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Revista Brasileira de Medicina do Esporte

Print version ISSN 1517-8692On-line version ISSN 1806-9940

Rev Bras Med Esporte vol.23 no.4 São Paulo July/Aug. 2017 

Original Article




Henry Humberto León-Ariza1 

Daniel Alfonso Botero-Rosas1 

Aura Catalina Zea-Robles2 

1University of La Sabana, School of Medicine, Program in Biosciences, Morphophysiology area, Chía, Colombia.

2Santo Tomas University, Department of Humanities, Bogotá, Colombia.



The maximum oxygen consumption (VO2max) is the gold standard in the cardiorespiratory endurance assessment.


This study aimed to develop a mathematical model that contains variables to determine the VO2max of sedentary people.


Twenty participants (10 men and 10 women) with a mean age of 19.8±1.77 years were included. For each participant, body composition (percentage of fat and muscle), heart rate variability (HRV) at rest (supine and standing), and VO2max were evaluated through an indirect test on a cycloergometer. A multivariate linear regression model was developed from the data obtained, and the model assumptions were verified.


Using the data obtained, including percentage of fat (F), percentage of muscle (M), percentage of power at very low frequency (VLF), α-value of the detrended fluctuation analysis (DFAα1), heart rate (HR) in the resting standing position, and age of the participants, a model was established for men, which was expressed as VO2max = 4.216 + (Age*0.153) + (F*0.110) - (M*0.053) - (VLF*0.649) - (DFAα1*2.441) - (HR*0.014), with R2 = 0.965 and standard error = 0.146 L/min. For women, the model was expressed as VO2max = 1.947 - (Age*0.047) + (F*0.024) + (M*0.054) + (VLF*1.949) - (DFAα1*0.424) - (HR*0.019), with R2 = 0.987 and standard error = 0.077 L/min.


The obtained model demonstrated the influence exerted by body composition, the autonomic nervous system, and age in the prediction of VO2max.

Keywords: body composition; autonomic nervous system; oxygen consumption; linear models



O consumo máximo de oxigênio (VO2máx) é o padrão-ouro na avaliação da resistência cardiorrespiratória. Objetivo: Este estudo visou desenvolver um modelo matemático com as variáveis usadas na determinação do VO2máx em indivíduos sedentários.


Vinte indivíduos (10 homens e 10 mulheres) com média de idade 19,8±1,77 anos foram incluídos. Para cada participante, foram avaliados composição corporal (percentual de gordura e de músculo), variabilidade da frequência cardíaca (VFC) em repouso (em decúbito dorsal e em pé) e o VO2máx, empregando-se o protocolo em cicloergômetro, método indireto. A partir dos dados obtidos, desenvolveu-se um modelo de regressão linear multivariado e os pressupostos do modelo foram verificados.


Usando os dados obtidos, incluindo percentual de gordura (G), porcentagem de músculos (M), porcentagem de energia em frequência muito baixa (FMB), valor de α da análise de flutuação sem tendências (DFAα1), frequência cardíaca (FC) em repouso na posição em pé e a idade dos participantes, estabeleceu-se um modelo para homens, expresso como: VO2máx = 4,216 + (Idade*0,153) + (G*0,110) - (M*0,053) - (FMB0,649*) - (DFAα1*2,441) - (FC*0,014) com R2 = 0,965 e erro padrão = 0,146 L/min. Para as mulheres, o modelo foi expresso como: VO2máx = 1,947 - (Idade*0,047) + (G*0,024) + (M*0,054) + (FMB*1,949) - (DFAα1*0,424) - (FC*0,019) com R2 = 0,987 e erro padrão de 0,077 L/min.


O modelo desenvolvido demonstrou a influência exercida pela composição corporal, pelo sistema nervoso autônomo e pela idade na predição do VO2máx.

Descritores: composição corporal; sistema nervoso autônomo; consumo de oxigênio; modelos lineares



El consumo máximo de oxígeno (VO2max) es el patrón áureo en la evaluación de la resistencia cardiorrespiratoria.


Este estudio tuvo como objetivo desarrollar un modelo matemático con las variables que participan en la determinación del VO2max en individuos sedentarios.


