Identification of heart rate dynamics during treadmill and cycle ergometer exercise: the role of model zeros and dead time

Kenneth J. Hunt; Hanjie Wang

doi:10.12688/f1000research.153397.2

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Brief Report

Revised

Identification of heart rate dynamics during treadmill and cycle ergometer exercise: the role of model zeros and dead time

[version 2; peer review: 2 approved]

Kenneth J. Hunt ¹, Hanjie Wang¹

PUBLISHED 01 Nov 2024

Author details Author details

¹ rehaLab - the Laboratory for Rehabilitation Engineering ,Institute for Human Centred Engineering HuCE School of Engineering and Computer Science, Bern University of Applied Sciences,, Biel/Bienne, 2501, Switzerland

Kenneth J. Hunt
Roles: Conceptualization, Data Curation, Formal Analysis, Funding Acquisition, Investigation, Methodology, Project Administration, Resources, Software, Supervision, Validation, Visualization, Writing – Original Draft Preparation, Writing – Review & Editing

Hanjie Wang
Roles: Conceptualization, Data Curation, Formal Analysis, Investigation, Methodology, Software, Visualization, Writing – Review & Editing

OPEN PEER REVIEW

REVIEWER STATUS

Abstract

Background

The response of heart rate to changes in exercise intensity is comprised of several dynamic modes with differing magnitudes and temporal characteristics. Investigations of empirical identification of dynamic models of heart rate showed that second-order models gave substantially and significantly better model fidelity compared to the first order case. In the present work, we aimed to reanalyse data from previous studies to more closely consider the effect of including a zero and a pure delay in the model.

Methods

This is a retrospective analysis of 22 treadmill (TM) and 54 cycle ergometer (CE) data sets from a total of 38 healthy participants. A linear, time-invariant plant model structure with up to two poles, a zero and a dead time is considered. Empirical estimation of the free parameters was performed using least-squares optimisation. The primary outcome measure is model fit, which is a normalised root-mean-square model error.

Results

A model comprising parallel connection of two first-order transfer functions, one with a dead time and one without, was found to give the highest fit (56.7 % for TM, 54.3 % for CE), whereby the non-delayed component appeared to merely capture initial transients in the data and the part with dead time likely represented the true dynamic response of heart rate to the excitation. In comparison, a simple first-order model without dead time gave substantially lower fit than the parallel model (50.2 % for TM, 47.9 % for CE).

Conclusions

This preliminary analysis points to a linear first-order system with dead time as being an appropriate model for heart rate response to exercise using treadmill and cycle ergometer modalities. In order to avoid biased estimates, it is vitally important that, prior to parameter estimation and validation, careful attention is paid to data preprocessing in order to eliminate transients and trends.

Keywords

heart rate dynamics, system identification, treadmill exercise, cycle ergometer exercise

Corresponding author: Kenneth J. Hunt

Competing interests: No competing interests were disclosed.

Grant information: This work was supported by the Swiss National Science Foundation (Principal Investigator KH, Grant Ref. 320030-185351).
The funders had no role in study design, data collection and analysis, decision to publish, or preparation of the manuscript.

Copyright: © 2024 Hunt KJ and Wang H. This is an open access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

How to cite: Hunt KJ and Wang H. Identification of heart rate dynamics during treadmill and cycle ergometer exercise: the role of model zeros and dead time [version 2; peer review: 2 approved]. F1000Research 2024, 13:894 (https://doi.org/10.12688/f1000research.153397.2) First published: 06 Aug 2024, 13:894 (https://doi.org/10.12688/f1000research.153397.1) Latest published: 01 Nov 2024, 13:894 (https://doi.org/10.12688/f1000research.153397.2)

Revised Amendments from Version 1

The article has been revised in response to reviewer comments and suggestions. The major differences are:
(i) We provide more information regarding the clinical and physiological perspectives, and practical applications;
(ii) We proved more details on the study population, such that readers do not need to refer to the original, background publications;
(iii) Several futher details have been provided including participants' training status, average maximal heart rate etc.

See the authors' detailed response to the review by Przemysław Seweryn Kasiak
See the authors' detailed response to the review by Lauren Miutz

1. Introduction

Heart rate is an easy-to-measure physiological variable that can be used to characterise the intensity of exercise, both quantitatively and qualitatively using categories such as light, moderate and vigorous.¹ The ability to regulate heart rate during exercise using feedback control would therefore allow accurate prescription of training regimes in both clinical and non-clinical settings. Since feedback controllers require a model of the dynamic response of heart rate during exercise, it is important to first consider the fidelity of different model structures.

It has been proposed that heart rate response to changes in exercise intensity comprises three main phases²^,³: an immediate, relatively small and fast Phase I; a slower, delayed and larger Phase II; and, if the exercise intensity exceeds the anaerobic threshold, a later and very slow Phase III drift.

This observation led to the investigation of empirical identification of dynamic models of heart rate using first- and second-order transfer functions for treadmill (TM)⁴ and cycle ergometer (CE)⁵ exercise. The second-order case was anticipated to capture Phase I and II response modes; but, since the models were intended to be used for analytical design of feedback controllers for heart rate, where integral action would cancel very slow drift, Phase III was not considered; in addition, to simplify feedback design, dead time was neglected.

Thus, in both of the preceding investigations of heart rate dynamics⁴^,⁵ the dynamic response of heart rate was modelled as nominal linear transfer functions $P_{o} (s)$ of first (P1) and second (P2) order:

(1)

P 1 : P_{o} (s) = \frac{k}{τs + 1}, P 2 : P_{o} (s) = \frac{k}{(τ_{1} s + 1) (τ_{2} s + 1)}

where

k

is a steady-state gain and the

τ

’s are time constants.

It was found that second-order models gave substantially and significantly better model fidelity compared to the first order case (TM,⁴ CE⁵) and that feedback control of heart rate was more accurate when based on second-order models (TM,⁶ CE⁵).

But the classical Phase I - Phase II model of heart rate response²^,³ comprises the parallel connection of two first-order models, i.e. the sum of a first-order transfer function of the form $P 1$ above and a P1 with pure delay. Theoretically, this would lead to a second-order model with two poles, but also—when dead time is neglected—with a single zero. The effect of this (theoretical) zero was not reported in the previous studies⁴^,⁵ as it was found not to lead to any difference in empirical model fit, presumably due to overfitting. Furthermore, since the classical sources propose the addition of a dead time to one of the modes to capture the slightly later onset of the Phase II component, the inclusion of a pure delay warrants further attention.

The present work therefore aimed to perform a retrospective analysis of the previous investigations of heart rate dynamics during treadmill⁴ and cycle ergometer⁵ exercise to more closely consider the effect of including a zero and a dead time in the model. The respective datasets are available on the OLOS repository.⁷^,⁸

2. Methods

2.1 Data collection

Full details of experimental procedures employed for data collection in the preceding treadmill and cycle ergometer investigations can be found in the respective publications.⁴^,⁵ Essential elements of the protocols are summarised in this Brief Report.

For both exercise modalities, healthy, able-bodied participants exercised at moderate-to-vigorous intensity: in the treadmill analysis⁴ there were 11 participants (8 male, 3 female; overall mean age 32.5 years, mean body mass 75.5 kg, mean height 1.79 m); for the cycle ergometer⁵ there were 27 participants (20 male, 7 female; overall mean age 30.8 years, mean body mass 76.3 kg, mean height 1.79 m). Participants were required to be regular exercisers (30-min bouts, 3 times per week), non-smokers, and to be free of injury and illness.

A similar pseudo-random binary sequence (PRBS) input signal was employed in both cases to excite relevant modes of heart rate response dynamics. All participants performed two identical open-loop identification tests to facilitate counterbalanced cross-validation of model parameter estimates: consequently, there were 22 TM data sets and 54 CE data sets. All of these data sets were included in the present retrospective analysis.

To aid the following Discussion (Sec. 4), all existing heart rate measurements that were included in the parameter estimation and validation procedures are illustrated (Figure 1).

Figure 1. Heart rate measurements.

In each plot, thin lines are the individual measurements (22 for TM, 54 for CE); the thick red lines are averages of the individual measurements. Data are plotted as deviations $△$ HR around mean heart rate levels. (a) Treadmill, (b) Cycle Ergometer.

2.2 Model structures

In the present work, we consider a linear, time-invariant (LTI) plant model structure with up to two poles, a zero and a dead time, that maps an input signal u to the output y, namely

(2)

P_{o} (s) = \frac{k (T_{z} s + 1)}{(τ_{1} s + 1) (τ_{2} s + 1)} e^{- T_{d} s} : u \mapsto y

where

k

is the steady-state gain,

τ_{1}

and

τ_{2}

are time constants (corresponding to real poles at

s = - 1 / τ_{1}

and

s = - 1 / τ_{2}

T_{z}

admits a zero

(s = - 1 / T_{z})

, and

T_{d}

is a pure delay. The general model Eq. (2) can be constrained by choice of the

τ

’s,

T_{z}

and

T_{d}

to several simpler structures as summarised in tabular form (Table 1): in total, the seven model structures listed were considered in the present analysis; this includes one formed by the parallel connection of two first-order transfer functions, one with a dead time and one without, viz.

P 1 ∥ P 1 D

Table 1. Model structures.

Model	$P_{o} (s)$	constraints (cf. Eq. (2))
P1	$\frac{k}{τ s + 1}$	$T_{z} = 0, T_{d} = 0, τ_{1} = τ, τ_{2} = 0$
P1D	$\frac{k}{τ s + 1} e^{- T_{d} s}$	$T_{z} = 0, τ_{1} = τ, τ_{2} = 0$
P2	$\frac{k}{(τ_{1} s + 1) (τ_{2} s + 1)}$	$T_{z} = 0, T_{d} = 0$
P2D	$\frac{k}{(τ_{1} s + 1) (τ_{2} s + 1)} e^{- T_{d} s}$	$T_{z} = 0$
P2Z	$\frac{k (T_{z} s + 1)}{(τ_{1} s + 1) (τ_{2} s + 1)}$	$T_{d} = 0$
P2ZD	$\frac{k (T_{z} s + 1)}{(τ_{1} s + 1) (τ_{2} s + 1)} e^{- T_{d} s}$	none
$P 1 ∥ P 1 D$	$\frac{k_{p 1}}{τ_{p 1} s + 1} + \frac{k_{p 2}}{τ_{p 2} s + 1} e^{- T_{d} s}$	N/A

The generic plant output signal $y$ corresponds to heart rate [beats/min, bpm] while the input $u$ depends on the exercise modality: for the treadmill, it is speed [m/s]; for the cycle ergometer, it is work rate [W]. As noted above, the input for both modalities took the form of a PRBS signal.

