Modelling Heart Rate and Power Output

This is the first part of what will be a three part blog about modelling heart rate and power output. The three blogs together will try to follow the structure of a scientific article:

  • Part I (introduction). A general overview of the background.
  • Part II (methodology). A technical deep-dive into data analysis for this problem
  • Part III (results, conclusion and discussion in one). An overview and a discussion of the collected results.


In high performance endurance sports it is imperative that your aerobic fitness is being monitored. In general, this is done by doing tests: e.g. a VO2max test, 20 minute test or time trial. The main drawbacks of these options is that they interfere with your training regime, are not very time efficient and only offer a snapshot of your fitness. The goal of these three blogs is to discuss a novel way to monitor fitness and progress without the aforementioned drawbacks.

A novel way to monitor fitness and progress

The alternative for monitoring fitness and progress that will be discussed here is to build a model of the relationship between heart rate and power output. To some extent, heart rate can be estimated from the power output and vice versa. In the most simplest form, this relationship looks like this:

When there is no power output, heart rate already is at a base level which we will call the intercept. When increasing power, the heart rate rises in a linear relationship with power (1). We will call the amount the heart rate rises per extra Watt power output the slope. From a certain point, the rise in heart rate per Watt (the slope) will decreases with respect to power, this inflection point will occur around the threshold of the cyclist but that is outside the scope of this blog. Up to the threshold the relationship between heart rate and power can be described using a linear model:

heart rate = intercept + (power * slope)

It is beyond the scope of this blog to explain in detail why this linear relationship exists, but it comes down to the following. During aerobic exercise, an increase in power must be covered by an increase in oxygen consumption (causal relationship A), which results in an increase in heart rate (causal relationship B). Because both A and B can be considered as linear relationships, the relationship between heart rate and power output during aerobic exercise can be considered linear.

If a model for this relationship is calculated for every workout and is monitored over time, it might be possible to see trends in the intercept and slope parameters of this model. An example for monitoring changes in the parameters of this model is that a long term decrease of intercept could be interpreted as an increase in fitness. This means that a gradually decreasing intercept over the course of months can be a sign of improved fitness.


Besides the benefits there are of course some limitations to this approach. The first is that although this model is good representation of aerobic fitness, it doesn’t say much about anaerobic fitness because of a lack of linearity between heart rate and power above threshold. Other testing methods like a VO2max test or a power profile test need to be used to test your anaerobic fitness.

The second drawback is that this model needs enough steady state data points to be reliable. In our next blog it will be shown why this is necessary and that this can be a challenge.


This model makes it possible to analyse normal training rides, given that there is power and heart rate data available. The open source application Golden Cheetah makes this analysis possible in its HrPw plot in the Activities view. It looks like this:

All data points are plotted from the start (yellow) of the activity to the end (red). Power is on the horizontal axis, heart rate on the vertical axis. The light blue diagonal line is the calculated linear relationship between power and heart rate. If you look very closely, there is some text at the bottom of the plot:

This plot shows the formula of this model:

Slope: 0.211
Intercept: 103.4

So, heart rate can be estimated using the formula:

heart rate = 103.4 + (power * 0.211)

It also shows the estimated power output at heart rate 150 bpm: 220W. Nice, this is exactly the data we wanted to see!


In conclusion, you can see that at least in theory the linear relationship between power output and heart rate can be used to quantify aerobic fitness. In our next blog some of the challenges that come with calculating this model will be covered and an alternative algorithm to the one used in Golden Cheetah will be presented. After that, another blog about this topic will be published where it will be shown how to use this model in high performance training using a case study with real world data of a competitive cyclist. Stay tuned: Follow us here on Medium, Twitter and Facebook.

  1. The relation between power output, oxygen uptake and heart rate in male athletes. FJ Arts et al. Int J Sports Med. 1994 Jul;15(5):228–31