Veinte sujetos (10 hombres y 10 mujeres) con edad promedio de 19,8 ± 1,77 años se han incluido. Para cada participante, se evaluó la composición corporal (porcentaje de grasa y músculo), variabilidad de la frecuencia cardiaca (VFC) en reposo (decúbito supino y en pie) y VO2max mediante un test indirecto en cicloergómetro. A partir de los datos obtenidos se desarrolló un modelo de regresión lineal multivariado y se comprobaron los supuestos del modelo.


Usando los datos obtenidos, incluyendo porcentaje de grasa (G), porcentaje de músculos (M), porcentaje de energía en frecuencia muy baja (FMB), valor de α del análisis de fluctuación sin tendencias (DFAα1), frecuencia cardiaca (FC) en reposo en la posición en pie y la edad de los participantes, se estableció un modelo para hombres que se expresa como VO2max = 4,216 + (Edad*0,153) + (G*0,110) - (M*0,053) - (FMB*0,649) - (DFAα1*2,441) - (FC*0,014) con R2 = 0,965 y error típico = 0,146 L/min. Para las mujeres el modelo se expresó como VO2max = 1,947 - (Edad*0,047) + (G*0,024) + (M*0,054) + (FMB*1,949) - (DFAα1*0,424) - (FC*0,019) con R2 = 0,987 y error típico de 0,077 L/min.


El modelo desarrollado demostró la influencia ejercida por la composición corporal, el sistema nervioso autónomo y la edad en la predicción del VO2max.

Descriptores: composición corporal; sistema nervioso autónomo; consumo de oxígeno; modelos lineales


For the determination of cardiorespiratory endurance, the gold standard has been maximum oxygen consumption (VO2max), which corresponds to the maximum transport capacity and use of oxygen during high intensity exercise1. From a physiological point of view, VO2max is determined by central conditions associated with the transport of atmospheric oxygen to the muscles (lung function, cardiac output, and blood volume), while the use of oxygen is determined peripherally by conditions such as muscular capillarity, diffusion capacity, and mitochondrial activity2. In addition, there is a significant genetic component of VO2maX3.

From a central viewpoint, one of the main factors for the determination of VO2max is cardiac output, which corresponds to the volume of blood ejected by the heart in one minute. This value can be increased by as much as six times in the case of high-intensity exercise in well-trained athletes4. Cardiac output exhibits rapid variation due to the action of the autonomic nervous system, and it can be greatly increased in the ventricular cavities, which has been observed in athletes5.

To evaluate the action of the autonomic nervous system, Heart Rate Variability (HRV) has been used, which analyzes a time series (TS) of the variation in the duration of a beat with respect to the next. In addition, due to the use of digital signal processing systems, this TS can be analyzed by characteristics of its spectrum and by nonlinear methods6. Similarly, both HRV and heart rate (HR) at rest account for cardiovascular adaptations such as left ventricular hypertrophy and increased parasympathetic tone7.

From a peripheral point of view, conditions exist at the muscular level itself that are related to VO2max such as muscular capillarity and mitochondrial density8. This finding demonstrates the directly proportional relationship between VO2max and muscle mass and an inversely proportional relationship to percentage of fat9.

Given the significant role that VO2max plays in predicting cardio-respiratory fitness and its limitations for direct evaluation in physically inactive people, the aim of this study is to contribute to the development of a mathematical model that determines the variables that are directly involved in predicting the VO2max in sedentary people.


After approval of the project by the ethics committee of the University of La Sabana, 20 volunteer subjects (10 men and 10 women) with a mean age of 19.8 ±1.77 were selected for convenience. The inclusion criteria consisted of a sedentary lifestyle, no history of cardiovascular disease, no musculoskeletal pathological conditions, and not taking any medications that affect the response of the autonomic nervous system. All participants signed an informed consent form. In addition, a specialist with the capacity to evaluate obtained biological signals was consulted, and those subjects whose signals were excessively contaminated and were impossible to clean were excluded. Participants were evaluated in the morning between 7 and 9 a.m. and were asked about quality of sleep the night before and the use of caffeine, alcohol, or cigarettes during the previous 24 hours; inclusion decisions were postponed for those who did not meet the established criteria.