2.3 Parameter estimation and outcome measure

Empirical parameter estimation was performed using the Matlab System Identification Toolbox (The MathWorks, Inc., USA), wherefore, in the table (Table 1), we have adopted model names corresponding to the terminology used in the toolbox. In general, models of the form Eq. (2) are referred to in the toolbox as “process models”.

Estimation of the free model parameters— $k$ , the $τ$ ’s, $T_{z}$ and $T_{d}$ in Eq. (2), constrained for the different model structures as indicated in Table 1—was done with the Matlab procest function using least-squares optimisation with regularly sampled time-domain data.⁹ To focus the search algorithm, model parameters were constrained to lie in physiologically plausible ranges. As in our previous work⁴^,⁵ separate models were identified for each individual data set and counterbalanced cross-validation was employed by pairing the two measurements for each participant.

The primary outcome measure is model fit, which is a normalised root-mean-square model error (NRMSE):

(3)

fit = (1 - \sqrt{\frac{\sum_{i = 1}^{N} {(y (i) - y_{sim} (i))}^{2}}{\sum_{i = 1}^{N} {(y (i) - \bar{y})}^{2}}})

where

\bar{y}

is the mean heart rate and

y_{sim}

is the heart rate that was simulated using the estimated models. The summations range over the evaluation period up to the number of discrete data points included,

N

. A sample period of 5 s was used. Model fit was computed using the Matlab compare function.

3. Results

Goodness-of-fit values for the seven model structures and two exercise modalities are summarised in Table 2; the estimated model parameters are also tabulated (Table 3). Average maximal heart rate for the treadmill was 158.4 bpm; for the cycle ergometer it was 140.2 bpm (this is in line with our setting the mean target heart rate for the CE to be 20 bpm lower than for the TM in order to achieve a similar level of perceived exertion).¹⁰

Table 2. Mean model fit (normalised RMSE, Eq. (3), [%]).

Modality	P1	P1D	P2	P2D	P2Z	P2ZD	$P 1 ∥ P 1 D$
TM	50.2	54.0	54.5	54.5	53.9	55.2	56.7
CE	47.9	51.9	51.0	52.1	50.4	52.8	54.3

Table 3. Model parameters for treadmill (TM) and cycle ergometer (CE).

Model	Modality	$k / (bpm / [u])$	$τ_{1} / s$	$τ_{2} / s$	$T_{z} / s$	$T_{d} / s$
P1	TM	28.6	70.6	-	-	-
P1	CE	0.46	68.8	-	-	-
P1D	TM	25.0	47.7	-	-	13.1
P1D	CE	0.40	45.9	-	-	13.8
P2	TM	24.7	18.6	37.8	-	-
P2	CE	0.39	19.6	37.7	-	-
P2D	TM	23.9	13.7	37.8	-	5.4
P2D	CE	0.38	15.5	33.2	-	6.9
P2Z	TM	24.1	24.9	40.2	7.3	-
P2Z	CE	0.38	31.2	46.2	18.7	-
P2ZD	TM	23.7	33.2	50.6	38.4	11.1
P2ZD	CE	0.39	33.5	59.4	50.0	12.5
$P 1 ∥ (P 1 D)$ *	TM	7.0	141.5	-	-	-
$P 1 ∥ (P 1 D)$ *	CE	0.09	180.7	-	-	-
$(P 1) ∥ P 1 D$ *	TM	20.2	34.3	-	-	17.9
$(P 1) ∥ P 1 D$ *	CE	0.35	37.9	-	-	17.1

* For the $P 1 ∥ P 1 D$ model structure, parameters are shown separately for the P1 (second-bottom row) and P1D (bottom row) components: $k$ and $τ_{1}$ correspond respectively to $k_{p 1}$ and $τ_{p 1}$ , or $k_{p 2}$ and $τ_{p 2}$ , in the bottom row of Table 1.

4. Discussion

Goodness-of-fit outcomes for the treadmill and cycle ergometer followed a similar pattern. There was a substantial improvement in fit for P1D vs. P1, indicating the clear presence of dead time in heart rate response; $T_{d}$ for P1D was similar for TM and CE at 13.1 s and 13.8 s, respectively (Table 3).

Model fit for P2, P2D and P2Z was similar to P1D, while P2ZD showed a further slight improvement. It has to be remarked, however, that estimated $T_{z}$ values for individual models varied widely on the range -15 s to 100 s, thus displaying in part negative-phase behaviour (i.e. $T_{z} < 0$ ). Furthermore, fit for P2Z was slightly lower than for P1D, P2 and P2D. Taken together, these observations point to a degree of overfitting when a plant zero is included.

Having excluded further consideration of models with a zero, we note a further substantial increase in fit for the parallel $P 1 ∥ P 1 D$ model structure when compared to P1D, P2 and P2D. A critical observation in this regard is that the P1 parameters in the $P 1 ∥ P 1 D$ structure displayed very small gains and very large time constants when compared to the parallel-models’ P1D parameters (Table 3): for the TM, the gains were 7.0 bpm/(m/s) and 20.2 bpm/(m/s), (P1 vs. P1D), and the time constants 141.5 s vs. 34.3 s; for the CE, gains were 0.09 bpm/W vs. 0.35 bpm/W and time constants 180.7 s vs. 37.9 s.

A likely explanation for this apparent anomaly can be gleaned by perusal of the heart rate measurements (Figure 1). It can be seen that there is a small yet clearly discernible drift in heart rate during the first few minutes of the responses, with the duration of drift in line with the observed P1 time constants 141.5 s (TM, Figure 1a) and 180.7 s (CE, Figure 1b). It is therefore plausible that the P1 part of the $P 1 ∥ P 1 D$ model merely reflects the initial transient, while the P1D part represents the true dynamic response of heart rate to the excitation. Care should therefore be taken in future investigations to exclude initial transients and slow trends prior to parameter estimation and validation.

The gains and time constants are seen to be somewhat lower for the P1D part of the $P 1 ∥ P 1 D$ model than for the P1D-only model (gains 20.2 bpm/(m/s) vs. 25.0 bpm/(m/s) for TM, 0.35 bpm/W vs. 0.40 bpm/W for CE; time constants 34.3 s vs. 47.7 s for TM, 37.9 s vs. 45.9 s for CE; Table 3), and the dead times somewhat higher (17.9 s vs. 13.1 s for TM, 17.1 s vs. 13.8 s for CE). These differences are likely due to model bias introduced in the P1D-only model as a consequence of the initial drift in heart rate, as discussed above.

As noted in previous reports⁴^,⁵ second-order models of the form P2 gave substantially and significantly better fidelity than first-order models P1 (cf. Table 2). However, the identification here of a substantial dead time, coupled with the observed superiority of the P1D part of the parallel $P 1 ∥ P 1 D$ model (following elimination of heart rate drift), suggests that the second time constant in the P2 model may simply have partially absorbed the neglected time delay rather than having modelled any underlying dynamic mode in the heart rate response.

A final observation is that the time constants for the TM and CE, when compared for all seven model structures, are in strikingly close agreement (Table 3). This is in line with a previous comparison of heart rate dynamics between the TM and CE modalities that showed no significant difference in the time constant of heart rate response.¹⁰

Due to the retrospective nature of this investigation—that used existing data sets—the results and conclusions are considered to be provisional, but they do provide insights for the design of future studies: to avoid the confounding effect of initial transients, the plant input test signal should be designed to ensure that a physiological steady state has been reached in advance of the data evaluation period; a formal, statistical study design should be employed for comparison of the different model structures - the results of the present work provide effect-size estimates for statistical power and sample size calculations.

5. Conclusions

This preliminary analysis points to the P1D structure—that is to say, a linear first-order system with dead time—as being an appropriate model for heart rate response to exercise using treadmill and cycle ergometer modalities. In order to avoid biased estimates, it is vitally important that, prior to parameter estimation and validation, careful attention is paid to data preprocessing in order to eliminate transients and trends.

Ethical considerations

The study that generated both the treadmill and cycle ergometer datasets was performed in accordance with the Declaration of Helsinki; the study was reviewed and approved by the Ethics Committee of the Swiss Canton of Bern (Ref. 2019-02184; approval date 16 January 2020). Participants provided written, informed consent prior to inclusion in the study.

Authors’ contributions

Both authors made substantial contributions to the conception and design of the study; HW did the treadmill data acquisition; KH and HW performed the data analysis; both authors contributed to the interpretation of the data. KH drafted the manuscript; HW reviewed it critically for important intellectual content. Both authors read and approved the final manuscript.

Data availability

The datasets analysed in this research are available in the OLOS repository as follows:

Identification of heart rate dynamics during treadmill exercise: comparison of first- and second-order models - Treadmill dataset, https://doi.org/10.34914/olos:bivq3dcebff5dfrqtbf3v5y7si.⁷

Heart rate dynamics identification and control in cycle ergometer exercise: Comparison of first- and second-order performance --Cycle ergometer dataset, https://doi.org/10.34914/olos:xtyv7akiu5bzdba3oemrarg4ru.⁸

Please click on the link, then click on the “Files” tab at the bottom right of the screen to access the data.

Data is available under the terms of the Creative Commons Attribution 4.0 International license.

Software

Data analysis was conducted using third-party proprietary software (Matlab, Release 2024a, The MathWorks, Inc., USA). There is no open-source alternative that performs the specific functions employed. The Methods section of this article provides sufficient information to allow replication of the analysis, i.e. Matlab function names and mathematical definitions of the outcome measures.

Acknowledgements

Alexander Spörri (Bern University of Applied Sciences) did the cycle ergometer data acquisition.

