Supplementary information to:
Calculated Performance: Using Quantitative Models to Optimize Your Training
Triathlete Magazine, January 2009, p.119
by Dave Clarke and Michael Ricci

Dave Clarke is a M.Sc. in kinesiology, a Ph.D. in biological engineering and a top-tier age-group triathlete.

Michael Ricci is a Level 3 USAT coach and head coach and founder of D3 Multisport based in Boulder, Colorado.

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The scientific basis of the impulse-response models

In the mid-1970s, Eric Banister and his colleagues at Simon Fraser University in Vancouver, Canada pioneered quantitative studies seeking to model the relationship between training and performance. A number of studies have also been conducted by researchers in France. The models used in these studies are typically referred to as “impulse-response models”, whereby training doses are the impulse and performance is the response. A version of this model is used in PhysFarm’s Raceday™ software. A simpler modified model called the Performance Manager™ (PMC), developed by exercise scientist Dr. Andrew Coggan, is a tool in the TrainingPeaks WKO+ software. We discuss both the impulse-response model and the PMC in this article.

Most models of training and performance can be expressed in general terms as

Performance = Fitness – Fatigue

(or in the words of Dr. Coggan, Form = Fitness + Freshness). Fitness and fatigue are calculated from the individual training doses comprising all the workouts in the training block. One form of the impulse-response model equation can be written as follows:

where p(t) = performance on day t, po = initial performance level, wi = training dose on day i, k1 = weighting factor for fitness, k2 = weighting factor for fatigue, T1 = decay constant for fitness, T2 = decay constant for fatigue. The inputs to the model are the initial performance level and the training doses from each day during the training period. The effect of the exponential terms is to reduce the influence of the training doses that were performed at longer times prior to the day for which performance is predicted. In other words, a given training dose performed closer to the day of performance will more strongly influence the predicted performance than training performed earlier.

The impulse-response model has four parameters (the two weighting factors, k1 and k2, and the decay constants, T1 and T2) whose values must be determined for each individual. These parameters are determined by “fitting” the model to training and performance data. The parameter values that allow the model to most accurately fit the performance data as a function of the training data are used. Calibrating the model from one’s performance and training data ensures that the model is specific to the individual. Including all four parameters allows the model to predict performance in absolute terms (e.g., the wattage that one could hold on the bike in a given race). However, the downside is that a large amount of data is required to reliably determine the parameters. The onus is therefore on the user to test his/herself regularly throughout the training period to keep the model performing well.

To overcome the parameter estimation obstacle, Dr. Coggan developed the PMC, which can be thought of as a simpler but less powerful version of the full impulse-response model. Specifically, Dr. Coggan eliminated the two gain parameters (k1 and k2) and replaced the exponentially weighted sums with simpler exponentially weighted moving averages of the daily training dose (Coggan, [date unknown]). He also incorporated default values for the two time constants (T1 = 42 days and T2 = 7 days) (Coggan, [date unknown]), such that no parameter estimation is required to use the PMC, although the user can change these if desired. The PMC does, however, depend on an accurate estimate of functional threshold power (FTP, defined as the maximum power that a rider can sustain for an hour without fatiguing; FTP corresponds to the lactate threshold). If one races frequently, there are means to analyze the power data to estimate this value. Otherwise, testing is required. In the PMC, the corresponding fitness term is called the “chronic training load” (CTL) and the fatigue term is the “acute training load” (ATL). The difference between the CTL and ATL is called the “training stress balance” ( TSB ), which is an indicator of freshness. The output of the model is “form”, which is defined as some combination of CTL and TSB that must be empirically determined on an individual basis.

Both models have advantages and limitations. The beauty of the PMC is its simplicity, but its biggest downfall is that performance is not predicted per se and it can be difficult to interpret the model outputs without previous data or experience as guidance. Conversely, the full impulse-response model can predict performance in absolute terms, but the user must measure performance on a frequent (i.e., weekly) basis. In both cases, diligent downloading of power meter or GPS data or tedious manual quantification of training data is required for either model to perform well.

