Assessment of physical tness of endurance athletes is usually performed by means of standardized exercise protocols in specialized laboratories. The recent development of devices measuring heart rate, power output, speed, etc. raises the possibility to assess the tness level from data collected on the eld. We propose a model based on cardiac parameters identi cation on training activities. Identi ed parameters prove to allow for heart rate simulations that match measurements with an average root mean square error of 4 beats/min for cycling activities for which power output and heart rate were provided and 6 beats/min for running activities for which the heart rate was provided and power output was estimated based on global positioning system (GPS) tracking.
Improvement of sport performance is dependent on physiological adaptations
amongst others. These training adaptations are of particular importance for
endurance athletes like cyclists and runners. Training optimization requires
knowledge about how humans adapt to workout sessions. This knowledge can be seen
as a model linking training workout characteristics to tness level. Such system
modeling approaches have been re ned for at least four decades [
The athlete adaptation model takes as inputs quanti able training
workload w(t) to predict the tness level f l(t) evolution over time as represented in
the upper part of Fig. 1. Building and evaluating such a model requires
inputoutput instances. Inputs are easy to gather thanks to the emerging tendency of
athletes to log all their activities on a server by using their smartphone, sport
watch or bike computer. On the other hand, endurance tness is multifactorial
(cardiovascular, metabolic, endocrine, ...) and quantitative metrics are not
directly available. Currently, the evaluation of the endurance tness is assessed by
incremental exercise protocols performed in specialized laboratories[
This work aims to provide a methodology that will help inferring the tness level from workout sessions data themselves. For this purpose, a parametric heart rate model is proposed (see bottom part of Fig. 1).
The main principle is that a lower heart rate observed for a given intensity
of exercise indicates a better endurance tness level [
The validity of the proposed heart rate model will be assessed by simulation with cyclists' data for which we have instant power output and heart rate measurements. The model will then be re-used with runners' activities. For the latter, the power output needs to be estimated as it is not directly measured during the run. 2
This paper proposes a parametric heart rate model that describes the relationship between an athlete instant power output po(t) and his heart rate hr(t) as it is illustrated in Fig. 2. The model parameters can be identi ed to best reproduce heart rate measurements on a single activity. 2.1
Steady state The steady state heart rate HRss refers to the heart rate that
is reached after stabilisation at constant power output P O. The relationship
between steady-state heart rate and power output is athlete-dependent and is
known to be very close to linear as long as the heart rate is below its maximum
value called maximum heart rate HRmax [
P O; if HRrest + m P O < HRmax
otherwise:
Transient Response It appears from exercise laboratory measurements on ergometers (a segment of measurement is illustrate in Fig. 3) that a power step upward leads to a new steady state heart rate that is reached after a few seconds. The exponential-looking shape of the heart rate curve in response to power steps suggests this phenomenon can be roughly described by a rst order di erential equation : dHR(t) dt + 1
HR(t) = P O(t) r with r an athlete-speci c time constant. It will be shown below that this assumption allows for accurate simulations.
In a time frame [t0; t] where power output is constant, the solution of this equation is given by HR(t) = HR(t0) + (HRss(P O(t)) t HR(t0))e r : In the discrete over-sampled time domain, an iterative form given by 1 r HR(t + 1) = HR(t) + (HRss(P O(t))
HR(t)) can be used.
As there is no reason to assume equality between rise time r and fall time f . The equation is allowed to di er for increasing and decreasing heart rate and becomes HR(t+1) = (HR(t) + 1 (HRss(po(t))
r HR(t) + 1 (HRss(po(t)) f
HR(t)); if HRss(po(t))
HR(t) HR(t)); if HRss(po(t)) < HR(t):
The heart rate transient response is thus captured by two athlete-time speci c parameters which are r and f . 2.2
Running activities are provided as timestamped geolocalized points with elevation that was corrected using elevation maps. The elevation can be derived with respect to the curvilinear horizontal distance to get the slope. The runner velocity is also derived from locations and timestamp. Both are smoothed using factors that were chosen to give best simulation accuracy. It is assumed that the smoothing factors are not athlete- or activity- speci c.