References

1. Riebe D, Ehrman JK, Liguori G, et al.: ACSM’s guidelines for exercise testing and prescription. 10th ed.Philadelphia: Wolters Kluwer; 2018.
2. Whipp BJ, Ward SA, Lamarra N, et al.: Parameters of ventilatory and gas exchange dynamics during exercise. J. Appl. Physiol. 1982; 52(6): 1506–1513. Publisher Full Text
3. Bearden SE, Moffat RJ: VO2 and heart rate kinetics in cycling: transitions from an elevated baseline. J. Appl. Physiol. 2001; 90(6): 2081–2087. PubMed Abstract | Publisher Full Text
4. Wang H, Hunt KJ: Identification of heart rate dynamics during treadmill exercise: comparison of first- and second-order models. Biomed. Eng. Online. 2021; 20(37): 10–37. PubMed Abstract | Publisher Full Text | Free Full Text
5. Spörri AH, Wang H, Hunt KJ: Heart rate dynamics identification and control in cycle ergometer exercise: comparison of first- and second-order performance. Front. Control Eng. 2022; 3: 894180. Publisher Full Text
6. Wang H, Hunt KJ: Feedback control of heart rate during treadmill exercise based on a two-phase response model. PLoS One. 2023; 18(10): e0292310. PubMed Abstract | Publisher Full Text | Free Full Text
7. Wang H, Hunt KJ: Identification of heart rate dynamics during treadmill exercise: comparison of first- and second-order models. [Dataset]. OLOS Repository. 2023. Publisher Full Text
8. Spörri AH, Wang H, Hunt KJ: Heart rate dynamics identification and control in cycle ergometer exercise: comparison of first- and second-order performance. [Dataset]. OLOS Repository. 2023. Publisher Full Text
9. Ljung L: System Identification: theory for the user. 2nd ed.Upper Saddle River, New Jersey, USA: Prentice Hall; 1998.
10. Hunt KJ, Grunder R, Zahnd A: Identification and comparison of heart-rate dynamics during cycle ergometer and treadmill exercise. PLoS One. 2019; 14(8): e0220826. PubMed Abstract | Publisher Full Text | Free Full Text

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Version 2

VERSION 2 PUBLISHED 06 Aug 2024

Author details Author details

Hanjie Wang
Roles: Conceptualization, Data Curation, Formal Analysis, Investigation, Methodology, Software, Visualization, Writing – Review & Editing

Competing interests

No competing interests were disclosed.

Grant information

This work was supported by the Swiss National Science Foundation (Principal Investigator KH, Grant Ref. 320030-185351).
The funders had no role in study design, data collection and analysis, decision to publish, or preparation of the manuscript.

Article Versions (2)

version 2

Revised

Published: 01 Nov 2024, 13:894

https://doi.org/10.12688/f1000research.153397.2

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Published: 06 Aug 2024, 13:894

https://doi.org/10.12688/f1000research.153397.1

© 2024 Hunt KJ and Wang H. This is an open access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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Hunt KJ and Wang H. Identification of heart rate dynamics during treadmill and cycle ergometer exercise: the role of model zeros and dead time [version 2; peer review: 2 approved]. F1000Research 2024, 13:894 (https://doi.org/10.12688/f1000research.153397.2)

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Reviewer Report 09 Nov 2024

Lauren Miutz, Department of Health and Sport Science, University of Dayton, Dayton, Ohio, USA; Faculty of Kinesiology, University of Calgary, Calgary, Canada

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https://doi.org/10.5256/f1000research.173966.r336843

Appreciate the authors addressing my previous comments ... Continue reading

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Reviewer Report 06 Nov 2024

Przemysław Seweryn Kasiak, Department of Internal Medicine and Cardiology, Medical University of Warsaw, Warsaw, Poland

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https://doi.org/10.5256/f1000research.173966.r336844

I'm grateful for the Authors for applying necessary revision.
I also congratulate them on their article. Finally, please reconsider adding a proposed reference.

References
1. Kasiak PS, Wiecha S, Cieśliński I, Takken T, et al.: Validity of the Maximal Heart ... Continue reading

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Reviewer Report 24 Oct 2024

Lauren Miutz, Department of Health and Sport Science, University of Dayton, Dayton, Ohio, USA; Faculty of Kinesiology, University of Calgary, Calgary, Canada

Approved with Reservations

https://doi.org/10.5256/f1000research.168288.r331353

Brief description of article: Reanalyzed retrospective data to better identify a model of best fit for heart rate changes during exercise. The present study included a zero point and a pure delay in the heart rate model.

Relevance: Although the research reported that the model including dead time (compared to the model without dead time) provided a “better fit,” is the this clinically meaningful. The researchers reported a change in percentage of fit, however in the field of clinical or non-clinical exercise physiology, heart rate (especially regarding exercise) is largely provided as a range. Thus, are the percentages provided, between the two models, clinically relevant within the field.

Major points:

• Additional detail around the sample is needed, simply referencing the original work is not sufficient.

• Information around resting average heart rate (cycle and treadmill) and average maximal heart rate (cycle and treadmill) should be provided.

Minor point:

• Training status of the participants should be reports as heart rate (and potentially the delay and amount of dead time) are often linked to one’s training status, especially aerobic capacity.

• Interesting findings, however when reanalyzing retrospective data the researchers need to be careful the questions being answered is relevant and that adequate detail is provided to the reader, regardless of their knowledge surrounding the original study/data set.

Is the work clearly and accurately presented and does it cite the current literature?

Yes
Is the study design appropriate and is the work technically sound?

Yes
Are sufficient details of methods and analysis provided to allow replication by others?

Partly
If applicable, is the statistical analysis and its interpretation appropriate?

Yes
Are all the source data underlying the results available to ensure full reproducibility?

Yes
Are the conclusions drawn adequately supported by the results?

Yes

Competing Interests: No competing interests were disclosed.

Reviewer Expertise: Exercise physiology; cerebrovascular physiology; sport-related concussion & exertional measures

I confirm that I have read this submission and believe that I have an appropriate level of expertise to confirm that it is of an acceptable scientific standard, however I have significant reservations, as outlined above.

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Author Response 01 Nov 2024

Kenneth Hunt, rehaLab - the Laboratory for Rehabilitation Engineering ,Institute for Human Centred Engineering HuCE School of Engineering and Computer Science, Bern University of Applied Sciences,, Biel/Bienne, 2501, Switzerland

01 Nov 2024

Author Response

Dear Reviewers

Many thanks for your detailed and constructive comments on our manuscript. We are pleased to provide a point-by-point response below. Changes to the manuscript have been implemented ... Continue reading Dear Reviewers

Many thanks for your detailed and constructive comments on our manuscript. We are pleased to provide a point-by-point response below. Changes to the manuscript have been implemented as noted below and submitted using “track changes” in the revised manuscript.

With best wishes.

Yours faithfully
The authors

Reviewer #2 Report and Response: Lauren Miutz

Brief description of article: Reanalyzed retrospective data to better identify a model of best fit for heart rate changes during exercise. The present study included a zero point and a pure delay in the heart rate model.

RESPONSE: We would like to thank you for your careful reading of our article and for your constructive comments. Please find below our point-by-point response.

Relevance: Although the research reported that the model including dead time (compared to the model without dead time) provided a “better fit,” is the this [sic] clinically meaningful. The researchers reported a change in percentage of fit, however in the field of clinical or non-clinical exercise physiology, heart rate (especially regarding exercise) is largely provided as a range. Thus, are the percentages provided, between the two models, clinically relevant within the field.

RESPONSE: Thank you for this remark regarding relevance for clinical or non-clinical exercise physiology. In a similar vein, Reviewer #1 requested more information regarding the clinical and physiological perspectives, and practical applications. To address these points, we added the following paragraph at the start of the Introduction:

“Heart rate is an easy-to-measure physiological variable that can be used to characterise the intensity of exercise, both quantitatively and qualitatively using categories such as light, moderate and vigorous [10]. The ability to regulate heart rate during exercise using feedback control would therefore allow accurate prescription of training regimes in both clinical and non-clinical settings. Since feedback controllers require a model of the dynamic response of heart rate during exercise, it is important to first consider the fidelity of different model structures.”

Furthermore, we would like to remark that, in the field of systems and control engineering—which is the field within which we authors work—model fidelity is usually provided either in absolute terms as a root-mean-square error (RMSE) or as a normalised RMSE, i.e. model fit, expressed in percent. We cannot definitively answer the question of whether the reported improvement in fit will be relevant to researchers or practitioners in other fields (e.g. exercise physiologists) and must leave it up to colleagues in those fields to assess our findings for themselves.

Major points:

• Additional detail around the sample is needed, simply referencing the original work is not sufficient.

RESPONSE: Thank you for this recommendation (a similar request was made by Reviewer #1). We do agree that it would be helpful to the reader to have more information in the current article. As noted in our response to Reviewer #1, we have added more details to the Methods, as follows:

“For both exercise modalities, healthy, able-bodied participants exercised at moderate-to-vigorous intensity: in the treadmill analysis ³there were 11 participants (8 male, 3 female; overall mean age 32.5 years, mean body mass 75.5 kg, mean height 1.79 m); for the cycle ergometer ⁴there were 27 participants (20 male, 7 female; overall mean age 30.8 years, mean body mass 76.3 kg, mean height 1.79 m). Participants were required to be regular exercisers (30-min bouts, 3 times per week), non-smokers, and to be free of injury and illness.”

• Information around resting average heart rate (cycle and treadmill) and average maximal heart rate (cycle and treadmill) should be provided.

RESPONSE: Thank you for this suggestion. We have added information on average maximal heart rate (cycle and treadmill) to the Results section, as quoted below. Unfortunately, we cannot provide average resting heart rate because recording was started at the onset of the PRBS input signal: resting heart rate was not recorded (resting HR played no role in the analysis).

Text added to Results:
“Average maximal heart rate for the treadmill was 158.4 bpm; for the cycle ergometer it was 140.2 bpm (this is in line with our setting the mean target heart rate for the CE to be 20 bpm lower than for the TM in order to achieve a similar level of perceived exertion [9]).”

