Static models

In this module we explore a set of practical applications and models that are widely spread in the sport science world.

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

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 Anaerobic Power Reserve model focuses on short-duration, high-intensity efforts, describing how performance declines from maximal sprint 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 < 300 s, such as sprints or high-intensity interval work.

The 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.

The Omni-Domain Power–Duration model is a flexible, data-driven formulation capable of fitting 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 ODP model does not require strict physiological assumptions, but it consistently provides an excellent fit to empirical power–duration data across the full spectrum of effort durations.

VO₂ Kinetics (static version)

The one-exponential VO₂ kinetic model is a simple yet widely used representation of how oxygen uptake 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.

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, slow, and very slow components) to capture the complexity of oxygen uptake in prolonged or high-intensity exercise.
• Time delays between exercise onset and the measurable VO₂ response.
• Asymmetries between on-transients and off-transients during 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.

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 is no longer considered, and only the clearance phase is modeled.
This allows us to focus on the post-exercise decline in blood lactate concentration, which reflects the balance between:

• Fast clearance from the blood to oxidative tissues.
• Slower clearance from deeper compartments or slower metabolic pathways.

The equation below represents this biphasic decay, with $k_1$ and $k_2$ describing the respective clearance rate constants, $A$ as a scaling factor, and $\mathrm{BLC}_0$ as the baseline lactate concentration:

Supercompensation Model (static version)

The supercompensation principle describes how the body adapts to training stimuli over time.
After a training session, performance capacity temporarily decreases due to fatigue.
During recovery, the body not only returns to its baseline state but may exceed it, resulting in supercompensation — an improved capacity compared to pre-training levels.

If a new training stimulus is applied at the right time, the adaptation curve can progressively shift upward, leading to long-term performance improvements.
Conversely, if training is too frequent (insufficient recovery) or too infrequent (loss of adaptation), performance gains are reduced or absent.

Mathematically, this concept is often described using the Banister model (also known as the fitness–fatigue model).
In this model:

• Fitness represents the positive adaptations to training.
• Fatigue represents the negative, transient effects of training load.
• Performance is modeled as the net effect of fitness minus fatigue, each with its own time constant and sensitivity.

To replicate this on your spreadsheets you can use the following equations. For a given training load (TL), we generically refer to CTL as the chronic training load (or fitness, i.e. the positive and long-lasting effects of training) and to ATL as the acute training load (or fatigue, i.e. the short-term effects of training that are negative for the performance). With TSB we refer to the training stress balance (or form, i.e. the balance between the positive and negative effects of training on the performance potential).

In the next module you will learn about dynamics models, and therefore you will be able to use the super-compensation model with much more flexibility. To compute the TSB you can combine these equations:

The parameters $\tau_1$, $\tau_2$, $k_1$, and $k_2$ have been empirically defined. In some case you will find that $\tau_1=42$ days and $\tau_2=7$ days, and $k_1=2$ and $k_2=1$ (it depends on the software which implements the model).

Checkout

Well done, you were able to complete the third module of this course.