Acquisition variables

Body composition was evaluated anthropometrically, and the measures included body weight (kg), height (cm) and six skin folds (triceps, suprailiac, subscapular, abdominal, thigh, and leg). To calculate the fat percentage, the Yuhasz10 formula was used, and for muscle mass calculations, the Doupe et al.11 formula was used. The perimeters, diameters, and folds were obtained by a standardized protocol applied by a certified expert of the International Society for the Advancement of Kinanthropometry (ISAK). In addition, waist circumference was measured midway between the seventh rib and the iliac spine12. The instruments used included a Harpenden® caliper, a Stanley® metric tape, a Berfer® pachymeter, a Faga® stadiometer, and an Omron® scale. HR was measured in subjects in the physiology laboratory of the Biomedical Campus of the University of La Sabana at an average temperature of 20°C without any influence from drafts, noise or lights, which can alter the response of the autonomic nervous system. For these measurements, a Polar RS800CX heart rate monitor was used for 10 minutes (five minutes in the supine position and five minutes in the standing position), which is enough time for signal analysis13. After warming up for five minutes, and after having established 75% of the maximum theoretical HR by the Tanaka et al.14 formula the subjects performed an incremental test on the cycloergometer until they reached the established HR. At that point, the power was recorded, and this value was used to estimate VO2max with the Astrand nomogram of the cycloergometer15. The general characteristics of the participants are described in Table 1.

Table 1 General characteristics of body composition and oxygen consumption.  

Men Women Difference
Age 19.50 1.08 20.10 2.28 p = 0.47
Weight (kg) 68.22 8.25 57.27 8.08 p < 0.01*
Height (cm) 177.90 8.62 161.10 3.63 p < 0.00*
% Fat 13.17 4.19 31.67 10.00 p < 0.00*
% Muscle 51.28 3.67 37.30 6.29 p < 0.00*
Waist circumference (cm) 83.70 7.87 78.74 6.18 p = 0.14
Power (W) 112.01 17.97 71.10 15.24 p < 0.00*
HR Test (beats/min) 147.85 4.57 145.30 4.30 p = 0.22
% max HR 76.07 2.24 74.94 2.04 p = 0.25
VO2max (L/m) 2.57 0.45 2.23 0.39 p = 0.08
VO2max (ml/kg/min) 37.88 6.09 39.70 9.27 p = 0.61

SD: Standard Deviation, HR: Heart rate, VO2max: Maximum oxygen consumption, *Statistically significant difference.

Signal analysis

Analysis of HRV was performed using Kubios HRV software (University of Kuopio)16. The pre-processing phase of the HR signal consisted of removal of artifacts (RR interval variations greater than 0.45 sec with respect to the average) and filtering of the signal using a Smoothness priors high-pass filter with a Lambda of 500 and a cutoff frequency of 0.035 Hz17. From the noise-free tachogram obtained, the ST segment of the HRV was analyzed in the time domain, and the relevant parameters were calculated (average HR, standard deviation of the heart rate (STDHR), root mean square of the differences between successive RR intervals (RMSSD), and the number of successive RRs that differ by more than 50 ms divided by the total RR intervals (pNN50).

For analysis of the frequency domain, a Fast Fourier transform (FFT) and autoregressive (AR) analysis were used. To obtain an ST segment with equidistant samples, piecewise cubic spline interpolation was used at a rate of 4 Hz; then an FFT was applied to obtain the power spectral density (PSD) and power parameters (RMS and total percentage values) in the very low frequency (VLF, 0-0.04 Hz), low frequency (LF, 0.04 - 0.15 Hz), and high frequency (HF, 0.15 - 0.4 Hz) ranges. Subsequently, a 16th order AR model was implemented to obtain these same values in the VLF, LF, and HF ranges. For the nonlinear analysis, a Poincaré diagram was used to establish the SD1 and SD2 parameters, and detrended fluctuation analysis (DFA) was used to establish α1 and α2. The overall results of the HRV analysis are shown in Table 2.

Table 2 Variables used for analysis of Heart Rate Variability. Values in parentheses correspond to the standard deviation. 

