Engineers use mathematical models to design and optimize the performance of complex systems such as chemical plants and airplanes. Similarly, exercise scientists have developed mathematical models to optimize sport performance. While these models used to suffer from limited practicality outside laboratory studies, the advent of the power meter for cycling has spurred the development of new metrics of training dose, modifications to published models, and excellent software that renders the power of these models accessible to any triathlete. We believe that such models will eventually serve as a cornerstone tool for coaches and we seek to expose these models to the triathlon community by discussing their use from scientific and practical standpoints.
The promise of mathematical models of athletic training and performance
Imagine that you or your coach could design a training program that guaranteed peak performance for your goal race. Such guarantees are hard to come by, as the normal approach to training program design involves combining general knowledge from training textbooks, trial-and-error experimentation, and the experience of the coach and/or athlete. While this approach can work, much guesswork is still required and a more precise method is desired. Enter mathematical modeling. Mathematical models provide a quantitative framework for rational, systematic and objective design and analysis of training. Importantly, because the model inputs and parameters are based on your own training data, the models are specific to you, such that reliable quantitative prediction of performance is now possible.
The models: how they work
Most of the models used in modern scientific studies are based on the work of Eric Banister and his colleagues at Simon Fraser University in Vancouver, Canada. The models are typically referred to as “impulse-response models”, whereby daily training dose is the “impulse” (or input) and performance is the “response” (or output). The key underlying assumption of the models is that training induces both positive and negative effects, commonly known as fitness and fatigue, respectively. Performance, or one’s “form” on race day, is calculated as the balance between fitness and fatigue. As a coaching tool, the models are used to help design and schedule workouts such that an optimum of fitness and fatigue is achieved to maximize performance on the day of the goal race.
While the basic premise of these models has changed little in three decades, their practical utility has only recently been exploited largely due to the advent of the power meter for measuring power output during cycling. The power meter measures the true rate of work, such that exercise intensity can be precisely quantified regardless of the course or conditions. Dr. Andrew Coggan, an exercise scientist and competitive cyclist, developed a number of power-based metrics for quantifying training data. In particular, the training stress score (TSSTM) can serve as the input (or impulse) for the models. For the purpose of quantifying training, the TSS appears to be a much more useful metric than previously used heart-rate-based metrics and it has enjoyed widespread acceptance in the cycling and triathlon communities.
Two commercially available software packages make implementing these models easy and accessible for everyone. A version of the full impulse-response model is used in PhysFarm’s RaceDayTM software created by Dr. Philip Skiba, a physician, sports scientist, and triathlon coach. A simpler version of the model, called the Performance ManagerTM (PMC), was developed by Dr. Coggan and is featured in TrainingPeaks’ WKO+ software.
Both models have advantages and limitations. The model used in RaceDay is more powerful because it predicts performance in absolute terms (i.e., wattage that could be held on the bike during a race). However, much data is needed for the model calculations, which comes either from frequent (i.e., weekly) performance tests or data from hard workouts. The advantage 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. 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.
Practical application of the PMC model
With this background in hand, we discuss applying these models to analyzing and predicting triathlon training and performance. Specifically, we present two case studies of athletes in which the PMC was used to analyze and plan their cycling training. In the discussion, we refer to the terms CTL, ATL, and TSB, which are the variables involved in the PMC calculations. CTL is the “chronic training load”, which is a term used to quantify one’s fitness, ATL is the “acute training load”, which quantifies one’s level of fatigue, and TSB is the “training stress balance”, which quantifies one’s freshness and equals the difference between CTL and ATL. The output of the model is “form”, which is defined as some combination of CTL (fitness) and TSB (freshness). There are no scientifically established values of CTL and TSB that predict whether one will have good form or not. Instead, one must find their own range of values that correspond to peak performances. This is done through analysis of previous training data and/or through iterative use of the model to plan their training for goal races.
Case 1: Half-Ironman triathlon training analysis and planning
The PMC can be used to analyze older training and performance data. In this case, a female athlete raced a half-Ironman distance triathlon before the PMC was available to the public. Her training data was extracted from her log and charted using the PMC (Figure 1). We have annotated a few of the underlying reasons for the trends in the CTL, ATL, and TSB curves. We can see that in the first three or so months of training that the data was inconsistently logged, such that ATL remained artificially low and the TSB remained high. By March, the chart features more irregular spikes, which reflects daily training doses. A short taper prior to a sprint race caused a spike in TSB and this was followed by heavier training into late April and May. During this time, the CTL rose in a gradual fashion and the ATL tended to stay high until race day.