Minor point:

• Training status of the participants should be reports [sic] as heart rate (and potentially the delay and amount of dead time) are often linked to one’s training status, especially aerobic capacity.

RESPONSE: Thank you for this comment. As noted above, we have now provided information regarding participants’ training status: “Participants were required to be regular exercisers (30-min bouts, 3 times per week) …”. Beyond this requirement (it was a formal inclusion criterion), we did not record any further information in this regard.

• Interesting findings, however when reanalyzing retrospective data the researchers need to be careful the questions being answered is relevant and that adequate detail is provided to the reader, regardless of their knowledge surrounding the original study/data set.

RESPONSE: Many thanks for your observations, which are valuable. We hope that our responses and the corresponding revision of the manuscript now provide adequate detail independent of the original publications.
Dear Reviewers

Many thanks for your detailed and constructive comments on our manuscript. We are pleased to provide a point-by-point response below. Changes to the manuscript have been implemented as noted below and submitted using “track changes” in the revised manuscript.

With best wishes.

Yours faithfully
The authors

Reviewer #2 Report and Response: Lauren Miutz

Brief description of article: Reanalyzed retrospective data to better identify a model of best fit for heart rate changes during exercise. The present study included a zero point and a pure delay in the heart rate model.

RESPONSE: We would like to thank you for your careful reading of our article and for your constructive comments. Please find below our point-by-point response.

Relevance: Although the research reported that the model including dead time (compared to the model without dead time) provided a “better fit,” is the this [sic] clinically meaningful. The researchers reported a change in percentage of fit, however in the field of clinical or non-clinical exercise physiology, heart rate (especially regarding exercise) is largely provided as a range. Thus, are the percentages provided, between the two models, clinically relevant within the field.

RESPONSE: Thank you for this remark regarding relevance for clinical or non-clinical exercise physiology. In a similar vein, Reviewer #1 requested more information regarding the clinical and physiological perspectives, and practical applications. To address these points, we added the following paragraph at the start of the Introduction:

“Heart rate is an easy-to-measure physiological variable that can be used to characterise the intensity of exercise, both quantitatively and qualitatively using categories such as light, moderate and vigorous [10]. The ability to regulate heart rate during exercise using feedback control would therefore allow accurate prescription of training regimes in both clinical and non-clinical settings. Since feedback controllers require a model of the dynamic response of heart rate during exercise, it is important to first consider the fidelity of different model structures.”

Furthermore, we would like to remark that, in the field of systems and control engineering—which is the field within which we authors work—model fidelity is usually provided either in absolute terms as a root-mean-square error (RMSE) or as a normalised RMSE, i.e. model fit, expressed in percent. We cannot definitively answer the question of whether the reported improvement in fit will be relevant to researchers or practitioners in other fields (e.g. exercise physiologists) and must leave it up to colleagues in those fields to assess our findings for themselves.

Major points:

• Additional detail around the sample is needed, simply referencing the original work is not sufficient.

RESPONSE: Thank you for this recommendation (a similar request was made by Reviewer #1). We do agree that it would be helpful to the reader to have more information in the current article. As noted in our response to Reviewer #1, we have added more details to the Methods, as follows:

“For both exercise modalities, healthy, able-bodied participants exercised at moderate-to-vigorous intensity: in the treadmill analysis ³there were 11 participants (8 male, 3 female; overall mean age 32.5 years, mean body mass 75.5 kg, mean height 1.79 m); for the cycle ergometer ⁴there were 27 participants (20 male, 7 female; overall mean age 30.8 years, mean body mass 76.3 kg, mean height 1.79 m). Participants were required to be regular exercisers (30-min bouts, 3 times per week), non-smokers, and to be free of injury and illness.”

• Information around resting average heart rate (cycle and treadmill) and average maximal heart rate (cycle and treadmill) should be provided.

RESPONSE: Thank you for this suggestion. We have added information on average maximal heart rate (cycle and treadmill) to the Results section, as quoted below. Unfortunately, we cannot provide average resting heart rate because recording was started at the onset of the PRBS input signal: resting heart rate was not recorded (resting HR played no role in the analysis).

Text added to Results:
“Average maximal heart rate for the treadmill was 158.4 bpm; for the cycle ergometer it was 140.2 bpm (this is in line with our setting the mean target heart rate for the CE to be 20 bpm lower than for the TM in order to achieve a similar level of perceived exertion [9]).”

Minor point:

• Training status of the participants should be reports [sic] as heart rate (and potentially the delay and amount of dead time) are often linked to one’s training status, especially aerobic capacity.

RESPONSE: Thank you for this comment. As noted above, we have now provided information regarding participants’ training status: “Participants were required to be regular exercisers (30-min bouts, 3 times per week) …”. Beyond this requirement (it was a formal inclusion criterion), we did not record any further information in this regard.

• Interesting findings, however when reanalyzing retrospective data the researchers need to be careful the questions being answered is relevant and that adequate detail is provided to the reader, regardless of their knowledge surrounding the original study/data set.

RESPONSE: Many thanks for your observations, which are valuable. We hope that our responses and the corresponding revision of the manuscript now provide adequate detail independent of the original publications.
Competing Interests: No competing interests were disclosed. Close
Report a concern
Respond or Comment

COMMENTS ON THIS REPORT

Author Response 01 Nov 2024

Kenneth Hunt, rehaLab - the Laboratory for Rehabilitation Engineering ,Institute for Human Centred Engineering HuCE School of Engineering and Computer Science, Bern University of Applied Sciences,, Biel/Bienne, 2501, Switzerland

01 Nov 2024

Author Response

Dear Reviewers

Many thanks for your detailed and constructive comments on our manuscript. We are pleased to provide a point-by-point response below. Changes to the manuscript have been implemented ... Continue reading Dear Reviewers

Many thanks for your detailed and constructive comments on our manuscript. We are pleased to provide a point-by-point response below. Changes to the manuscript have been implemented as noted below and submitted using “track changes” in the revised manuscript.

With best wishes.

Yours faithfully
The authors

Reviewer #2 Report and Response: Lauren Miutz

Brief description of article: Reanalyzed retrospective data to better identify a model of best fit for heart rate changes during exercise. The present study included a zero point and a pure delay in the heart rate model.

RESPONSE: We would like to thank you for your careful reading of our article and for your constructive comments. Please find below our point-by-point response.

Relevance: Although the research reported that the model including dead time (compared to the model without dead time) provided a “better fit,” is the this [sic] clinically meaningful. The researchers reported a change in percentage of fit, however in the field of clinical or non-clinical exercise physiology, heart rate (especially regarding exercise) is largely provided as a range. Thus, are the percentages provided, between the two models, clinically relevant within the field.

RESPONSE: Thank you for this remark regarding relevance for clinical or non-clinical exercise physiology. In a similar vein, Reviewer #1 requested more information regarding the clinical and physiological perspectives, and practical applications. To address these points, we added the following paragraph at the start of the Introduction:

“Heart rate is an easy-to-measure physiological variable that can be used to characterise the intensity of exercise, both quantitatively and qualitatively using categories such as light, moderate and vigorous [10]. The ability to regulate heart rate during exercise using feedback control would therefore allow accurate prescription of training regimes in both clinical and non-clinical settings. Since feedback controllers require a model of the dynamic response of heart rate during exercise, it is important to first consider the fidelity of different model structures.”

Furthermore, we would like to remark that, in the field of systems and control engineering—which is the field within which we authors work—model fidelity is usually provided either in absolute terms as a root-mean-square error (RMSE) or as a normalised RMSE, i.e. model fit, expressed in percent. We cannot definitively answer the question of whether the reported improvement in fit will be relevant to researchers or practitioners in other fields (e.g. exercise physiologists) and must leave it up to colleagues in those fields to assess our findings for themselves.

Major points:

• Additional detail around the sample is needed, simply referencing the original work is not sufficient.

RESPONSE: Thank you for this recommendation (a similar request was made by Reviewer #1). We do agree that it would be helpful to the reader to have more information in the current article. As noted in our response to Reviewer #1, we have added more details to the Methods, as follows:

“For both exercise modalities, healthy, able-bodied participants exercised at moderate-to-vigorous intensity: in the treadmill analysis ³there were 11 participants (8 male, 3 female; overall mean age 32.5 years, mean body mass 75.5 kg, mean height 1.79 m); for the cycle ergometer ⁴there were 27 participants (20 male, 7 female; overall mean age 30.8 years, mean body mass 76.3 kg, mean height 1.79 m). Participants were required to be regular exercisers (30-min bouts, 3 times per week), non-smokers, and to be free of injury and illness.”

• Information around resting average heart rate (cycle and treadmill) and average maximal heart rate (cycle and treadmill) should be provided.

RESPONSE: Thank you for this suggestion. We have added information on average maximal heart rate (cycle and treadmill) to the Results section, as quoted below. Unfortunately, we cannot provide average resting heart rate because recording was started at the onset of the PRBS input signal: resting heart rate was not recorded (resting HR played no role in the analysis).

Text added to Results:
“Average maximal heart rate for the treadmill was 158.4 bpm; for the cycle ergometer it was 140.2 bpm (this is in line with our setting the mean target heart rate for the CE to be 20 bpm lower than for the TM in order to achieve a similar level of perceived exertion [9]).”

Minor point:

• Training status of the participants should be reports [sic] as heart rate (and potentially the delay and amount of dead time) are often linked to one’s training status, especially aerobic capacity.

RESPONSE: Thank you for this comment. As noted above, we have now provided information regarding participants’ training status: “Participants were required to be regular exercisers (30-min bouts, 3 times per week) …”. Beyond this requirement (it was a formal inclusion criterion), we did not record any further information in this regard.

• Interesting findings, however when reanalyzing retrospective data the researchers need to be careful the questions being answered is relevant and that adequate detail is provided to the reader, regardless of their knowledge surrounding the original study/data set.

RESPONSE: Many thanks for your observations, which are valuable. We hope that our responses and the corresponding revision of the manuscript now provide adequate detail independent of the original publications.
Dear Reviewers

Many thanks for your detailed and constructive comments on our manuscript. We are pleased to provide a point-by-point response below. Changes to the manuscript have been implemented as noted below and submitted using “track changes” in the revised manuscript.