More on quantifying training dose

The input to these models is training dose. Several means are available to quantify training dose, including the “training impulse” (TRIMP) and the “training stress score™” (TSS). TRIMP is a heart-rate-based method that is calculated using the following formula from the time course of heart rate (HR) data during a training session (Morton et al., 1990):

where Duration = duration of the training session, HRex = average HR during the session, HRrest = resting HR, HRmax = maximum HR, and b = correction factor (equals 1.92 for men and 1.67 for women). The exponential term is intended to correct for training done at higher intensity, because without the term exercise of long duration at slow pace would be over-emphasized.

A major advantage of using the TRIMP is that heart rate monitors automatically record HR data and calculates the average HR for each training session. Unfortunately, heart rate varies non-linearly with exercise intensity (in particular at very high intensities) and it depends on factors other than just exercise intensity. For example, performing a given workout at two different temperatures could confer different TRIMP scores. Power meters measure the true rate of work (Allan and Coggan, 2006), such that the training dose can be precisely quantified. The metrics necessary for calculating TSS include normalized power (NP™), FTP, and intensity factor (IF™) (Allan and Coggan, 2006).

The average power of a ride is an indicator of the ride’s intensity but it does not reflect how metabolic stress varies with exercise intensity. Specifically, the accumulation of lactate is a nonlinear function of power output, with increased accumulation occurring at higher power outputs. Therefore, training dose is not adequately assessed by average power. Instead, the NP metric is used. NP is calculated from power data as follows: 1) the power over each 30 second blocks during the training session is averaged, 2) the averages are raised to the 4th power (lactate vs. power output varies approximately with the 4th power), and 3) the 4th root of the grand average of these values is then calculated to obtain NP for the session (Allan and Coggan, 2006).

As previously mentioned, FTP is akin to the lactate threshold intensity and is estimated by analyzing power data or through testing. IF is the ratio between NP and FTP, which expresses the training session’s intensity as a percentage of one’s capability (Allan and Coggan, 2006). TSS is then calculated as follows:

TSS = IF2 x Duration x 100

where Duration is expressed in hours. TSS is a useful and intuitive metric for quantifying training dose. A TSS training session of 100 points corresponds to a session that lasts 1 hour and is carried out at FTP (Allan and Coggan, 2006). Training session doses are therefore expressed relative to this score.

While TSS was originally created for use with bicycle power meter data, calculation schemes to assess stress scores for swimming and running have emerged. For example, TrainingPeaks has the rTSS (run TSS) and Raceday software incorporates the SwimScore™, BikeScore™, and RunScore™ metrics. For both Raceday and TrainingPeaks software, GPS data files can also be downloaded and analyzed, making run training session quantification easy.

While the power-based metrics are theoretically superior to heart-rate-based metrics, none of the power-based metrics have been validated in peer-reviewed scientific studies. Objective and independent scientific validation of these metrics is required in order to fully appreciate their benefits and limitations. Despite this, the TSS has enjoyed widespread acceptance in the cycling and triathlon communities, suggesting that users find the metrics worthwhile.

* Performance Manager, training stress score, normalized power, and intensity factor are trademarks of Peaksware, LLC
* Raceday, SwimScore, BikeScore, and RunScore are trademarks of PhysFarm Training Systems, LLC.

References

Allan, H., Coggan, A.R. (2006). Training and Racing with a Power Meter. Boulder, CO: Velo Press.

Coggan, A.R. The scientific inspiration for the Performance Manager. http://www.cyclingpeakssoftware.com/power411/performancemanagerscience.asp. Accessed July 30, 2008.

Banister, E.W., Carter, J.B., Zarkadas, P.C. (1999). Training theory and taper: validation in triathlon athletes. European Journal of Applied Physiology and Occupational Physiology. 79: 182-191.

Morton, R.H., Fitz-Clarke, J.R., Banister, E.W. (1990). Modeling human performance in running. Journal of Applied Physiology. 69: 1171-1177.

Taha, T., Thomas, S.G. (2003). Systems modelling of the relationship between training and performance. Sports Medicine. 33: 1061-1073.

Thomas, L., Mujika, I., Busso, T. (2008). A model study of optimal training reduction during pre-event taper in elite swimmers. Journal of Sports Sciences. 26: 643-52.