Static models

In this module we explore a set of practical models that are widely used in sport science: the power-duration curve, the VO₂ response to a step in power, and the lactate curve after exercise.

They are all static models: fixed formulas that return a value for a given duration or for a given time since the start (or the end) of exercise. They have no memory of what happened before. That is enough for many questions, and it is the natural first step. Module 4 is about dynamic modelsM4, which can follow an input that keeps changing.

Modeling the relationship between intensity and sustainable effort duration

Quantifying how long an athlete can sustain a given intensity (power or speed) is fundamental for assessing performance capacity and prescribing training.
This relationship between intensity and sustainable duration is commonly expressed through the power-duration or speed-duration curve.
Several mathematical models have been developed to represent this curve, each with advantages in specific time domains (e.g., sprint, middle-distance, endurance).

In the power-time domain, Critical Power (CP) represents the horizontal asymptote of the curve.
It is often interpreted as the boundary between the heavy and severe exercise-intensity domains.
The heavy domain lies between the lactate threshold (the lowest intensity at which blood lactate rises above its resting level) and CP; the severe domain lies above CP.

The parameter W' represents the finite amount of work that can be performed above CP, often visualized as the area between the CP line and the power-duration curve.

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In the power vs. 1/time domain, the CP relationship becomes linear:

• Y-intercept: CP (power when duration approaches infinity)
• Slope: W'

In the figure below, data points from a 3-min all-out test are fitted with the CP model to estimate CP and W'.

The left panel shows the fit as a curve in the power-time domain; the right panel shows the same fit as a straight line in the power vs. 1/time domain, where you can read CP as the intercept and W' as the slope.

The Anaerobic Power Reserve (APR) model focuses on short-duration, high-intensity efforts, describing how performance declines from maximal peak power (MPP) to maximal aerobic power (MAP) over time.

Equation (power version):

Where:

• MAP = maximal aerobic power
• MPP = maximal peak power (e.g., best 10-second power)
• k = rate constant describing how quickly power declines toward MAP

This model is particularly suitable for efforts shorter than 300 s, such as sprints or high-intensity interval work.

The three-parameter critical power (3-PCP) model extends the classic CP framework by introducing a third parameter (τ), improving the fit for very short durations where curvature in the early part of the curve is more pronounced:

Compared with the classic CP model, $P(t) = W'/t + CP$, adding τ to the time keeps the predicted power finite as $t$ approaches zero.

The Omni-Domain Power-Duration (OmniPD) model is a flexible, data-driven formulation that fits performance data over the full range of durations, from sprints to ultra-endurance events.

It uses two equations:

For $t \le TCP_{MAX}$:

For $t > TCP_{MAX}$:

Where:

• CP, W', MPP retain their conventional meanings
• TCP_MAX = time at which the transition from the short-duration to the long-duration equation occurs
• A = parameter governing the slow decline beyond TCP_MAX

The OmniPD model does not rest on strict physiological assumptions, but it fits empirical power-duration data well across the whole range of effort durations.

The chart below compares the four models of this section (CP, 3-parameter CP, APR, and OmniPD) on a log time axis: use the tabs to change each model's parameters and the checkboxes to show or hide a model, and look at the durations where the models agree and where they diverge.

VO₂ kinetics (static version)

The one-exponential VO₂ kinetics model is a simple, widely used representation of how oxygen uptake (VO₂) responds to the onset of exercise.
In this formulation, the increase in $\mathrm{VO}_2$ from a baseline value ($A$) toward a higher steady-state level is described by a single exponential function.
The amplitude of the response ($\Delta$) reflects the total increase in oxygen consumption, while the time constant ($\tau$) represents how quickly the system approaches its new steady state.

The time constant has a simple meaning you can read off any exponential response: after one τ, the response has covered about 63% of the way to its new value; after three τ, about 95%. A small τ means a fast response, a large τ a slow one.

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Mathematically, the model is expressed as:

This static version assumes an immediate response at the onset of exercise and a single dominant time constant.
However, more sophisticated models exist that account for:

• Multiple components (e.g., primary and slow components) to describe oxygen uptake in prolonged or high-intensity exercise.
• Time delays between exercise onset and the measurable VO₂ response.
• Asymmetries between the on-transient (exercise onset) and the off-transient (recovery).

These extended models are particularly relevant when studying high-performance athletes, prolonged exercise, or transitions above the lactate threshold, where VO₂ kinetics may deviate from the idealized single-exponential shape.

In the chart below, move the τ slider until the model curve follows the data and the error (RSSE) is as small as possible. This is the same knob-turningM2 you did in Module 2, with τ as the knob.

Lactate kinetics (static version)

The two-pool lactate model describes how blood lactate concentration evolves over time following exercise.
It is based on the concept that lactate is distributed between two main compartments:

1. The muscle compartment, where lactate is primarily produced during high-intensity exercise.
2. The blood compartment, where lactate accumulates and is eventually removed.

In the static version of the model used here, lactate production during exercise is not modeled: the model starts at the end of exercise.
From there, blood lactate first rises, because lactate keeps moving from the muscle into the blood, and then falls, as lactate is removed from the blood faster than it arrives.

The equation below has this rise-then-fall shape. $k_1$ is the rate constant of lactate appearance in the blood (it shapes the rise), $k_2$ is the rate constant of lactate removal from the blood (it shapes the slower decline), $A$ is a scaling factor that sets the amplitude of the curve, and $\mathrm{BLC}_0$ is the baseline lactate concentration:

In the chart below, move the four sliders until the model curve follows both the rise and the fall of the data and the RMSE is as small as possible.

Wrap-up

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