With best wishes.

Yours faithfully
The authors

Reviewer #2 Report and Response: Lauren Miutz

Brief description of article: Reanalyzed retrospective data to better identify a model of best fit for heart rate changes during exercise. The present study included a zero point and a pure delay in the heart rate model.

RESPONSE: We would like to thank you for your careful reading of our article and for your constructive comments. Please find below our point-by-point response.

Relevance: Although the research reported that the model including dead time (compared to the model without dead time) provided a “better fit,” is the this [sic] clinically meaningful. The researchers reported a change in percentage of fit, however in the field of clinical or non-clinical exercise physiology, heart rate (especially regarding exercise) is largely provided as a range. Thus, are the percentages provided, between the two models, clinically relevant within the field.

RESPONSE: Thank you for this remark regarding relevance for clinical or non-clinical exercise physiology. In a similar vein, Reviewer #1 requested more information regarding the clinical and physiological perspectives, and practical applications. To address these points, we added the following paragraph at the start of the Introduction:

“Heart rate is an easy-to-measure physiological variable that can be used to characterise the intensity of exercise, both quantitatively and qualitatively using categories such as light, moderate and vigorous [10]. The ability to regulate heart rate during exercise using feedback control would therefore allow accurate prescription of training regimes in both clinical and non-clinical settings. Since feedback controllers require a model of the dynamic response of heart rate during exercise, it is important to first consider the fidelity of different model structures.”

Furthermore, we would like to remark that, in the field of systems and control engineering—which is the field within which we authors work—model fidelity is usually provided either in absolute terms as a root-mean-square error (RMSE) or as a normalised RMSE, i.e. model fit, expressed in percent. We cannot definitively answer the question of whether the reported improvement in fit will be relevant to researchers or practitioners in other fields (e.g. exercise physiologists) and must leave it up to colleagues in those fields to assess our findings for themselves.

Major points:

• Additional detail around the sample is needed, simply referencing the original work is not sufficient.

RESPONSE: Thank you for this recommendation (a similar request was made by Reviewer #1). We do agree that it would be helpful to the reader to have more information in the current article. As noted in our response to Reviewer #1, we have added more details to the Methods, as follows:

“For both exercise modalities, healthy, able-bodied participants exercised at moderate-to-vigorous intensity: in the treadmill analysis ³there were 11 participants (8 male, 3 female; overall mean age 32.5 years, mean body mass 75.5 kg, mean height 1.79 m); for the cycle ergometer ⁴there were 27 participants (20 male, 7 female; overall mean age 30.8 years, mean body mass 76.3 kg, mean height 1.79 m). Participants were required to be regular exercisers (30-min bouts, 3 times per week), non-smokers, and to be free of injury and illness.”

• Information around resting average heart rate (cycle and treadmill) and average maximal heart rate (cycle and treadmill) should be provided.

RESPONSE: Thank you for this suggestion. We have added information on average maximal heart rate (cycle and treadmill) to the Results section, as quoted below. Unfortunately, we cannot provide average resting heart rate because recording was started at the onset of the PRBS input signal: resting heart rate was not recorded (resting HR played no role in the analysis).

Text added to Results:
“Average maximal heart rate for the treadmill was 158.4 bpm; for the cycle ergometer it was 140.2 bpm (this is in line with our setting the mean target heart rate for the CE to be 20 bpm lower than for the TM in order to achieve a similar level of perceived exertion [9]).”

Minor point:

• Training status of the participants should be reports [sic] as heart rate (and potentially the delay and amount of dead time) are often linked to one’s training status, especially aerobic capacity.

RESPONSE: Thank you for this comment. As noted above, we have now provided information regarding participants’ training status: “Participants were required to be regular exercisers (30-min bouts, 3 times per week) …”. Beyond this requirement (it was a formal inclusion criterion), we did not record any further information in this regard.

• Interesting findings, however when reanalyzing retrospective data the researchers need to be careful the questions being answered is relevant and that adequate detail is provided to the reader, regardless of their knowledge surrounding the original study/data set.

RESPONSE: Many thanks for your observations, which are valuable. We hope that our responses and the corresponding revision of the manuscript now provide adequate detail independent of the original publications.
Competing Interests: No competing interests were disclosed. Close
Report a concern

Views

Reviewer Report 27 Aug 2024

Przemysław Seweryn Kasiak, Department of Internal Medicine and Cardiology, Medical University of Warsaw, Warsaw, Poland

Approved with Reservations

https://doi.org/10.5256/f1000research.168288.r317159

Firstly, I would like to thank for inviting me as a Referee of this study. This research touches an important topic of heart rate during exercise and provide useful piece of knowledge for clinicians and sport practitioners. I am providing ... Continue reading

I suggest more precise description of the rationale for this study from clinical and physiological perspective. Perhaps, an additional paragraph in the introduction would be welcome.
Similarly, there is a lack of information about practical applications of this study in the discussion.

Minor issues:

I suggest providing a brief description of the study population in the abstract.
If the Authors will meet the word limit, I strongly recommend enhancing the information about study group (their demographics, fitness level etc.) and testing protocols.
Did the Authors evaluate sample size to ensure credibility of their analysis and conclusions? There are only 38 participants in this study.

In summary, this study is interesting. Further analysis of the preliminary report would be helpful for deep understanding of heart rate response to exercises.

Is the work clearly and accurately presented and does it cite the current literature?

Partly
Is the study design appropriate and is the work technically sound?

Yes
Are sufficient details of methods and analysis provided to allow replication by others?

No
If applicable, is the statistical analysis and its interpretation appropriate?

Partly
Are all the source data underlying the results available to ensure full reproducibility?

Yes
Are the conclusions drawn adequately supported by the results?

Partly

Competing Interests: No competing interests were disclosed.

Reviewer Expertise: Sports Cardiology

CITE

Report a concern

Author Response 01 Nov 2024

Kenneth Hunt, rehaLab - the Laboratory for Rehabilitation Engineering ,Institute for Human Centred Engineering HuCE School of Engineering and Computer Science, Bern University of Applied Sciences,, Biel/Bienne, 2501, Switzerland

01 Nov 2024

Author Response
Dear Reviewers

Many thanks for your detailed and constructive comments on our manuscript. We are pleased to provide a point-by-point response below. Changes to the manuscript have been implemented ... Continue reading
Dear Reviewers

Many thanks for your detailed and constructive comments on our manuscript. We are pleased to provide a point-by-point response below. Changes to the manuscript have been implemented as noted below and submitted using “track changes” in the revised manuscript.

With best wishes.

Yours faithfully
The authors

Reviewer #1 Report and Response: Przemysław Seweryn Kasiak

Firstly, I would like to thank for inviting me as a Referee of this study. This research touches an important topic of heart rate during exercise and provide useful piece of knowledge for clinicians and sport practitioners. I am providing my comments for the Authors below.

RESPONSE: Dziękuję bardzo—and thank you very much for your helpful comments, especially for your observation of relevance for clinicians and sport practitioners.
Major issues:

I suggest more precise description of the rationale for this study from clinical and physiological perspective. Perhaps, an additional paragraph in the introduction would be welcome.

Similarly, there is a lack of information about practical applications of this study in the discussion.

RESPONSE: Thank you for these suggestions. We agree it is very useful to provide more information regarding the clinical and physiological perspectives, and practical applications. To address both of these points, we have added the following paragraph at the start of the Introduction:
“Heart rate is an easy-to-measure physiological variable that can be used to characterise the intensity of exercise, both quantitatively and qualitatively using categories such as light, moderate and vigorous [10]. The ability to regulate heart rate during exercise using feedback control would therefore allow accurate prescription of training regimes in both clinical and non-clinical settings. Since feedback controllers require a model of the dynamic response of heart rate during exercise, it is important to first consider the fidelity of different model structures.”
New reference [10]:
10. Riebe D, Ehrman JK, Liguori G, Magal M, editors. ACSM’s guidelines for exercise testing and prescription. 10th ed. Philadelphia: Wolters Kluwer; 2018.

Minor issues:

I suggest providing a brief description of the study population in the abstract.

RESPONSE: Regrettably, the Abstract already has 298 words, with a limit of 300 words. But we have provided more detailed information on the study population in the main text as noted below.

If the Authors will meet the word limit, I strongly recommend enhancing the information about study group (their demographics, fitness level etc.) and testing protocols.

RESPONSE: Thank you for this suggestion. As mentioned in the text, this is a retrospective analysis of data from two previously-published studies. We pointed out that full information regarding the study groups and testing protocols can be found in the corresponding citations, viz. references [3] and [4], as follows:
“Full details of experimental procedures employed for data collection in the preceding treadmill and cycle ergometer investigations can be found in the respective publications. ³^, ⁴Essential elements of the protocols are summarised in this Brief Report.”
Nevertheless, we do agree it would be helpful to the reader to have more information in the current article, for which reason we have added more details to the Methods, as follows:
“For both exercise modalities, healthy, able-bodied participants exercised at moderate-to-vigorous intensity: in the treadmill analysis ³there were 11 participants (8 male, 3 female; overall mean age 32.5 years, mean body mass 75.5 kg, mean height 1.79 m); for the cycle ergometer ⁴there were 27 participants (20 male, 7 female; overall mean age 30.8 years, mean body mass 76.3 kg, mean height 1.79 m). Participants were required to be regular exercisers (30-min bouts, 3 times per week), non-smokers, and to be free of injury and illness.”

Did the Authors evaluate sample size to ensure credibility of their analysis and conclusions? There are only 38 participants in this study.

RESPONSE: Thank you for this important query. As noted above, the 38 participants came from two previous studies: a pilot treadmill study, [3], with 11 participants, and a cycle ergometer study, [4], with 27 participants.

For the treadmill study, the sample size of n = 11 was chosen in accordance with the requirements of pilot studies, and no formal statistical power analysis was conducted.

