Trail running has become increasingly popular in recent years, and the number of runners taking part in off-road races continues to grow. Unlike road running, trail races often take place in the mountains or on uneven terrain, where athletes face steep climbs and descents, technical surfaces, and changing weather conditions. These challenges mean that runners need to adapt their pacing and energy use throughout the race.
In our study, we build on Keller’s classical model of running [1] by adapting it to better describe the specific demands of trail running. We take into account several important factors that affect performance, such as fatigue from long effort, carbohydrate intake during the race, altitude, and terrain difficulty, see Figure 1. This allows the model to more accurately reflect what happens during a typical trail race.
Because of the complexity of trail running, traditional methods for finding an optimal race strategy are not suitable. Our model uses a different approach, combining physical effort, energy supply, and course features to find the best way to run a given race.
To check how well the model works, we tested it on well-known races, including events from the Golden Trail Series and the Trail World Championships. These included courses of various lengths and profiles—vertical races, half-marathons, and marathons. We compared the predicted finish times for an elite male runner to the actual course records, and in most cases, the model was very accurate, with relative errors under 5%, see Table 1.
Table 1: Comparison of model output with route records of selected trail tuning events.
The model also produces a detailed pacing plan, showing where a runner should speed up or slow down during the race. This can help runners improve their performance by managing their effort more effectively. While we used data from elite athletes, the model can be adapted for anyone by entering individual values like VO₂ max, weight, or the maximum force the runner can produce [2].
Literature
[1] Keller, Joseph B. “A theory of competitive running.” Physics today 26.9 (1973): 42-47.
[2] Jaszczak, Bogna, and Łukasz Płociniczak. “Optimal strategy for trail running with nutrition and fatigue factors.” SIAM Journal on Applied Mathematics 85.2 (2025): 730-754.
By Bogna Jaszczak and Łukasz Płociniczak (Wrocław University of Science and Technology)