Popular endurance models fail to predict runners’ interval performance, study finds

A widely used family of mathematical models borrowed from cycling collapses when applied to high-intensity interval running, according to a new study of trained adult runners. The research, published in Physiological Reports, tested the four main formulations of the so-called W′-balance framework — models that track how a finite reserve of work capacity above critical power is drained and refilled during intermittent exercise — and found that every one of them, used exactly as published, grossly overestimated how many intervals athletes could complete. In a cohort of 27 trained endurance runners performing exhausting interval sessions on an outdoor track, the pooled prediction errors ranged from roughly 26 to 72 intervals, while the athletes actually completed an average of only 7.6 intervals per session.

The W′-balance framework rests on the two-parameter critical power model, a cornerstone of endurance physiology. Critical power (CP) represents the upper boundary of sustainable effort, the highest power output at which a metabolic steady state can be maintained, while W′ describes the finite amount of work that can be performed above CP before task failure. In cycling, where bike-mounted power meters have long provided second-by-second measurement of mechanical work, these parameters and the models built on them have transformed how training intensity is prescribed and monitored. In running, comparable opportunities have emerged only recently, with foot-mounted inertial measurement units such as the Stryd sensor enabling real-time estimation of running power, whose reliability and validity have been documented across a range of speeds, gradients and terrains.

W′-balance models promise something coaches genuinely want: the ability to anticipate, before a session begins, how many intervals an athlete should be able to complete at a given target power. By tracking the dynamic depletion and reconstitution of W′ during intermittent exercise, these formulations allow high-intensity interval training to be prescribed with sub-interval accuracy, with individualized progression and objective monitoring of acute and chronic responses. That appeal is sharpened by the established efficacy of high-intensity interval training for improving performance in endurance runners. Yet the development and empirical validation of these models have been almost exclusively confined to cycling, typically using ergometer-based protocols with active recovery between work bouts, and the few attempts to apply them to running have produced disappointing results.

The new study set out to close that gap with a cross-sectional, within-subject repeated-measures design. Twenty-seven trained endurance runners — 25 men and 2 women aged 19 to 39, with an average of eight years of running experience and weekly training volumes of about 35 kilometers — attended three sessions spaced three to five days apart. In the first, the researchers determined each athlete’s CP and W′ using a validated two-point field test consisting of maximal 9-minute and 3-minute running time trials separated by 30 minutes of rest, with power measured by a foot-mounted Stryd sensor. The athletes then completed two open-ended interval sessions to volitional exhaustion at 108% of their individual critical power, an intensity associated with maximal oxygen uptake and typical of HIIT workouts in endurance runners.

The two interval protocols differed in how severely each work bout drained the W′ reserve. In the 40% condition, each work interval was individualized so that running at 108% CP depleted, on average, 40% of the athlete’s W′ — about 152 seconds of work — followed by passive rest lasting half the work duration. In the 60% condition, each bout consumed roughly 60% of W′ over about 233 seconds, again with a 2:1 work-to-rest ratio and fully passive recovery. Participants were instructed to keep going, interval by interval, for as long as they could sustain the target power, and the number of completed intervals served as the primary outcome. Across the 54 sessions analyzed, runners completed an average of 9.6 intervals in the 40% protocol and 5.5 in the 60% protocol.

When the four original models were applied with their published parameters, the results were strikingly poor. The Morton–Billat linear model, the simplest formulation, assumes that W′ is restored during rest at a rate equal to the power deficit below CP — which, under fully passive recovery, equals CP itself. Because every rest interval in the study, ranging from 26 to 260 seconds, exceeded the athletes’ average W′/CP time of roughly 32 seconds, the model predicted complete replenishment between intervals and returned the simulation cap of 80 intervals in all 54 sessions, overpredicting by an average of 72.4 intervals. The differential-form W′BAL-ODE model, whose apparent recovery time constant emerges directly from W′/CP and averaged only about 31 seconds, produced an identical failure: the cap in every session, with the same mean absolute error of 72.4 intervals.

The integral-form W′BAL-INT model, governed by a much slower time constant of roughly 337 seconds in this cohort, still saturated at the simulation cap in 46 of 54 sessions, with a mean overprediction of 60.9 intervals. The researchers attribute this to the model’s central assumption of ongoing recovery, in which every previously expended joule of W′ is reconstituted at every subsequent moment, even while additional W′ is still being spent above CP — an assumption that lets a steady-state W′ balance well above zero persist across successive work–rest cycles. The biexponential Caen model, which splits recovery into fast and slow components with parameters derived from trained cyclists, fared best but still overpredicted by an average of 20.1 intervals, saturating in 17 of 54 sessions. Its errors were also strongly protocol-dependent: an average overprediction of 34.2 intervals in the 40% condition versus 5.5 in the 60% condition.

To address the failure, the team proposed a simple modification of the linear model, called Linear_mod, which re-parameterizes the recovery rate during rest as a fitted multiple of the work-rate deficit — the difference between work-interval power and CP — rather than fixing it to CP alone. This single dimensionless recovery coefficient, R, avoids the degeneracy that makes the original linear model useless under passive recovery. Even with its off-the-shelf setting of R = 1, Linear_mod outperformed all four original models by an order of magnitude, with a mean absolute error of 4.85 intervals. When R was fitted once across the entire cohort, the optimal value was 1.63, cutting the error to 2.85 intervals and largely eliminating systematic bias — suggesting trained runners need a recovery rate roughly 1.6 times the work-rate deficit per second of passive rest, substantially slower than the original formulations imply.

Crucially, when R was fitted individually for each athlete using both of their sessions, the mean absolute error fell to just 0.55 intervals, and the individually fitted values — averaging about 1.54 — proved stable across the two very different protocols, with a within-subject coefficient of variation below 10%. That stability suggests R behaves as a reproducible, athlete-specific calibration coefficient rather than a protocol-dependent artifact, though the authors caution it should be regarded as an empirical tolerance coefficient rather than direct evidence of an underlying physiological trait. The findings also extend the only previous running study, which found the integral model could not predict intermittent treadmill performance unless its time constant was re-estimated from the runners’ own data. The authors note key limitations: the model was calibrated only for sessions at 108% CP with passive recovery and a 2:1 work-to-rest ratio in trained runners, it cannot yet distinguish passive from active recovery, the cohort-specific value of R should not be transferred to other populations without dedicated calibration, and the reported errors represent in-sample calibration rather than out-of-sample predictive accuracy, which future studies should test with an independent session.

Subject of Research: Transfer of W′-balance recovery models from cycling to high-intensity interval running in trained adults

Article Title: Original W′‐balance models do not transfer to high‐intensity interval training in trained adult runners: Proposal of a single‐coefficient linear alternative

Article References: García‐Pinillos, F., Jaén‐Carrillo, D., Riquelme‐Sebastián, L., Ruiz‐Alias, S. A., Fernández‐Navarrete, I., González‐Mohino, F., Muñoz de la Cruz, V., & Olaya‐Cuartero, J. (2026). Original W′‐balance models do not transfer to high‐intensity interval training in trained adult runners: Proposal of a single‐coefficient linear alternative. Physiological Reports, 14(20), Article e71152. https://doi.org/10.14814/phy2.71152

Image Credits: AI Generated

DOI: 10.14814/phy2.71152

Keywords: W′-balance, critical power, high-intensity interval training, endurance running, running power meters, passive recovery, exercise physiology, Stryd sensor, training prescription, mathematical modeling, trained runners, performance prediction