Mathematical models

A model is an informative and simplified representation of a system, aimed at understanding, communicating, explaining, or designing aspects of interest of that system. In sports science, mathematical models are used for:

• Analyzing athlete/human performance in exercise
• Evaluating training effects
• Predicting training effects or, more generally, performance
• Optimizing training protocols, training effects, and/or exercise performance

Today, models are everywhere. They estimate our VO₂max (maximal oxygen uptake), our calories, our steps, our sleep quality, and many other metrics. The models inside your smartwatch, fitness app, or training software all share a few common elements, which we will discover throughout this course.

🥱 Boring stuff first: essential definitions

A helpful way to think about a model is like a shipping service: you provide an input, and it delivers an output. You give something in to the model, and you get something out.

All models share a few essential components:

• Variables: symbols used to represent elements that can vary. These are the quantities that change over time or between scenarios.

• Parameters: symbols used to represent fixed values within a given context. These do not vary during a specific application of the model, although they can be adjusted from one application to another.

• Input variable: typically something easy to measure, obtain, or control. We usually denote this with x.

• Output variable: usually the goal of the model, a quantity that is difficult to measure directly or that we want to predict. This is typically represented with y.

• Relationships (governing equations): mathematical expressions that link x to y in a well-defined (univocal) way, meaning that each input corresponds to a single output.

The easiest of the examples

I don't want you to get lost in details, semantics, or complex explanations that don't make sense just yet.
We'll introduce more terminology and formal language later. For now, let's start with something that everyone can relate to: an example so simple that it makes sense immediately.

Let's consider this: maximum heart rate declines with age. How?
A widely known model suggests that maximum heart rate decreases by 1 beat per minute (bpm) for every year of age.
It also proposes that, for any given age, the maximum heart rate can be estimated using the formula:

This is a model that takes age as input and returns the predicted maximum heart rate as output.

In this case, the model is linear, as it follows the general form of a straight-line equation:

Here we have two parameters:

• b = 220 is the intercept (in units of bpm),
• a = -1 is the slope (in units of bpm/year).

As we'll discuss later in the course, this model may not be appropriate for every population, age group, or gender.
That's why we often adjust the parameters to better fit specific data or populations.
When we change the values of a and b, the model's predictions will change too.

Try it yourself!
Use the sliders to manually adjust the parameters and observe how the predicted maximum heart rate changes.

Notice that the input never changes (a 50-year-old), yet the predicted HR_max moves with every slider.

Comes back inModule 2 · Model example: the HRmax = 220 − age model

This example should help you understand that behind any model, there is a structure dictated by its underlying equation.
Once that structure is defined, the values of the parameters can still strongly influence the model's response.
In this case, the model was linear, and the two parameters were the slope and intercept of the line described by the single equation.
You will learn throughout this course that this is far from the only structure available, but it is the most common and the simplest to understand.

🪄 The process of “creating” a model

Modeling is a structured process that starts with observation of a phenomenon in the real world, such as acute or chronic physiological adaptations during exercise. From there, we move into conceptualization, where we ask: What kind of process or relationship do I want to describe? This leads to formulation, where we select the appropriate mathematical tools (e.g., differential equationsM4, regression models, neural networksM5) to express the concept rigorously.

Next comes identification, where we fit the model to experimental data. In this course, identification, calibration, fitting and training all mean the same thing (in Module 2 you will also find it called parameter estimation): finding the parameter values that make the model match the data. We then need verification, which asks: Am I solving the equations correctly? In other words, does my implementation (a spreadsheet, a piece of code) really compute what the equations say?

Then we proceed to validation, which asks: Am I solving the correct equations? Here we test whether the model's predictions agree with data that we did not use during the identification process.

Finally, accreditation answers the practical question: Can I use this model in everyday practice? A model is only useful if it performs well enough to support decisions or improve understanding in a real-world context.

Comes back inModule 5

Model example: muscle force-length relationship

Let's consider a classic example in biomechanics: how torque generation ability during an isometric leg extension exercise varies with knee angle. Torque is primarily generated by the knee extensor muscles (like the quadriceps), and their force production depends heavily on muscle length, which is determined by joint angle.

We often observe that torque is maximal at a specific "optimal" knee angle, and decreases when the joint is either too extended or too flexed. This behavior reflects a fundamental physiological principle: if a muscle operates at a length different from its optimal length, its ability to generate force is reduced.

This phenomenon, known as the force-length relationship, is nonlinear (a nonlinear model is one whose graph is not a straight line, like the parabola below), and a simple way to describe it mathematically is with a parabolic (quadratic) function.

A quadratic function is symmetric and can model a peak, so it suits this context.

where x is the knee angle (in degrees), y is the torque generated at that angle, and a, b, and c are parameters to be identified from the data.

