Fat_{min}, 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˙O_{2max}). 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 Fat_{max} 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 Fat_{max} and between *translation* and Fat_{min} 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 Fat_{min} in trained people than those with lower V˙O_{2max} 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 Fat_{max}, Fat_{min}, and MFO. These relationships suggest that when Fat_{max} occurs at a higher intensity, the curve becomes more dilated, and the Fat_{max} 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˙O_{2max}, Fat_{max}, 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 Fat_{max} (range, 29%-68% V˙O_{2max}), MFO (0.14-0.80 g·min^{−1}), or V˙O_{2max} (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.

## CONCLUSIONS

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 Fat_{max} and MFO but in addition allows calculation of Fat_{min}. 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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#### APPENDIX

Development of the global equation, including by substitution the three variables (*dilatation*, *symmetry*, and *translation*), whose parameters are summed up in Table 1.

Basic equation:

with

and

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˙O_{2max}:

with the constant of intensity *K* = *π* / 100.

**Keywords:**

FAT_{max}; MAXIMAL FAT OXIDATION; INDIRECT CALORIMETRY; EXERCISE INTENSITY