Dynamical System Modeling of Self-Regulated Systems Undergoing Multiple Excitations: First Order Differential Equation Approach

Decrease of heart rate variability during exercise: An index of cardiorespiratory fitness

The present study proposes to measure and quantify the heart rate variability (HRV) changes during effort as a function of the heart rate and to test the capacity of the produced indices to predict cardiorespiratory fitness measures. Therefore, the beat-to-beat cardiac time interval series of 18 adolescent athletes (15.2 ± 2.0 years) measured during maximal graded effort test were detrended using a dynamical first-order differential equation model. HRV was then calculated as the standard deviation of the detrended RR intervals (SDRR) within successive windows of one minute. The variation of this measure of HRV during exercise is properly fitted by an exponential decrease of the heart rate: the SDRR is divided by 2 every increase of heart rate of 20 beats/min. The HR increase necessary to divide by 2 the HRV is linearly inversely correlated with the maximum oxygen consumption (r = -0.60, p = 0.006), the maximal aerobic power (r = -0.62, p = 0.006), and, to a lesser extent, to the power at the ventilatory thresholds (r = -0.53, p = 0.02 and r = -0.47, p = 0.05 for the first and second threshold). It indicates that the decrease of the HRV when the heart rate increases is faster among athletes with better fitness. This analysis, based only on cardiac measurements, provides a promising tool for the study of cardiac measurements generated by portable devices.

The complex relationship between effort and heart rate: a hint from dynamic analysis

OBJECTIVE

Dynamic analysis can be used to study the changes of self-regulated biological processes driven by external stimuli. Recently, the changes of heart rate during effort tests has successfully been adjusted using a simple first-order differential equation model driven by body power expenditure. Although this approach produces valid estimates and yields pertinent indices for the analysis of such measurements, it suffers from an inability to model the saturation of the heart-rate increase at high power expenditures and the change of heart-rate equilibrium following effort.

APPROACH

We propose a new analysis allowing the estimation of changes of the heart rate in response to effort (gain) as a function of the power expenditure value.

MAIN RESULTS

When applied to the measured heart rates of 30 amateur athletes performing a maximum graded-effort treadmill test, the proposed model was able to predict 99% of the heart rate change measured during exercise. The estimated gains decreased with a power increase above the first ventilatory threshold. This trend was stronger above the second ventilatory threshold and was strongly correlated with the maximum oxygen consumption.

SIGNIFICANCE

The proposed approach yields a highly precise model of heart rate dynamics during variable effort that reflects the changes of metabolic energy systems at play during exercise.

Validity of dynamical analysis to characterize heart rate and oxygen consumption during effort tests

Performance is usually assessed by simple indices stemming from cardiac and respiratory data measured during graded exercise test. The goal of this study is to characterize the indices produced by a dynamical analysis of HR and VO₂ for different effort test protocols, and to estimate the construct validity of these new dynamical indices by testing their links with their standard counterparts. Therefore, two groups of 32 and 14 athletes from two different cohorts performed two different graded exercise testing before and after a period of training or deconditioning. Heart rate (HR) and oxygen consumption (VO₂) were measured. The new dynamical indices were the value without effort, the characteristic time and the amplitude (gain) of the HR and VO₂ response to the effort. The gain of HR was moderately to strongly associated with other performance indices, while the gain for VO₂ increased with training and decreased with deconditioning with an effect size slightly higher than VO₂ max. Dynamical analysis performed on the first 2/3 of the effort tests showed similar patterns than the analysis of the entire effort tests, which could be useful to assess individuals who cannot perform full effort tests. In conclusion, the dynamical analysis of HR and VO₂ obtained during effort test, especially through the estimation of the gain, provides a good characterization of physical performance, robust to less stringent effort test conditions.