The New Power-Duration Model in WKO4 November 19, 2013 | Andrew R. Coggan, Ph.D.

Importance of new power-duration model

Performance Manager Chart in WKO4

How do you evaluate mathematical models? Statistical criteria F test Runs test AIC Residuals Distribution Magnitude Bias Parameter estimates Number Independence Precision External validation

Models of the power-duration relationship CP1-10 CP3-30 CP 3-parameter 𝑃 𝑡 =𝐶𝑃+ 𝑊 ′ ∗( 1 𝑡 − 𝑊 ′ 𝐶𝑃 −𝑃𝑚𝑎𝑥 ) 𝑃 𝑡 =𝐶𝑃+ 𝑊 ′ /𝑡 𝑃 𝑡 =𝐶𝑃+ 𝑊 ′ /𝑡 AIS Ward-Smith Pinot and Grappe 𝑃 𝑡 =𝐶𝑃+ 𝑊 ′ 1 𝑡 +( 𝑊 ′ 2 𝑡𝑎𝑢 −𝐶𝑃)(1− 𝑒 − 𝑊 ′ 1 𝑡𝑎𝑢 )( 𝑡𝑎𝑢 𝑡 ) 𝑃 𝑡 =𝑃𝑚𝑎𝑥∗ 𝑡 𝑏 𝑃 𝑡 =𝑃𝑚𝑎𝑥∗ 𝑒 −𝑡/𝑡𝑎𝑢 +𝐶𝑃(1− 𝑒 −𝑡/𝑡𝑎𝑢 )

Limitations of other models: residuals CP1-10 CP3-30 CP 3-parameter AIS Ward-Smith Pinot and Grappe

Limitations of other models: validation CP1-10 CP3-30 CP 3-parameter AIS Ward-Smith Pinot and Grappe (n/a)

Conclusions None of the presently-available models provide a satisfactory description of the entire power-duration relationship. A new model must therefore be developed.

The new power-duration model in WKO4 Part III: Introduction of a superior approach

Modeled vs. actual data: WKO4 model 𝑃 𝑡 =𝑓(𝑃𝑚𝑎𝑥, 𝐹𝑅𝐶, 𝐹𝑇𝑃,…) Golden Cheetar, SportsTracks, Strava, etc. 7 min

Definitions of terms Pmax – the maximal power that can be generated for a very short period of time. Units are W or W/kg. Functional reserve capacity (FRC) – the total amount of work that can be done during continuous exercise above FTP before fatigue occurs. Units are kJ or J/kg. Functional threshold power (FTP) – the highest power that can be sustained in a quasi-steady-state for a prolonged period of time. Units are W or W/kg

Intellectual property rights Registered trademarks are being pursued for the following new terms – unauthorized use is expressly prohibited: Power-Duration Curve™ Power Duration Curve™ P-D curve™ / PD Curve™ Power-Duration History Chart™ Power Duration History Chart™ Functional Reserve Capacity™ FRC™ Dynamic Functional Reserve Capacity™ dFRC™ Rider Phenotyping™

Modeled vs. actual data: WKO4 model 𝑃 𝑡 =𝑓(𝑃𝑚𝑎𝑥, 𝐹𝑅𝐶, 𝐹𝑇𝑃,…)

Distribution of normalized residuals: WKO4 model

Normalized residuals vs. duration: WKO4 model

Independence of param. estimates: WKO4 model Correlation matrix for WKO4 model Pmax FRC FTP 1.00 0.58 0.41 0.15 Values shown in bolded red font are statistically significant.

CVs of model fit parameters Pmax FRC FTP 6.8 ±3.2% 4.7 ±3.4% 1.2 ±0.5% Tightness of parameter estimates supports conclusion that Pmax, FRC, and FTP are different things.

Model fit parameters in WKO4

External validation: WKO4 model Intercept and slope not significantly different from 0 and 1, respectively. SEE = +/- 0.8 W, or +/- 0.2%.