Men Women Difference
Time domain HR Supine 69.2 ±(8.9) 76.8 ±(8.4) p = 0.04*
Standing 89.4 ±(9.0) 92.7 ±(5.6) p = 0.40
STDHR Supine 6.4 ±(1.5) 6.8 ±(1.9) p = 1.00
Standing 7.2 ±(1.9) 7.2 ±(13) p = 0.63
RMSSD Supine 89.5 ±(54.2) 82.9 ±(25.5) p = 0.18
Standing 21.9 ±(11.6) 22.8 ±(7.7) p = 0.82
pNN50 Supine 48.6 ±(22.5) 35.6 ±(18.1) p = 0.17
Standing 4.9 ±(4.9) 5.0 ±(5.6) p = 0.96
Frequency Domain %VLF Supine 29.6 ±(15.9) 32.5 ±(14.5) p = 0.68
Standing 50.0 ±(21.8) 46.6 ±(18.3) p = 0.71
%LF Supine 23.5 ±(7.0) 26.6 ±(8.9) p = 0.41
Standing 40.7 ±(18.1) 42.2 ±(16.8) p = 0.85
%HF Supine 46.7 ±(17.9) 40.7 ±(18.6) p = 0.47
Standing 9.3 ±(6.1) 11.2 ±(4.2) p = 0.42
LF/HF Supine 0.67 ±(0.59) 0.94 ±(0.82) p = 0.41
Standing 6.0 ±(3.6) 4.1 ±(2.0) p = 0.16
Nonlinear analysis SD1 Supine 63.4 ±(38.4) 44.6 ±(18.0) p = 0.88
Standing 16.6 ±(7.4) 16.1 ±(5.5) p = 0.19
SD2 Supine 100.2 ±(33.7) 88.2 ±(22.9) p = 0.33
Standing 78.8 ±(23.3) 70.6 ±(11.7) p = 0.38
DFAα1 Supine 0.78 ±(0.25) 0.78 ±(0.22) p = 0.31
Standing 1.56 ±(0.18) 1.49 ±(0.13) p = 0.34
DFAα2 Supine 0.89 ±(0.23) 0.23 ±(0.17) p = 0.45
Standing 0.90 ±(0.21) 0.90 ±(0.19) p = 0.94

HR: Heart Rate; STDHR: Standard deviation of heart rate values; RMSSD: Root mean square of the differences between successive RR intervals; pNN50: Number of successive RR intervals that differ by more than 50 ms divided by total RR intervals; VLF: Very Low Frequency; LF: Low Frequency; HF: High Frequency; SD1 and SD2: Standard deviations of the Poincaré graph; DFA: Detrended fluctuation analysis.

Statistical analysis

Statistical analyses were performed with IBM SPSS Statistics 21. Initially, the means of the general parameters (age, weight, height, body composition, VO2max, and HR) of men and women were statistically compared with a two-tailed t-test for unpaired data. The level of statistical significance was set at a p value ≤ 0.05. Next, the Pearson correlation coefficients (r) were calculated between VO2max and each of the variables including body composition, age, and HRV. An association was established according to the r value (no correlation r = 0.0, weak r = 0.1, medium r = 0.5, significant r = 0.75, very strong r = 0.9, and perfect r = 1) for further ordering using the association value18. From this ordering, the best predictors of the dependent variable (VO2max) in men and women were recorded and used to implement a multivariate linear regression model, according to equation 1 below. In the development of this model, simple linear regression and stepwise regressions were performed to discriminate each of the predictors and eliminate collinearity.

Equation 1:

Where VO2max is the value estimated by the Astrand nomogram, X1, X2… Xk are the independent variables, B0 is the initial condition, β1, β2... βk correspond to the coefficients of each of the variables, and ε is the residual or unpredictable value.

To determine the validity of the model, compliance with the assumptions was evaluated independently, including Homoscedasticity, (standardized residual error values vs. standardized predictions), Normality (behavior of probabilities), Non-collinearity (inflation factors of the variance), Linearity (correlations between independent variables and the dependent variable), and Independence (Durbin Watson model)19.