For the cycle ergometer study, an a priori statistical power and sample size estimate was included in the study protocol that was approved by our ethics committee. The details can be found in reference [4], as follows: “The sample size of n = 27 participants was estimated a priori by a statistical power calculation that used estimates of expected effect sizes and sample standard deviations obtained from previous studies in this lab, with the significance level of 5 % and a statistical power of 80 % (1 − β = 0.8).”

In both cases, no post-hoc statistical power calculation was performed because observed effect sizes and their uncertainty bounds supersede any a priori estimates, and because “Post-hoc power estimates … have been shown to be logically invalid and practically misleading”; quotation from:

Dziak JJ, Dierker LC, Abar B. The Interpretation of Statistical Power after the Data have been Gathered. Curr Psychol. 2020 Jun;39(3):870-877.
https://doi.org/10.1007/s12144-018-0018-1
Dear Reviewers

Many thanks for your detailed and constructive comments on our manuscript. We are pleased to provide a point-by-point response below. Changes to the manuscript have been implemented as noted below and submitted using “track changes” in the revised manuscript.

With best wishes.

Yours faithfully
The authors

Reviewer #1 Report and Response: Przemysław Seweryn Kasiak

Firstly, I would like to thank for inviting me as a Referee of this study. This research touches an important topic of heart rate during exercise and provide useful piece of knowledge for clinicians and sport practitioners. I am providing my comments for the Authors below.

RESPONSE: Dziękuję bardzo—and thank you very much for your helpful comments, especially for your observation of relevance for clinicians and sport practitioners.
Major issues:

I suggest more precise description of the rationale for this study from clinical and physiological perspective. Perhaps, an additional paragraph in the introduction would be welcome.

Similarly, there is a lack of information about practical applications of this study in the discussion.

RESPONSE: Thank you for these suggestions. We agree it is very useful to provide more information regarding the clinical and physiological perspectives, and practical applications. To address both of these points, we have added the following paragraph at the start of the Introduction:
“Heart rate is an easy-to-measure physiological variable that can be used to characterise the intensity of exercise, both quantitatively and qualitatively using categories such as light, moderate and vigorous [10]. The ability to regulate heart rate during exercise using feedback control would therefore allow accurate prescription of training regimes in both clinical and non-clinical settings. Since feedback controllers require a model of the dynamic response of heart rate during exercise, it is important to first consider the fidelity of different model structures.”
New reference [10]:
10. Riebe D, Ehrman JK, Liguori G, Magal M, editors. ACSM’s guidelines for exercise testing and prescription. 10th ed. Philadelphia: Wolters Kluwer; 2018.

Minor issues:

I suggest providing a brief description of the study population in the abstract.

RESPONSE: Regrettably, the Abstract already has 298 words, with a limit of 300 words. But we have provided more detailed information on the study population in the main text as noted below.

If the Authors will meet the word limit, I strongly recommend enhancing the information about study group (their demographics, fitness level etc.) and testing protocols.

RESPONSE: Thank you for this suggestion. As mentioned in the text, this is a retrospective analysis of data from two previously-published studies. We pointed out that full information regarding the study groups and testing protocols can be found in the corresponding citations, viz. references [3] and [4], as follows:
“Full details of experimental procedures employed for data collection in the preceding treadmill and cycle ergometer investigations can be found in the respective publications. ³^, ⁴Essential elements of the protocols are summarised in this Brief Report.”
Nevertheless, we do agree it would be helpful to the reader to have more information in the current article, for which reason we have added more details to the Methods, as follows:
“For both exercise modalities, healthy, able-bodied participants exercised at moderate-to-vigorous intensity: in the treadmill analysis ³there were 11 participants (8 male, 3 female; overall mean age 32.5 years, mean body mass 75.5 kg, mean height 1.79 m); for the cycle ergometer ⁴there were 27 participants (20 male, 7 female; overall mean age 30.8 years, mean body mass 76.3 kg, mean height 1.79 m). Participants were required to be regular exercisers (30-min bouts, 3 times per week), non-smokers, and to be free of injury and illness.”

Did the Authors evaluate sample size to ensure credibility of their analysis and conclusions? There are only 38 participants in this study.

RESPONSE: Thank you for this important query. As noted above, the 38 participants came from two previous studies: a pilot treadmill study, [3], with 11 participants, and a cycle ergometer study, [4], with 27 participants.

For the treadmill study, the sample size of n = 11 was chosen in accordance with the requirements of pilot studies, and no formal statistical power analysis was conducted.

For the cycle ergometer study, an a priori statistical power and sample size estimate was included in the study protocol that was approved by our ethics committee. The details can be found in reference [4], as follows: “The sample size of n = 27 participants was estimated a priori by a statistical power calculation that used estimates of expected effect sizes and sample standard deviations obtained from previous studies in this lab, with the significance level of 5 % and a statistical power of 80 % (1 − β = 0.8).”

In both cases, no post-hoc statistical power calculation was performed because observed effect sizes and their uncertainty bounds supersede any a priori estimates, and because “Post-hoc power estimates … have been shown to be logically invalid and practically misleading”; quotation from:

Dziak JJ, Dierker LC, Abar B. The Interpretation of Statistical Power after the Data have been Gathered. Curr Psychol. 2020 Jun;39(3):870-877.
https://doi.org/10.1007/s12144-018-0018-1
Competing Interests: No competing interests were disclosed. Close
Report a concern
Respond or Comment

COMMENTS ON THIS REPORT

Author Response 01 Nov 2024

Kenneth Hunt, rehaLab - the Laboratory for Rehabilitation Engineering ,Institute for Human Centred Engineering HuCE School of Engineering and Computer Science, Bern University of Applied Sciences,, Biel/Bienne, 2501, Switzerland

01 Nov 2024

Author Response
Dear Reviewers

Many thanks for your detailed and constructive comments on our manuscript. We are pleased to provide a point-by-point response below. Changes to the manuscript have been implemented ... Continue reading
Dear Reviewers

Many thanks for your detailed and constructive comments on our manuscript. We are pleased to provide a point-by-point response below. Changes to the manuscript have been implemented as noted below and submitted using “track changes” in the revised manuscript.

With best wishes.

Yours faithfully
The authors

Reviewer #1 Report and Response: Przemysław Seweryn Kasiak

Firstly, I would like to thank for inviting me as a Referee of this study. This research touches an important topic of heart rate during exercise and provide useful piece of knowledge for clinicians and sport practitioners. I am providing my comments for the Authors below.

RESPONSE: Dziękuję bardzo—and thank you very much for your helpful comments, especially for your observation of relevance for clinicians and sport practitioners.
Major issues:

I suggest more precise description of the rationale for this study from clinical and physiological perspective. Perhaps, an additional paragraph in the introduction would be welcome.

Similarly, there is a lack of information about practical applications of this study in the discussion.

RESPONSE: Thank you for these suggestions. We agree it is very useful to provide more information regarding the clinical and physiological perspectives, and practical applications. To address both of these points, we have added the following paragraph at the start of the Introduction:
“Heart rate is an easy-to-measure physiological variable that can be used to characterise the intensity of exercise, both quantitatively and qualitatively using categories such as light, moderate and vigorous [10]. The ability to regulate heart rate during exercise using feedback control would therefore allow accurate prescription of training regimes in both clinical and non-clinical settings. Since feedback controllers require a model of the dynamic response of heart rate during exercise, it is important to first consider the fidelity of different model structures.”
New reference [10]:
10. Riebe D, Ehrman JK, Liguori G, Magal M, editors. ACSM’s guidelines for exercise testing and prescription. 10th ed. Philadelphia: Wolters Kluwer; 2018.

Minor issues:

I suggest providing a brief description of the study population in the abstract.

RESPONSE: Regrettably, the Abstract already has 298 words, with a limit of 300 words. But we have provided more detailed information on the study population in the main text as noted below.

If the Authors will meet the word limit, I strongly recommend enhancing the information about study group (their demographics, fitness level etc.) and testing protocols.

RESPONSE: Thank you for this suggestion. As mentioned in the text, this is a retrospective analysis of data from two previously-published studies. We pointed out that full information regarding the study groups and testing protocols can be found in the corresponding citations, viz. references [3] and [4], as follows:
“Full details of experimental procedures employed for data collection in the preceding treadmill and cycle ergometer investigations can be found in the respective publications. ³^, ⁴Essential elements of the protocols are summarised in this Brief Report.”
Nevertheless, we do agree it would be helpful to the reader to have more information in the current article, for which reason we have added more details to the Methods, as follows:
“For both exercise modalities, healthy, able-bodied participants exercised at moderate-to-vigorous intensity: in the treadmill analysis ³there were 11 participants (8 male, 3 female; overall mean age 32.5 years, mean body mass 75.5 kg, mean height 1.79 m); for the cycle ergometer ⁴there were 27 participants (20 male, 7 female; overall mean age 30.8 years, mean body mass 76.3 kg, mean height 1.79 m). Participants were required to be regular exercisers (30-min bouts, 3 times per week), non-smokers, and to be free of injury and illness.”

Did the Authors evaluate sample size to ensure credibility of their analysis and conclusions? There are only 38 participants in this study.

RESPONSE: Thank you for this important query. As noted above, the 38 participants came from two previous studies: a pilot treadmill study, [3], with 11 participants, and a cycle ergometer study, [4], with 27 participants.

For the treadmill study, the sample size of n = 11 was chosen in accordance with the requirements of pilot studies, and no formal statistical power analysis was conducted.

For the cycle ergometer study, an a priori statistical power and sample size estimate was included in the study protocol that was approved by our ethics committee. The details can be found in reference [4], as follows: “The sample size of n = 27 participants was estimated a priori by a statistical power calculation that used estimates of expected effect sizes and sample standard deviations obtained from previous studies in this lab, with the significance level of 5 % and a statistical power of 80 % (1 − β = 0.8).”

In both cases, no post-hoc statistical power calculation was performed because observed effect sizes and their uncertainty bounds supersede any a priori estimates, and because “Post-hoc power estimates … have been shown to be logically invalid and practically misleading”; quotation from:

Dziak JJ, Dierker LC, Abar B. The Interpretation of Statistical Power after the Data have been Gathered. Curr Psychol. 2020 Jun;39(3):870-877.
https://doi.org/10.1007/s12144-018-0018-1
Dear Reviewers

Many thanks for your detailed and constructive comments on our manuscript. We are pleased to provide a point-by-point response below. Changes to the manuscript have been implemented as noted below and submitted using “track changes” in the revised manuscript.

