Methods & references

1. Grade-adjusted pace (GAP)

   GPX analyzer, grade-adjusted pace calculator

   We convert the energy cost of running on a slope to an equivalent flat-ground pace using Minetti’s fifth-order polynomial for the metabolic cost of running as a function of gradient, fitted across −45% to +45% slopes.

   Source: Minetti AE (2002)

2. Normalized Power, Intensity Factor & TSS

   Training stress score, GPX analyzer (power)

   Normalized Power weights a rolling 30-second average raised to the fourth power to reflect the non-linear physiological cost of variable efforts; Intensity Factor is NP ÷ FTP; Training Stress Score scales intensity and duration so that one hour at FTP equals 100 points.

   Source: Allen H (2010)

3. Fitness, Fatigue & Form (CTL / ATL / TSB)

   Training calendar & performance chart (on-device app)

   Chronic Training Load (fitness) and Acute Training Load (fatigue) are exponentially weighted moving averages of daily training stress with 42- and 7-day time constants; Form is their difference. This is the impulse–response model applied to a daily stress score.

   Sources: Banister EW (1991); Allen H (2010)

4. Critical power, W′ and W′bal

   Critical power calculator, power-duration analysis

   Critical power is the asymptote of the power–duration relationship; W′ is the fixed work available above it. We model the depletion and exponential reconstitution of W′ during variable efforts to show real-time “tank” balance.

   Sources: Monod H (1965); Skiba PF (2012); Jones AM (2017)

5. VDOT & equivalent race times

   VDOT training paces, race-time predictor

   A recent race is converted to an effective VO₂ (VDOT) that already folds in running economy, then used to read off equivalent performances and training paces.

   Source: Daniels J (1979)

6. Race-time prediction

   Race-time predictor

   We project a finish time at a new distance from a known result using Riegel’s endurance model, where time scales with distance raised to a fatigue exponent of about 1.06.

   Source: Riegel PS (1981)

7. Heart-rate zones & maximum heart rate

   Heart-rate zones, GPX analyzer (time-in-zone)

   Zones are computed from heart-rate reserve (the Karvonen method) or as a percentage of maximum heart rate; where age is the only input we estimate maximum heart rate with the Tanaka equation (208 − 0.7 × age) rather than the cruder 220 − age.

   Sources: Karvonen MJ (1957); Tanaka H (2001)

8. Heart-rate training load (TRIMP)

   Training calendar (HR-only activities)

   For sessions without power we quantify load from heart rate. We support Banister’s exponential TRIMP and Lucía’s zone-weighted TRIMP, which credits time in the heavy and severe domains more than easy time.

   Sources: Banister EW (1991); Lucía A (2003)

9. Intensity distribution (polarized training)

   Training calendar, zone summaries

   We summarize how training time is split across the easy / threshold / hard zones, the basis for evaluating polarized versus threshold training.

   Source: Seiler S (2010)

10. Carbohydrate & fluid targets

    Carb fueling, sweat-rate & hydration

    Hourly carbohydrate targets follow Jeukendrup’s intake guidance, including the higher ceiling reachable with multiple transportable carbohydrates; fluid targets follow the ACSM position stand’s sweat-rate test and the ~2%-body-mass dehydration threshold.

    Sources: Jeukendrup A (2014); Sawka MN (2007)

Full bibliography

1. Minetti AE, Moia C, Roi GS, Susta D, Ferretti G (2002). Energy cost of walking and running at extreme uphill and downhill slopes. Journal of Applied Physiology, 93(3):1039–1046. doi:10.1152/japplphysiol.01177.2001
2. Allen H, Coggan AR (2010). Training and Racing with a Power Meter (2nd ed.). VeloPress, Boulder, CO. Defines Normalized Power (NP), Intensity Factor (IF), and Training Stress Score (TSS).
3. Banister EW (1991). Modeling elite athletic performance. In: Green HJ, McDougal JD, Wenger HA (eds), Physiological Testing of Elite Athletes, Human Kinetics, pp. 403–424. The fitness–fatigue (impulse–response) model and the original TRIMP.
4. Monod H, Scherrer J (1965). The work capacity of a synergic muscular group. Ergonomics, 8(3):329–338. doi:10.1080/00140136508930810
5. Skiba PF, Chidnok W, Vanhatalo A, Jones AM (2012). Modeling the expenditure and reconstitution of work capacity above critical power. Medicine & Science in Sports & Exercise, 44(8):1526–1532. doi:10.1249/MSS.0b013e3182517a80
6. Jones AM, Vanhatalo A (2017). The “critical power” concept: applications to sports performance with a focus on intermittent high-intensity exercise. Sports Medicine, 47(Suppl 1):65–78. doi:10.1007/s40279-017-0688-0
7. Daniels J, Gilbert J (1979). Oxygen Power: Performance Tables for Distance Runners. Self-published, Tempe, AZ. Origin of VDOT and the VO₂-based equivalent-performance tables.
8. Riegel PS (1981). Athletic records and human endurance. American Scientist, 69(3):285–290. The endurance/fatigue exponent used for race-time prediction (t₂ = t₁·(d₂/d₁)^1.06).
9. Tanaka H, Monahan KD, Seals DR (2001). Age-predicted maximal heart rate revisited. Journal of the American College of Cardiology, 37(1):153–156. doi:10.1016/S0735-1097(00)01054-8
10. Karvonen MJ, Kentala E, Mustala O (1957). The effects of training on heart rate: a longitudinal study. Annales Medicinae Experimentalis et Biologiae Fenniae, 35(3):307–315. The heart-rate-reserve (Karvonen) method for zone calculation.
11. Lucía A, Hoyos J, Santalla A, Earnest C, Chicharro JL (2003). Tour de France versus Vuelta a España: which is harder?. Medicine & Science in Sports & Exercise, 35(5):872–878. doi:10.1249/01.MSS.0000064999.82036.B4 Lucía TRIMP, weighting time in three ventilatory-threshold-bounded zones.
12. Seiler S (2010). What is best practice for training intensity and duration distribution in endurance athletes?. International Journal of Sports Physiology and Performance, 5(3):276–291. doi:10.1123/ijspp.5.3.276 Polarized intensity distribution and the three-zone model.
13. Jeukendrup A (2014). A step towards personalized sports nutrition: carbohydrate intake during exercise. Sports Medicine, 44(Suppl 1):S25–S33. doi:10.1007/s40279-014-0148-z Carbohydrate-per-hour targets and multiple transportable carbohydrates.
14. Sawka MN, Burke LM, Eichner ER, Maughan RJ, Montain SJ, Stachenfeld NS (2007). American College of Sports Medicine position stand: exercise and fluid replacement. Medicine & Science in Sports & Exercise, 39(2):377–390. doi:10.1249/mss.0b013e31802ca597 Sweat-rate measurement and the ~2%-body-mass dehydration threshold.

These tools are for training and education, not medical advice. Models are simplifications of human physiology and carry real uncertainty; treat the outputs as informed estimates, not precise measurements.