To "learn" or "find" the values of the parameters (i.e., to calibrate the model), I collect torque data at a few selected knee angles. Using these measurements, I fit the parameters a, b, and c. A fitting process consists of finding the parameters that minimize the distance between the experimental data and the model. This distance is typically expressed as the root mean squared error (RMSEM2).

Once the model is calibrated, we can find the knee angle that maximizes torque.
This corresponds to the vertex of the parabola, the point where torque is highest:

At this angle x*, the torque reaches its maximum value.

In summary: a simple quadratic model can capture the main features of the force-length relationship in leg extension.
It lets you make informed decisions from fewer measurements and estimate quantities such as the optimal angle for torque production.

Try it yourself!
Use the sliders to manually adjust the parameters and observe how they affect the fit of the model. Watch how the RMSE changes as you move them. I bet you cannot find a better combination of parameters than the one I initially provided. How do I know? I did not fit the model manually, but automatically, by means of an optimization process. We will come back to this concept in the next module (how an optimizer turns the knobsM2).

Model example: muscle force-velocity relationship

When modeling muscle function, another key relationship to consider is how muscle velocity affects force generation.

▸ Observation: There is a clear relationship between how fast a muscle shortens (its contraction velocity) and how much force it can produce.
▸ Concept: The faster the contraction, the lower the force the muscle can express. This is particularly true for concentric (shortening) actions.
▸ Formulation: To capture this inverse relationship, we use the Hill equation, a classic model that resembles a hyperbolic curve.
▸ Identification & validation: Once data are collected (e.g., from isokinetic tests), we can calibrate the parameters and validate the model against observations.

where $F$ is the force produced by the muscle, $v_M$ is its shortening (contraction) velocity, ${F_0}^M$ is the maximal isometric force (the force at zero velocity), and $a$ (a force) and $b$ (a velocity) are the parameters that shape the hyperbola.

Solving for $F$ gives:

One useful performance indicator is the maximum shortening velocity of the muscle, $v_{MAX}$. We can estimate it by substituting $F = 0$ into the Hill model:

In practice, during isokinetic leg extension exercises, we typically test at different angular velocities (e.g., 90°/s, 180°/s).
If you have already conducted isometric tests, you can use the peak torque estimated from the quadratic model (explained previously) to provide ${F_0}^M$, the maximal isometric force.

Since power is force times velocity, the power-velocity relationship becomes:

The charts below use joint torque and angular velocity in place of muscle force and shortening velocity: the curve has the same hyperbolic shape. The first chart shows torque data at different angular velocities, the fitted Hill curve, and the estimated $v_{MAX}$. The second chart shows the power-velocity curve derived from the same fit.

Torque falls steadily as velocity increases, while power first rises and then falls: it peaks at an intermediate velocity, well below $v_{MAX}$.

Model example: VO₂max estimation and doping identification

▸ Observation: Professional cyclists sometimes climb at extraordinary speeds, raising questions about the physiological plausibility of such performances.
▸ Concept: We want to build a model that estimates VO₂max based on climbing performance (i.e., vertical ascent speed).
▸ Formulation: We apply a simplified version of the cyclist's equation of motion, focusing on the power needed to overcome gravity during a climb.

In professional cycling, climbing high mountain passes at competitive speeds requires exceptional physiological capacity: power outputs often exceeding 6 W/kg sustained for 40+ minutes.

▸ What kind of model can we use to estimate the required VO₂max?

▸ We make the simplifying assumption that most of the required power output is used to gain elevation (i.e., to increase gravitational potential energy).
▸ This gives us a basic mechanical model:

where $m$ is the mass of rider and bike (kg), $g$ is the gravitational acceleration (9.81 m/s²), $h$ is the elevation gain of the climb (m), and $\Delta t$ is the time taken to climb it (s).

This formula tells us that power $P$ is inversely proportional to the time taken to climb.
The faster the ascent (i.e., the shorter the time $\Delta t$), the higher the required power output. In Module 4 you will meet the full version of this model, the equation of motion of a cyclistM4, which also accounts for air drag and rolling resistance.

In the estimator below, enter the rider and bike mass and the distance, elevation gain, and time of the climb. The tool computes the required power, converts it into oxygen uptake using an efficiency value and a basal VO₂, and then estimates VO₂max from the fraction of VO₂max the rider is assumed to sustain.

The faster the climb, the higher the VO₂max the rider needs, and the higher the star sits on the chart. When the star lands in the red zone (a VO₂max above 95 mL/kg/min, or an efficiency outside 22–24.5%), the estimate falls outside the range measured in professional cyclists and the performance is physiologically implausible: either the inputs are wrong or the performance deserves a closer look. This is the reasoning behind using such estimates in discussions about doping in cycling.

Wrap-up

Well done, you have completed the first module of this course.