With best wishes.

Yours faithfully
The authors

Reviewer #1 Report and Response: Przemysław Seweryn Kasiak

Firstly, I would like to thank for inviting me as a Referee of this study. This research touches an important topic of heart rate during exercise and provide useful piece of knowledge for clinicians and sport practitioners. I am providing my comments for the Authors below.

RESPONSE: Dziękuję bardzo—and thank you very much for your helpful comments, especially for your observation of relevance for clinicians and sport practitioners.
Major issues:

I suggest more precise description of the rationale for this study from clinical and physiological perspective. Perhaps, an additional paragraph in the introduction would be welcome.

Similarly, there is a lack of information about practical applications of this study in the discussion.

RESPONSE: Thank you for these suggestions. We agree it is very useful to provide more information regarding the clinical and physiological perspectives, and practical applications. To address both of these points, we have added the following paragraph at the start of the Introduction:
“Heart rate is an easy-to-measure physiological variable that can be used to characterise the intensity of exercise, both quantitatively and qualitatively using categories such as light, moderate and vigorous [10]. The ability to regulate heart rate during exercise using feedback control would therefore allow accurate prescription of training regimes in both clinical and non-clinical settings. Since feedback controllers require a model of the dynamic response of heart rate during exercise, it is important to first consider the fidelity of different model structures.”
New reference [10]:
10. Riebe D, Ehrman JK, Liguori G, Magal M, editors. ACSM’s guidelines for exercise testing and prescription. 10th ed. Philadelphia: Wolters Kluwer; 2018.

Minor issues:

I suggest providing a brief description of the study population in the abstract.

RESPONSE: Regrettably, the Abstract already has 298 words, with a limit of 300 words. But we have provided more detailed information on the study population in the main text as noted below.

If the Authors will meet the word limit, I strongly recommend enhancing the information about study group (their demographics, fitness level etc.) and testing protocols.

RESPONSE: Thank you for this suggestion. As mentioned in the text, this is a retrospective analysis of data from two previously-published studies. We pointed out that full information regarding the study groups and testing protocols can be found in the corresponding citations, viz. references [3] and [4], as follows:
“Full details of experimental procedures employed for data collection in the preceding treadmill and cycle ergometer investigations can be found in the respective publications. ³^, ⁴Essential elements of the protocols are summarised in this Brief Report.”
Nevertheless, we do agree it would be helpful to the reader to have more information in the current article, for which reason we have added more details to the Methods, as follows:
“For both exercise modalities, healthy, able-bodied participants exercised at moderate-to-vigorous intensity: in the treadmill analysis ³there were 11 participants (8 male, 3 female; overall mean age 32.5 years, mean body mass 75.5 kg, mean height 1.79 m); for the cycle ergometer ⁴there were 27 participants (20 male, 7 female; overall mean age 30.8 years, mean body mass 76.3 kg, mean height 1.79 m). Participants were required to be regular exercisers (30-min bouts, 3 times per week), non-smokers, and to be free of injury and illness.”

Did the Authors evaluate sample size to ensure credibility of their analysis and conclusions? There are only 38 participants in this study.

RESPONSE: Thank you for this important query. As noted above, the 38 participants came from two previous studies: a pilot treadmill study, [3], with 11 participants, and a cycle ergometer study, [4], with 27 participants.

For the treadmill study, the sample size of n = 11 was chosen in accordance with the requirements of pilot studies, and no formal statistical power analysis was conducted.

For the cycle ergometer study, an a priori statistical power and sample size estimate was included in the study protocol that was approved by our ethics committee. The details can be found in reference [4], as follows: “The sample size of n = 27 participants was estimated a priori by a statistical power calculation that used estimates of expected effect sizes and sample standard deviations obtained from previous studies in this lab, with the significance level of 5 % and a statistical power of 80 % (1 − β = 0.8).”

In both cases, no post-hoc statistical power calculation was performed because observed effect sizes and their uncertainty bounds supersede any a priori estimates, and because “Post-hoc power estimates … have been shown to be logically invalid and practically misleading”; quotation from:

Dziak JJ, Dierker LC, Abar B. The Interpretation of Statistical Power after the Data have been Gathered. Curr Psychol. 2020 Jun;39(3):870-877.
https://doi.org/10.1007/s12144-018-0018-1
Competing Interests: No competing interests were disclosed. Close
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Version 2

VERSION 2 PUBLISHED 06 Aug 2024

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Przemysław Seweryn Kasiak, Medical University of Warsaw, Warsaw, Poland
Lauren Miutz, University of Dayton, Dayton, USA; University of Calgary, Calgary, Canada

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09 Nov 2024 | for Version 2

Lauren Miutz, Department of Health and Sport Science, University of Dayton, Dayton, Ohio, USA; Faculty of Kinesiology, University of Calgary, Calgary, Canada

1 Views Cite this report Responses(0)

Approved

Appreciate the authors addressing my previous comments and adding the additional information to the article.

Competing Interests

No competing interests were disclosed.

Reviewer Expertise

Exercise Physiology

I confirm that I have read this submission and believe that I have an appropriate level of expertise to confirm that it is of an acceptable scientific standard.

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Reviewer Report

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06 Nov 2024 | for Version 2

Przemysław Seweryn Kasiak, Department of Internal Medicine and Cardiology, Medical University of Warsaw, Warsaw, Poland

1 Views Cite this report Responses(0)

Approved

I'm grateful for the Authors for applying necessary revision.
I also congratulate them on their article. Finally, please reconsider adding a proposed reference.

References

1. Kasiak PS, Wiecha S, Cieśliński I, Takken T, et al.: Validity of the Maximal Heart Rate Prediction Models among Runners and Cyclists.J Clin Med. 2023; 12 (8). PubMed Abstract | Publisher Full Text

Competing Interests

No competing interests were disclosed.

Reviewer Expertise

Sports Cardiology

I confirm that I have read this submission and believe that I have an appropriate level of expertise to confirm that it is of an acceptable scientific standard.

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24 Oct 2024 | for Version 1

Lauren Miutz, Department of Health and Sport Science, University of Dayton, Dayton, Ohio, USA; Faculty of Kinesiology, University of Calgary, Calgary, Canada

12 Views Cite this report Responses(1)

Approved With Reservations

Is the work clearly and accurately presented and does it cite the current literature?

Yes
Is the study design appropriate and is the work technically sound?

Yes
Are sufficient details of methods and analysis provided to allow replication by others?

Partly
If applicable, is the statistical analysis and its interpretation appropriate?

Yes
Are all the source data underlying the results available to ensure full reproducibility?

Yes
Are the conclusions drawn adequately supported by the results?

Yes

Competing Interests

No competing interests were disclosed.

Reviewer Expertise

Exercise physiology; cerebrovascular physiology; sport-related concussion & exertional measures

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Author Response

01 Nov 2024

Kenneth Hunt, rehaLab - the Laboratory for Rehabilitation Engineering ,Institute for Human Centred Engineering HuCE School of Engineering and Computer Science, Bern University of Applied Sciences,, Biel/Bienne, 2501, Switzerland

Dear Reviewers

Many thanks for your detailed and constructive comments on our manuscript. We are pleased to provide a point-by-point response below. Changes to the manuscript have been implemented as noted below and submitted using “track changes” in the revised manuscript.

With best wishes.

Yours faithfully
The authors

Reviewer #2 Report and Response: Lauren Miutz

Brief description of article: Reanalyzed retrospective data to better identify a model of best fit for heart rate changes during exercise. The present study included a zero point and a pure delay in the heart rate model.

RESPONSE: We would like to thank you for your careful reading of our article and for your constructive comments. Please find below our point-by-point response.

Relevance: Although the research reported that the model including dead time (compared to the model without dead time) provided a “better fit,” is the this [sic] clinically meaningful. The researchers reported a change in percentage of fit, however in the field of clinical or non-clinical exercise physiology, heart rate (especially regarding exercise) is largely provided as a range. Thus, are the percentages provided, between the two models, clinically relevant within the field.

RESPONSE: Thank you for this remark regarding relevance for clinical or non-clinical exercise physiology. In a similar vein, Reviewer #1 requested more information regarding the clinical and physiological perspectives, and practical applications. To address these points, we added the following paragraph at the start of the Introduction:

“Heart rate is an easy-to-measure physiological variable that can be used to characterise the intensity of exercise, both quantitatively and qualitatively using categories such as light, moderate and vigorous [10]. The ability to regulate heart rate during exercise using feedback control would therefore allow accurate prescription of training regimes in both clinical and non-clinical settings. Since feedback controllers require a model of the dynamic response of heart rate during exercise, it is important to first consider the fidelity of different model structures.”

Furthermore, we would like to remark that, in the field of systems and control engineering—which is the field within which we authors work—model fidelity is usually provided either in absolute terms as a root-mean-square error (RMSE) or as a normalised RMSE, i.e. model fit, expressed in percent. We cannot definitively answer the question of whether the reported improvement in fit will be relevant to researchers or practitioners in other fields (e.g. exercise physiologists) and must leave it up to colleagues in those fields to assess our findings for themselves.

Major points:

• Additional detail around the sample is needed, simply referencing the original work is not sufficient.

RESPONSE: Thank you for this recommendation (a similar request was made by Reviewer #1). We do agree that it would be helpful to the reader to have more information in the current article. As noted in our response to Reviewer #1, we have added more details to the Methods, as follows:

“For both exercise modalities, healthy, able-bodied participants exercised at moderate-to-vigorous intensity: in the treadmill analysis ³there were 11 participants (8 male, 3 female; overall mean age 32.5 years, mean body mass 75.5 kg, mean height 1.79 m); for the cycle ergometer ⁴there were 27 participants (20 male, 7 female; overall mean age 30.8 years, mean body mass 76.3 kg, mean height 1.79 m). Participants were required to be regular exercisers (30-min bouts, 3 times per week), non-smokers, and to be free of injury and illness.”