The independent variables that showed a stronger association with the dependent variable (VO2max) for men were age, HR at rest in a supine position, and DFAα1 in the nonlinear analysis. For women, the variables included percentage of muscle mass, percentage of fat, percentage of VLF in a supine position analyzed by the AR method, and HR value at rest with patient in a standing position (Table 3).

Table 3 Correlation between the dependent variable (maximum oxygen consumption) and the independent variables used to develop the model.  

Age % Fat % Muscle %VLF DFAα1 Supine HR
Men n = 10 Correlation 0.423 0.162 0.076 -0.160 -0.362 -0.527
p - value 0.112 0.328 0.417 0.330 0.152 0.059
Women n = 10 Correlation 0.212 -0.315 0.668 0.641 0.466 -0.723
p - value 0.278 0.188 0.017* 0.023* 0.087 0.009*
All n = 20 Correlation 0.202 -0.465 0.609 0.206 0.136 -0.682
p - value 0.394 0.039* 0.004* 0.384 0.569 0.009*

*Statistically significant difference.

The model output showed an R2 of 0.965 for men with a standard error of estimation of VO2max of 0.146 L/min, while for women, the R2 was 0.987 with a standard error of estimation of VO2max of 0.077 L/min. The ANOVA for both men and women showed a linear relationship between the dependent variable and the independent variables, which was significant in men (p = 0.027) and women (p = 0.007). The Durbin Watson test for independence produced a value of 3.209 for men and 2.036 for women. The scatterplots of standardized predictions and standardized residuals (Figure 1A and 1B) did not show any pattern of association, which is consistent with homoscedasticity, while the normal probability graph (Figure 1C and 1D) showed a trend of aligned residues along the diagonal of the graph associated with normality.

Figura 1 Figure A and B represent scatterplots of the residuals, while Figures C and D represent the probability of the residuals.  

Inflation factors of the variance in the model for men ranged from 1.494 to 5.676 and were higher for the DFAα1 variable. In the case of women, the values ranged from 1.493 to 3.145, and the percentage of muscle mass variable was higher.

Finally, the equation obtained for the prediction of VO2max in men was:

Equation 2:

The equation for women was:

Equation 3:

The results for the coefficients, standard error, significance, and inflation factors of the variance for men and women are shown in Table 4.

Table 4 Characteristics of the constants developed for the model.  

Nonstandardized coefficients Standardized coefficients p - Value Inflation Factor of the Variance
B Standard error. Beta
Men constant 4.22 2.20 NA 0.15 NA
Age 0.15 0.06 0.37 0.09* 1.80
% Fat 0.11 0.02 1.02 0.01* 3.32
% Muscle -0.05 0.03 -0.43 0.16 4.61
%VLF -0.65 0.37 -0.23 0.18 1.49
DFAα1 -2.44 0.46 -1.36 0.01* 5.68
Standing HR -0.01 0.01 -0.28 0.19 2.41
Women (Constant) 1.95 0.72 NA 0.07 NA
Age -0.05 0.01 -0.28 0.04* 1.49
% Fat 0.02 0.00 0.63 0.01* 2.54
% Muscle 0.05 0.01 0.88 0.01* 3.14
%VLF 1.95 0.28 0.73 0.01* 2.41
DFAα1 -0.42 0.18 -0.26 0.09 2.54
Standing HR -0.02 0.00 -0.43 0.02* 1.86

NA: Not applicable. *statistically significant data (p < 0.05).

The results of the predicted values and actual values for each of the cases in men and women are shown in Table 5 and Figure 2.

Figure 2 Correlation graph for data obtained in men and women through the Astrand test and calculated oxygen consumption from equations 2 and 3.  

Table 5 Behavior of actual and predicted oxygen consumption values and their differences.  

