Identification of heart rate dynamics during treadmill and cycle ergometer exercise: the role of model zeros and dead time [version 1; peer review: awaiting peer review]

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 rootmean-
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 firstorder
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.