• Information around resting average heart rate (cycle and treadmill) and average maximal heart rate (cycle and treadmill) should be provided.

RESPONSE: Thank you for this suggestion. We have added information on average maximal heart rate (cycle and treadmill) to the Results section, as quoted below. Unfortunately, we cannot provide average resting heart rate because recording was started at the onset of the PRBS input signal: resting heart rate was not recorded (resting HR played no role in the analysis).

Text added to Results:
“Average maximal heart rate for the treadmill was 158.4 bpm; for the cycle ergometer it was 140.2 bpm (this is in line with our setting the mean target heart rate for the CE to be 20 bpm lower than for the TM in order to achieve a similar level of perceived exertion [9]).”

Minor point:

• Training status of the participants should be reports [sic] as heart rate (and potentially the delay and amount of dead time) are often linked to one’s training status, especially aerobic capacity.

RESPONSE: Thank you for this comment. As noted above, we have now provided information regarding participants’ training status: “Participants were required to be regular exercisers (30-min bouts, 3 times per week) …”. Beyond this requirement (it was a formal inclusion criterion), we did not record any further information in this regard.

• Interesting findings, however when reanalyzing retrospective data the researchers need to be careful the questions being answered is relevant and that adequate detail is provided to the reader, regardless of their knowledge surrounding the original study/data set.

RESPONSE: Many thanks for your observations, which are valuable. We hope that our responses and the corresponding revision of the manuscript now provide adequate detail independent of the original publications.

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Competing Interests

No competing interests were disclosed.

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Reviewer Report

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27 Aug 2024 | for Version 1

Przemysław Seweryn Kasiak, Department of Internal Medicine and Cardiology, Medical University of Warsaw, Warsaw, Poland

28 Views Cite this report Responses(1)

Approved With Reservations

I suggest more precise description of the rationale for this study from clinical and physiological perspective. Perhaps, an additional paragraph in the introduction would be welcome.
Similarly, there is a lack of information about practical applications of this study in the discussion.

Minor issues:

I suggest providing a brief description of the study population in the abstract.
If the Authors will meet the word limit, I strongly recommend enhancing the information about study group (their demographics, fitness level etc.) and testing protocols.
Did the Authors evaluate sample size to ensure credibility of their analysis and conclusions? There are only 38 participants in this study.

In summary, this study is interesting. Further analysis of the preliminary report would be helpful for deep understanding of heart rate response to exercises.

Is the work clearly and accurately presented and does it cite the current literature?

Partly
Is the study design appropriate and is the work technically sound?

Yes
Are sufficient details of methods and analysis provided to allow replication by others?

No
If applicable, is the statistical analysis and its interpretation appropriate?

Partly
Are all the source data underlying the results available to ensure full reproducibility?

Yes
Are the conclusions drawn adequately supported by the results?

Partly

Competing Interests

No competing interests were disclosed.

Reviewer Expertise

Sports Cardiology

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Author Response

01 Nov 2024

Kenneth Hunt, rehaLab - the Laboratory for Rehabilitation Engineering ,Institute for Human Centred Engineering HuCE School of Engineering and Computer Science, Bern University of Applied Sciences,, Biel/Bienne, 2501, Switzerland

I suggest more precise description of the rationale for this study from clinical and physiological perspective. Perhaps, an additional paragraph in the introduction would be welcome.
Similarly, there is a lack of information about practical applications of this study in the discussion.

RESPONSE: Thank you for these suggestions. We agree it is very useful to provide more information regarding the clinical and physiological perspectives, and practical applications. To address both of these points, we have added the following paragraph at the start of the Introduction:
“Heart rate is an easy-to-measure physiological variable that can be used to characterise the intensity of exercise, both quantitatively and qualitatively using categories such as light, moderate and vigorous [10]. The ability to regulate heart rate during exercise using feedback control would therefore allow accurate prescription of training regimes in both clinical and non-clinical settings. Since feedback controllers require a model of the dynamic response of heart rate during exercise, it is important to first consider the fidelity of different model structures.”
New reference [10]:
10. Riebe D, Ehrman JK, Liguori G, Magal M, editors. ACSM’s guidelines for exercise testing and prescription. 10th ed. Philadelphia: Wolters Kluwer; 2018.

Minor issues:

I suggest providing a brief description of the study population in the abstract.

RESPONSE: Regrettably, the Abstract already has 298 words, with a limit of 300 words. But we have provided more detailed information on the study population in the main text as noted below.

If the Authors will meet the word limit, I strongly recommend enhancing the information about study group (their demographics, fitness level etc.) and testing protocols.

RESPONSE: Thank you for this suggestion. As mentioned in the text, this is a retrospective analysis of data from two previously-published studies. We pointed out that full information regarding the study groups and testing protocols can be found in the corresponding citations, viz. references [3] and [4], as follows:
“Full details of experimental procedures employed for data collection in the preceding treadmill and cycle ergometer investigations can be found in the respective publications. ³^, ⁴Essential elements of the protocols are summarised in this Brief Report.”
Nevertheless, we do agree it would be helpful to the reader to have more information in the current article, for which reason we have added more details to the Methods, as follows:
“For both exercise modalities, healthy, able-bodied participants exercised at moderate-to-vigorous intensity: in the treadmill analysis ³there were 11 participants (8 male, 3 female; overall mean age 32.5 years, mean body mass 75.5 kg, mean height 1.79 m); for the cycle ergometer ⁴there were 27 participants (20 male, 7 female; overall mean age 30.8 years, mean body mass 76.3 kg, mean height 1.79 m). Participants were required to be regular exercisers (30-min bouts, 3 times per week), non-smokers, and to be free of injury and illness.”

Did the Authors evaluate sample size to ensure credibility of their analysis and conclusions? There are only 38 participants in this study.

RESPONSE: Thank you for this important query. As noted above, the 38 participants came from two previous studies: a pilot treadmill study, [3], with 11 participants, and a cycle ergometer study, [4], with 27 participants.

For the treadmill study, the sample size of n = 11 was chosen in accordance with the requirements of pilot studies, and no formal statistical power analysis was conducted.

For the cycle ergometer study, an a priori statistical power and sample size estimate was included in the study protocol that was approved by our ethics committee. The details can be found in reference [4], as follows: “The sample size of n = 27 participants was estimated a priori by a statistical power calculation that used estimates of expected effect sizes and sample standard deviations obtained from previous studies in this lab, with the significance level of 5 % and a statistical power of 80 % (1 − β = 0.8).”

In both cases, no post-hoc statistical power calculation was performed because observed effect sizes and their uncertainty bounds supersede any a priori estimates, and because “Post-hoc power estimates … have been shown to be logically invalid and practically misleading”; quotation from:

Dziak JJ, Dierker LC, Abar B. The Interpretation of Statistical Power after the Data have been Gathered. Curr Psychol. 2020 Jun;39(3):870-877.
https://doi.org/10.1007/s12144-018-0018-1

View more View less

Competing Interests

No competing interests were disclosed.

Alongside their report, reviewers assign a status to the article:

Approved - the paper is scientifically sound in its current form and only minor, if any, improvements are suggested

Approved with reservations - A number of small changes, sometimes more significant revisions are required to address specific details and improve the papers academic merit.

Not approved - fundamental flaws in the paper seriously undermine the findings and conclusions

[1] 1. Riebe D, Ehrman JK, Liguori G, et al.: ACSM’s guidelines for exercise testing and prescription. 10th ed.Philadelphia: Wolters Kluwer; 2018.

[2] 2. Whipp BJ, Ward SA, Lamarra N, et al.: Parameters of ventilatory and gas exchange dynamics during exercise. J. Appl. Physiol. 1982; 52(6): 1506–1513. Publisher Full Text

[3] 3. Bearden SE, Moffat RJ: VO2 and heart rate kinetics in cycling: transitions from an elevated baseline. J. Appl. Physiol. 2001; 90(6): 2081–2087. PubMed Abstract | Publisher Full Text

[4] 4. Wang H, Hunt KJ: Identification of heart rate dynamics during treadmill exercise: comparison of first- and second-order models. Biomed. Eng. Online. 2021; 20(37): 10–37. PubMed Abstract | Publisher Full Text | Free Full Text

[5] 5. Spörri AH, Wang H, Hunt KJ: Heart rate dynamics identification and control in cycle ergometer exercise: comparison of first- and second-order performance. Front. Control Eng. 2022; 3: 894180. Publisher Full Text

[6] 6. Wang H, Hunt KJ: Feedback control of heart rate during treadmill exercise based on a two-phase response model. PLoS One. 2023; 18(10): e0292310. PubMed Abstract | Publisher Full Text | Free Full Text

[7] 7. Wang H, Hunt KJ: Identification of heart rate dynamics during treadmill exercise: comparison of first- and second-order models. [Dataset]. OLOS Repository. 2023. Publisher Full Text

[8] 8. Spörri AH, Wang H, Hunt KJ: Heart rate dynamics identification and control in cycle ergometer exercise: comparison of first- and second-order performance. [Dataset]. OLOS Repository. 2023. Publisher Full Text

[9] 9. Ljung L: System Identification: theory for the user. 2nd ed.Upper Saddle River, New Jersey, USA: Prentice Hall; 1998.

[10] 10. Hunt KJ, Grunder R, Zahnd A: Identification and comparison of heart-rate dynamics during cycle ergometer and treadmill exercise. PLoS One. 2019; 14(8): e0220826. PubMed Abstract | Publisher Full Text | Free Full Text

Identification of heart rate dynamics during treadmill and cycle ergometer exercise: the role of model zeros and dead time

Abstract

Background

Methods

Results

Conclusions

Keywords

Revised Amendments from Version 1

1. Introduction

(1)

2. Methods

2.1 Data collection

Figure 1. Heart rate measurements.

2.2 Model structures

(2)

Table 1. Model structures.

2.3 Parameter estimation and outcome measure

(3)

3. Results

Table 2. Mean model fit (normalised RMSE, Eq. (3), [%]).

Table 3. Model parameters for treadmill (TM) and cycle ergometer (CE).

4. Discussion

5. Conclusions

Ethical considerations

Authors’ contributions

Data availability

Software

Acknowledgements

References

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