Number of cases Actual VO2max Predicted VO2max Difference
Men 1 2.44 2.49 -0.05
2 3.35 3.28 0.07
3 2.44 2.60 -0.16
4 2.47 2.42 0.05
5 2.39 2.32 0.07
6 3.06 3.15 -0.09
7 3.11 3.02 0.09
8 1.95 1.99 -0.04
9 2.12 2.05 0.07
10 2.40 2.39 0.01
Women 1 2.55 2.57 -0.02
2 1.78 1.82 -0.04
3 1.65 1.70 -0.05
4 2.63 2.61 0.02
5 2.59 2.60 -0.01
6 2.46 2.43 0.03
7 2.38 2.44 -0.06
8 2.51 2.48 0.03
9 1.84 1.76 0.08
10 1.91 1.91 0.00


The fundamental hypothesis of the study is based on the ability to predict VO2max from the variables body composition and HRV. The results obtained showed an inverse correlation between VO2max and adipose tissue (r = -0.315) and a directly proportional relationship to muscle mass in women (r = 0.668). Similar results were found in previous studies, which demonstrated that adipose tissue exhibited an inverse relationship with the measured VO2max (r = -0.40)9 and, in turn, was proportional to muscle mass (r = 0.68)20; the latter was analyzed in patients with heart failure. Additional studies presented fat free mass (r = 0.87) as a fundamental determinant of VO2max21, which was not found in our case. Other widely used variables such as body weight and body mass index showed poor prediction of VO2max, which was also shown in other studies22.

Physiologically, the relationships found are valid, as skeletal muscle is considered the most active tissue from a metabolic point of view, in addition to being highly vascularized and rich in mitochondria23, contrary to adipose tissue, which has poor metabolic activity.

Athletes and individuals highly trained in predominantly aerobic exercise have a decreased HR at rest as a result of both neural and anatomical adaptations24. For this reason, the HR at rest has been proposed as a predictor of VO2max25. In this study, we showed that for men (r = -0.527) and women (r = -0.723), there is an inverse relationship between HR at rest in a standing position and VO2max. With respect to HRV and prediction of VO2max, some authors have stated that HRV can only explain up to 20.1% of the behavior of VO2max26; however, our results demonstrated that variables such as percentage of power at VLF or DFA fluctuation analysis are highly associated with the prediction of VO2max. These findings present new research opportunities, as the power at VLF is related to changes in the humoral system27, while DFA fluctuations have been considered useful when analyzing the intrinsic behavior of a system28. Finally, the age of the participants influenced the final prediction model of VO2max, with correlations of r = 0.423 in men and r = 0.212 in women, which has been considered in the development of other prediction models29.

The Durbin Watson test for independence of variables showed, in the case of women, independence of the residuals, while in men, the residuals were autocorrelated. These finding are consistent because HRV is related to physical activity and body composition30. Changes in body composition can lead to changes in the autonomic response; thus, in obese patients, the percentage of fat is associated with a decrease in the behavior of LF (r = -0.43) and a lower RMSSD (r = -0.35)31. Similar results have been found in other studies in which adipose tissue has been shown to be related to the LF/HF (r = 0.56)32.

Although many studies have shown changes in HRV in athletes and physically active people33, few studies have linked these findings to the composition of muscle mass. Some authors have found that HRV in overweight patients is lower when it is associated with reduced muscle mass34, while other authors have shown that anthropometric behavior contributes to changes in the HRV35.

A small sample of individuals was used because this was considered a pilot study; the sample should be expanded to validate the equations.


The VO2max responds physiologically to multiple variables; in this study, body composition (percentage of fat and percentage of muscle) was shown to be important for establishing VO2max. Additionally, the behavior of the autonomic nervous system also contributes to the understanding of the physiological adaptations that accompany higher values of VO2max. Our model showed that other important relationships exist between variables, such as the role of body composition in HRV. The joint analysis of body composition (percent of muscle mass and percent of fat), HR at rest, HRV (percent VLF and α coefficient of DFA), and age can explain the high percentage of VO2max both in men and women.


We acknowledge the physiology laboratory of the biomedical campus of the University of La Sabana, which provided the equipment and infrastructure for the development of the research and the research participants.

All authors declare no potential conflict of interest related to this article.


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Received: July 16, 2015; Accepted: February 08, 2017

Correspondence: Daniel Alfonso Botero-Rosas Morphophysiology area, School of Medicine, University of La Sabana. Campus del Puente del Común, Km. 7, Autopista Norte de Bogotá, Edificio F, Segundo piso. 53753, Chia, Cundinamarca, Colombia.

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