Figure 1: PMC chart for a six month training block for a female triathlete leading up to her goal race of a half-Ironman distance triathlon.
In the race, the athlete had a good swim and ran close to her desired pace but had a subpar bike ride. In analyzing her race, if all we knew was that the swim and run were good but the bike was bad, then it would be logical to think that the athlete was properly tapered and had a bad day on the bike perhaps due to insufficient training or some other factor. The PMC tells a different story, however, in that the TSB was only +1.1 just prior to race day. Generally, one would hope for a higher TSB after a taper, implying that this athlete was insufficiently tapered for the bike portion of the event. Had the TSB been known before this race, her training could have been adjusted to rest her more in the last few weeks before her race. In addition, this TSB value indicates that for this athlete a TSB higher than +1.1 is probably necessary for optimal performance.
The athlete had another ‘A’ race in 4 months and this time the PMC was used to optimize her training and taper. In particular, the program was designed to achieve a TSB score ranging from +10 to +17 while maintaining a reasonable CTL. The athlete was limited to three bike workouts per week, so two of them emphasized intensity, with one of the workouts inducing a high ATL. In Figure 2, it can be seen that the training worked as the CTL steadily increased during the training block until about three weeks before the event, at which time the taper was started. Three weeks may seem lengthy for a half Ironman-distance taper, but for this athlete it was required to attain the desired TSB. Her final TSB before the race was 14.9, and not only did she have a PR bike split, she also ran a PR as well. Afterwards, the athlete commented that using the PMC was almost like cheating. Obviously, she had to do the work, but knowing how many TSS points to accumulate in each workout and during each week in order to precisely specify her taper duration greatly facilitated optimizing her training program.
Figure 2. PMC chart for the same female triathlete as in Figure 1 for a subsequent three month block of training leading to a half-Ironman distance triathlon.
A typical week for the half-Ironman athlete:
• Average weekly TSS: 217 points
• Maximum weekly TSS: 388 points in week 4
• Minimum weekly TSS: 121 two weeks before her race.
Typical weekly workouts:
• 1 ride each week that included a sustained climb of 20-40 minutes, which accumulated 125-225 TSS points.
• 1 longer ride between 2-3 hours, which typically accumulated 100-150 TSS points.
• 1 other ride each week either on the mountain bike or done as an easy recovery ride.
Case 2: Training program charting and performance prediction for an Ironman triathlete
In our second example, the PMC was used to plan an optimal training program for an Ironman triathlete. In this case, the training plan was devised for the 12 weeks leading into an IM distance race. The TSS was estimated for each workout and the model was calculated using a Microsoft Excel spreadsheet. The planned program was structured to ensure a reasonably high CTL and high TSB by raceday. Thus, the athlete had the confidence that if the program was executed as planned, a peak performance would likely result. As the weeks progressed, the predicted output was compared with the actual PMC created in WKO+. On race day, the predicted TSB was within 1 point of the actual TSB, which was +25. During the race, the athlete rode a 5:21 bike split, which was a PR. We should also note that the athlete used the IF metric to pace his ride, which his power meter automatically provides, thus eliminating guesswork as to whether the ride was correctly paced. Therefore, quantifying training and pacing conferred substantial confidence to the athlete such that he could focus on achieving his goals.
Figure 3. PMC chart for a four month training block leading up to an Ironman triathlon.
A typical week for the Ironman athlete:
• Average weekly TSS: 470 points
• Maximum weekly TSS: 945 points, two weeks out
• Minimum weekly TSS: 153 points in week 3.
Typical weekly workouts:
• 1 ride each week of 4:30 or longer, which accumulated 200-300 TSS points.
• 1 ride each week with VO2max work, which usually accumulated 60-80 TSS points.
• 1 ride each week featuring big gear work, which usually accumulated 90-120 points.
Recent progress in the science of modeling training and performance, motivated largely by technological advances, is spurring a revolution in triathlon coaching and training planning. While these models cannot replace hard work (in fact, their utility depends on it!), they can indicate how much hard work to do and when to do it. In this way, quantitative models can inspire confidence that one’s hard work will lead to peak performance.
* Performance Manager and training stress score are trademarks of Peaksware, LLC.
* RaceDay is a trademark of PhysFarm Training Systems, LLC.
* Additional information, including further reading and a more technical discussion of the models, is available atwww.d3multisport.com.
Calculated Performance: Using Quantitative Models to Optimize Your Training
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.