Exponential modelling of heart rate recovery after a maximal exercise
Authors: Joseph Aneesh , Kanthakumar Praghalathan , Tharion Elizabeth 
Published: July 2023

Objectives:
Heart rate recovery (HRR) after exercise is clinically important as a predictor of mortality. In addition, HRR is an indicator of cardiac autonomic activity, since increased vagal activity and diminished sympathetic activity return the heart rate to resting conditions after exercise. The previous attempts to model HRR using polynomial, first-order and second-order modelling have produced mixed results. In this study, we hypothesised that the double-exponential fit would model the HRR more accurately than the single-exponential fit as it would capture the activity of both autonomic arms responsible for heart rate decay and investigated the outcome of these two models on the HRR data following a maximal exercise.

Materials and Methods:
Exponential curve fitting was done on a set of previously published data from our laboratory. The HRR data were acquired from 40 male participants (19–38 years) after a maximal treadmill exercise. The normalised HRR data from a 5-min time window from maximal heart rate were fitted using single and double-exponential curves, to obtain, respectively, the time constants Tau and, Tau 1 and Tau 2. The goodness-of-fit of the model was assessed with Chi-square values computed for each participant data set with both models. Considering that Chi-square of zero is a perfect fit, and therefore, smaller Chi-square values indicate a better fit than larger values, we computed the difference in the Chi-square values (Δχ2) between the models by subtracting the Chi-square value of the double-exponential fit from the Chi-square value of the single-exponential fit. This was based on the premise that if the calculated Δχ2 is positive, it would indicate a better fit with double-exponential than single-exponential decay model. The data are presented as mean ± standard deviation. Comparisons were made with Student’s t-test.

Results:
Data from four participants were excluded for technical reasons. The Tau of the single-exponential fit was 65.50 ± 12.13 s, while Tau 1 and Tau 2 of the double-exponential fit were 43.75 ± 18.96 s and 120.30 ± 91.32 s, respectively, the Tau 1 value being significantly lower than the Tau 2 value (P < 0.0001). Remarkably among the 36 participants, the difference in the Chi-square value was positive (127.2 ± 171.04) in 22 subjects and zero or marginally negative (−0.17 ± 0.31) in 14 subjects.

Conclusion:
Our results indicate that the double-exponential model fitted the HRR data better than the single-exponential model in almost two-thirds (61%) of our study population. In the remaining participants, the goodness-of-fit was nearly equivalent for both fits with no evidence of superior modelling with the single-exponential fit. Our data show that while the single-exponential fit is sufficient for modelling the HRR of 14 subjects, it was less efficient for fitting the data of most participants. In comparison, the double-exponential curve fit effectively modelled 100% of our study population. Given our findings, we conclude that the double-exponential model is more inclusive and better represented the HRR data of our study population than the single-exponential model.