To design an optimal training program we must understand the how training volumes and intensity result in improved performance and how to that performance can be increased while limiting the risk of illness and injury. Models have been developed the mathematically describe this relationship, and their feedback assists in program creation, improve taper strategies and adds helps to describe to periodization.

**Banister’s Impulse-Response Model**

By far the most popular approach is the Impulse-Response model, first proposed by Dr. Eric W. Banister in 1975. It tries to model athletic performance by comparing the positive long-term effects of training (“Fitness”) versus the negative short-term effects of training (“Fatigue”).

The model is described by the following equation:

Where an athlete’s performance at any time *pt *is given by their initial performance *p0* and the contributions of both the positive long-term effects and negative short-term effects of their training since that initial time. Those effects are given by the sum of the daily training loads (*ws*) coupled with terms that describe the rate of decay of those effects over time: *τ1* and *τ2*. Finally, those effects are translated into magnitudes of performance through the gain terms *k1 *and *k2*.

The limitations of looking at performance this way are firstly that it’s essentially a “black box” around physiological processes. It assumes a single number can represent adaption to performance when the reality is that many factors play a role. Furthermore, it assumes there is no upper limit to performance, and then more training eventually leads to increased performance. Finally, the constants used in the model are specific to each individual athlete and can vary between sports.

**Variations of the Impulse-Response model**

Banister’s work has been the inspiration for several models used to monitor training. These models differ on a few points:

**Rolling Averages vs. EWMA**

Rather than compute the complex integral terms for the long-term (chronic) and short-term (acute) effects of training in Banister’s model, other easier to calculate options are used, typically either simple rolling averages or exponentially weighted moving averages (EWMA).

Rolling averages take the average of the training loads of the past X number of days. Typically 4-6 weeks is used for the chronic window, and something in the range of 1-week is used for the acute. The limitation of rolling averages is that they assume there is no decay in training effects in time during the length of the window, followed by an instant 100% decay.

Exponentially weighted moving averages, however, include time decay constants so that recent training has more impact than that done after several days. These constants are typically 42-50 days for the chronic and 7-10 days for the acute.

**Ratio vs. Difference**

When comparing the chronic and the acute effects of training, some models look at a ratio of the training loads (typically: Acute/Chronic) while some take the difference between the two values (typically: Chronic – Acute).

We’ll review two implementations that use different combinations of these options.

**ACWR: Acute-Chronic Workload Ratio**

This model uses rolling averages for the short-term ‘Fatigue’ and long-term ‘Fitness’ aspects of training and compares them using a ratio. 1-week and 4-week windows are used for the acute and chronic portions of the ratio, respectively. Here’s an example using arbitrary units (AU) for training load:

Week 1: 700 AU

Week 2: 400 AU

Week 3: 700 AU

Week 4: 300 AU

Acute Training Load = 700 AU / 7 = 100 AU/day

Chronic Training Load = (700 AU + 400 AU + 700 AU + 300 AU)/(4*7) = 75 AU/day

**ACWR** = 100 / 75 **= 1.33**

With this model, **0.80 – 1.30 **is considered the training “safe zone” while ratios in excess of **1.50** put the athlete at an increased injury risk, but these values will change depending on the time constants and training load metrics used to “tune” the model.

For more information on the ACWR, read **this article **by Ryan White.

**PMC: Performance Management Chart**

Dr. Andrew Coggan, PHD, created a very popular variation of the impulse-response model with a cycling focus in mind. Coggan set the gain constants *k1* and *k2* equal to 1, making the changes in Chronic and Acute Training Load (CTL and ATL respectively) track relative changes in performance rather than attempting to be absolute predictors. CTL and ATL are calculated using exponentially weighted moving averages, with time constants of 42 and 7 as suggested defaults when applied to cycling. These same parameters have been adopted by other endurance sports and used to great effect as well.

Coggan defines a Training Stress Balance (TSB) as a difference between the Chronic and Acute Training Loads, and this whole concept is known as the Performance Management Chart or PMC for short. It’s been utilized by many endurance training applications, and it’s described in greater detail **in this blog post**.

The PMC is the first variation of the impulse response model used by Bereda’s Annual Planning product.

**Training Load Options**

All these models can be fed different training load metrics. Sometimes, they take multiple training load metrics at once.

When Banister introduced the impulse-response model, he also developed the Training Impulse Score or **TRIMP** for short which was a heart rate based training load metric. TRIMP, like many Training Load metrics, takes both training duration and intensity into account when “scoring” a training session, and as such TRIMP are accumulated more quickly with increased heart rate during exercise.

Other metrics can be broken down into:

**External Loads**

- Distance Covered
- Number of sprints
- Time spent running in soccer

**Internal Loads**

- Heart Rate
- RPE
- Blood Lactate Concentrations

When developing the PMC, Andrew Coggan also developed a power based metric for cycling, and that has spawned the development of many other power-based metrics ever since.