A conditional power–duration curve for the triathlon run: formulation, identifiability and a three-athlete proof of concept

Background. The running power–duration curve (PDC) estimated from fresh efforts describes the rested athlete. In triathlon the run starts after one to six hours of swimming and cycling, and what is needed to set paces, choose the cycling intensity and predict the run split is the PDC conditional on the state in which the run segment begins. Objective. This is a methodological study. It aims to formulate, estimate and validate an extension of the critical power model with durability that conditions the running PDC on the preceding cycling work, and to determine which quantities are identified by the field data of a triathlete instrumented with a power meter and a running power pod. Validation is longitudinal and individual (N-of-1) in three athletes: a proof of concept, not an estimation of population values. Methods. The slow-fatigue state at the start of the run is defined as the sum of the athlete's own running work and the cycling work converted into running-equivalent kJ/kg by a transfer coefficient τ. With that, the whole machinery of the base model (critical capacity and finite capacity decreasing with prior work, fixed point, stochastic frontier) applies unchanged and yields a family of curves indexed by the cycling dose. The model was estimated in two triathletes (400 days, 42 bike→run sessions with a transition ≤ 60 min, seven races) with pseudo-likelihood profiles by continuation, a per-session jackknife, refits with four sets of own-run durability records, five for B (one cleaned by reconstructing the date of each record), leave-one-race-out validation and causal validation with data preceding each race. In a third triathlete without a power pod the same model was applied to the pseudo-power P* = m·v (CP becomes critical speed and W' becomes distance capacity) and, with the fit frozen on the previous day, an Ironman run three days before the close of the analysis was predicted. Results. The bike→run pairs identify the 20-min running power ceiling in watts, conditional on the cycling dose, within the observed range (14 to 65 kJ/kg): the ceiling after 45 kJ/kg varied by ±1.2% (athlete A) and ±3.9% (athlete B) across decompositions of the data, and the 60- to 100-min windows of three races out of four were predicted within ±3%. In contrast, τ (0.34 to 2.5), the run's own durability (0.1 to 13% per 10 kJ/kg) and their product are not separated by training data: the pseudo-likelihood is bimodal and the mode is decided by the aggregated own-run durability records, which in a triathlete's training mix submaximal easy runs with runs that follow a bike. Reconstructing the dates of those records shows that, in each athlete, all own-run durability in the 30 and 40 kJ/kg bins comes from a single long effort (an Ironman in one, a run of more than 3 h in the other), and that this effort decides the mode. The extrapolation to the long-distance marathon is not identified either. In the athlete without a pod, the two modes of the pseudo-likelihood bracket what was observed in the Ironman: the fit without own-run durability predicted the 100-min window 8% faster than run, and the fit with own-run durability by time 3.5% slower; the 20-min ceiling was within 2 to 5%. Conclusions. The transition datum is the cleanest durability datum available in the field, and it suffices to predict the 20- to 100-min running ceiling after a known bike. Separating the transfer from the run's own durability requires a protocol of maximal efforts after known doses of running and of cycling, which is proposed. The model runs unchanged in the currency of pace, which makes it applicable to the triathlete without a pod. The total-time curve against cycling intensity that the model produces is monotonic: the optimal split is set by the sustainability of the bike leg, not by this model.