External validation: WKO4 model

“It ain’t braggin’ if you can back it up” WKO4 vs. other models “It ain’t braggin’ if you can back it up”

Limitations of other models: residuals CP1-10 CP3-30 CP 3-parameter AIS Ward-Smith Pinot and Grappe

Normalized residuals vs. duration: WKO4 model

Comparison of root mean squared errors Ɵ= 𝑣𝑎𝑟𝑖𝑎𝑛𝑐𝑒 +𝑏𝑖𝑎𝑠 Model RMSE CP1-10 8.5±2.6% CP3-30 12.3±4.5% CP 3-parameter 1.1±0.4% AIS 4.6±0.9% Ward-Smith 2.5±0.6% Pinot and Grappe 1.1±0.6% WKO4 model 0.3±0.1% Note that 1) while the RMSE is lower for models that better fit the beginning (CP 3-parameter) or end (Pinot and Grappe) of the power-duration relationship, both of these models are still biased (as evidenced by the distribution of the residuals), and 2) the RMSE is much lower for the WKO4 model (ideal is zero).

Comparison of FTP prediction Model Slope (unitless) Intercept (W) S.E.E. (W) CV (%) CP1-10 0.87 24 ±4.6 ±1.5 CP3-30 0.89 33 ±2.4 ±0.8 CP 3-parameter 46 ±4.8 ±1.6 AIS 1.01 9 ±20.5 ±6.8 Ward-Smith 0.74 23 ±8.6 ±2.8 Pinot and Grappe n/a WKO4 model 0.93 21 ±0.5 Bolded red values for slope and intercept are significantly different from 1 and 0, respectively.

Comparison of parameter estimates Model Pmax (W) W’ or FRC (kJ) CP or FTP (W) CP1-10 ∞ 13.9±4.4 303±56 CP3-30 19.2±6.1 285±52 CP 3-parameter 1219±309 22.4±8.1 277±53 AIS 12.0±1.3 276±45 Ward-Smith 1145±318 n/a 360±64 Pinot and Grappe 1438±413 WKO4 model 1132±273 19.3±6.8 286±51 Numbers shown in bold red font are significantly different from corresponding values from WKO4 model.

CP2-15 overestimates maximal steady-state McLellan TM, Cheung KSY. A comparative evaluation of the individual anaerobic threshold and the critical power. Med Sci Sports Exerc 1992; 24:543-550.

CP1-10 overestimates maximal steady-state Brickely G, Doust J, Williams CA. Physiological responses during exercise to exhaustion at critical power. Eur J Appl Physiol 2002; 88:146-151.

CP2-15 overestimates maximal steady-state Pringle JSM, Jones AM. Maximal lactate steady state, critical power and EMG during cycling. Eur J Appl Physiol 2002; 88:214-226.

Comparison of parameter estimates Model Pmax (W) W’ or FRC (kJ) CP or FTP (W) CP1-10 ∞ 13.9±4.4 303±56 CP3-30 19.2±6.1 285±52 CP 3-parameter 1219±309 22.4±8.1 277±53 AIS 12.0±1.3 276±45 Ward-Smith 1145±318 n/a 360±64 Pinot and Grappe 1438±413 WKO4 model 1132±273 19.3±6.8 286±51 Numbers shown in bold red font are significantly different from corresponding values from WKO4 model.

FRC vs. W’ "What's in a name? That which we call a rose by any other name would smell as sweet"

Pmax vs. 1 s power

Model is sensitive to changes across years 2000-2003: trained for mass start road races. 2004-2005: trained for 3 km pursuit. 2006-2007: JRA 4-6x/wk. 2008-2009: trained for 40 km TT (at altitude). 2010: JRA 3x/wk (plus weights). 2011-2013: JRA 6-7x/wk.

Model is sensitive to changes across years

Model is sensitive to changes w/in a year

Power-duration history chart in WKO4

Application of the model to running

Application of the model to swimming

Application of the model to skating

The new power-duration model in WKO4 The most advanced mathematical model of the exercise intensity-duration relationship ever developed.

Next week: the final installment! Current (i.e., WKO4) and possible future applications of the new power-duration model will be presented and discussed.