Minetti et al [
P O(t) = EC(slope):v(t) in [w/kg].
Cardiovascular Drift Intra-session workload results in fatigue that induces
increased heart rate for the same power output[
P O(t)dt Z t t0 with kf being the athlete's sensitivity to fatigue. This intuitive formulation might not be accurate but proved to help the heart rate model to better t activities measurements.
The model that was described contains three parameters that account for the
steady state relationship (HRrest [bpm],HRmax [bpm] and m [bpm/watt]); two
that account for heart rate transient response ( r and f [s]); and one that
accounts for the athlete's sensitivity to fatigue (kf coe cient). Those parameters
are identi ed on an activity from which heart rate HR(t) and power output
P O(t) are known or estimated. Given the power output P O(t), a set of cardiac
parameters can result in a simulation H\R(t) that can be compared to the heart
rate measurement HR(t). The six parameters are tuned to minimize the mean
square error between simulation and measurement using a non-linear
optimization algorithm known as Nelder-Mead method [
In most modern activity tracking platforms (Nike+, Runkeeper, Strava, Garmin Connect, Endomondo, TrainingPeaks, ...), activities are shared as tuples of geolocalized points associated with timestamps recorded by athletes with their device. They can potentially record more parameters like heart rate, cycling power, cadence, accelerations, temperature or baro-metric pressure.
The cardiac model was rst tted to 72 cycling activities of three cyclists containing instant power output and heart rate measures sampled every second. The power was measured with a torque meter in the crankshaft of their bikes.
The same was done with 234 running activities of two runners containing heart rate and geolocalized points. Instant power output was estimated based on based on global positioning system (GPS) tracking coordinates. 3
Cardiac parameters that were identi ed on activities with the described methodology enable accurate heart rate simulation on the same activity taking solely the instant power measure or estimation as input. Heart rate simulations di er from heart rate measurements with an average root mean square error of 4 bpm for the 72 cycling activities recorded with the use of a power meter. On average, a root mean square error of 6 bpm was observed for the 234 running activities.
Fig. 4 and 5 show respectively cycling and running activities simulation examples with measurement curves that were used.
In most activities, identi ed heart rate rising time constant r was found to be smaller than heart rate falling time constant f ; with respective average values of 24 and 30 seconds. Sensitivity to fatigue was modeled with an additional power output in the range [1; 6]:10 5 [w/J]. Steady state heart rate parameters were more subject to intra- and inter-athlete variability. Resting heart rate HRrest was found to be in the range [60; 100] [bpm] that does not necessarily correspond to the conventional resting heart rate that is taken on a person laying down. The slope coe cient m was found in the range [0:15; 0:45] [bpm/w].
Although parameters identi cation proved to result in accurate simulations,
their variability seems higher than what is expected from real cardiac parameters.
The parameter variability over di erent activities taking place at di erent times
during the year can be imputed to the athlete state (which is of interest) but
other factors can be invoked:
1. The day-to-day heart rate variability that is believed to be around 2-4 bpm
according to [
How cardiac parameters are exactly linked to tness levels or performances is not considered in this work. We rely on the fact that part of them are measured by coaches during standardized exercise in laboratory for the periodic follow-up of their athletes. 4
We identi ed a pressing need for objective tness assessments based on data acquired on the eld in training situations, in order to better understand human adaptation to physical training and to enable adaptive training plans. We propose a framework that provides the basis of a potential solution through cardiac parameters identi cation. Identi ed parameters prove to be su cient for athletes heart rate simulation based on athletes power output for running and cycling activities.
In further research, parameters accuracy can be improved by including exogenous meteorological information in the heart rate model. Fixing parameters that are easy to obtain, like resting heart rate, might also help accuracy of the other parameters.
The natural continuation of this work would compare identi ed parameters to exercise laboratory measurements, or even more interesting, to athletes target performances such as race times.