The New Power-Duration Model in WKO4November 19, 2013 | Andrew R. Coggan, Ph.D.

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Importance of new power-duration model

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Performance Manager Chart in WKO4

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How do you evaluate mathematical models?

1. Statistical criteriaa. F testb. Runs testc. AIC

2. Residualsa. Distributionb. Magnitudec. Bias

3. Parameter estimatesa. Numberb. Independencec. Precision

4. External validation

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Models of the power-duration relationship

CP1-10 CP3-30 CP 3-parameter

AIS Ward-Smith

𝑃 (𝑡 )=𝐶𝑃+𝑊 ′ /𝑡 𝑃 (𝑡 )=𝐶𝑃+𝑊 ′∗( 1𝑡−

𝑊 ′

𝐶𝑃−𝑃𝑚𝑎𝑥)

)

𝑃 (𝑡 )=𝐶𝑃+𝑊 ′ /𝑡

𝑃 (𝑡 )=𝐶𝑃+𝑊 ′1𝑡

+(𝑊 ′2𝑡𝑎𝑢

−𝐶𝑃 )(1−𝑒−𝑊

′ 1𝑡𝑎𝑢 )(

𝑡𝑎𝑢𝑡

)

Pinot and Grappe𝑃 (𝑡 )=𝑃𝑚𝑎𝑥∗𝑡𝑏

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Limitations of other models: residuals

CP1-10 CP3-30 CP 3-parameter

AIS Ward-Smith Pinot and Grappe

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Limitations of other models: validation

CP1-10 CP3-30 CP 3-parameter

AIS Ward-Smith Pinot and Grappe

(n/a)

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Conclusions

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

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The new power-duration model in WKO4

Part III: Introduction of a superior approach

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Modeled vs. actual data: WKO4 model

𝑃 (𝑡 )= 𝑓 (𝑃𝑚𝑎𝑥 ,𝐹𝑅𝐶 ,𝐹𝑇𝑃 ,…)

Andrew R. Coggan, Ph.D., Peaks Coaching Group, and/or TrainingPeaks are the owner(s) and/or licensee(s) of all intellectual property rights to this presentation and the material it contains. These works are protected by copyright laws and treaties around the world. All such rights are reserved. 11

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

Andrew R. Coggan, Ph.D., Peaks Coaching Group, and/or TrainingPeaks are the owner(s) and/or licensee(s) of all intellectual property rights to this presentation and the material it contains. These works are protected by copyright laws and treaties around the world. All such rights are reserved. 12

Intellectual property rights

• Registered trademarks are being pursued for the following new terms – unauthorized use is expressly prohibited:

1. Power-Duration Curve™2. Power Duration Curve™3. P-D curve™ / PD Curve™4. Power-Duration History Chart™5. Power Duration History Chart™6. Functional Reserve Capacity™7. FRC™8. Dynamic Functional Reserve Capacity™9. dFRC™10. Rider Phenotyping™

Andrew R. Coggan, Ph.D., Peaks Coaching Group, and/or TrainingPeaks are the owner(s) and/or licensee(s) of all intellectual property rights to this presentation and the material it contains. These works are protected by copyright laws and treaties around the world. All such rights are reserved. 13

Modeled vs. actual data: WKO4 model

𝑃 (𝑡 )= 𝑓 (𝑃𝑚𝑎𝑥 ,𝐹𝑅𝐶 ,𝐹𝑇𝑃 ,…)

Andrew R. Coggan, Ph.D., Peaks Coaching Group, and/or TrainingPeaks are the owner(s) and/or licensee(s) of all intellectual property rights to this presentation and the material it contains. These works are protected by copyright laws and treaties around the world. All such rights are reserved. 14

Distribution of normalized residuals: WKO4 model

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Normalized residuals vs. duration: WKO4 model

Andrew R. Coggan, Ph.D., Peaks Coaching Group, and/or TrainingPeaks are the owner(s) and/or licensee(s) of all intellectual property rights to this presentation and the material it contains. These works are protected by copyright laws and treaties around the world. All such rights are reserved. 16

Independence of param. estimates: WKO4 model

Correlation matrix for WKO4 modelPmax FRC FTP

Pmax 1.00 0.58 0.41

FRC 1.00 0.15

FTP 1.00

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CVs of model fit parameters

Pmax FRC FTP6.8

±3.2%4.7

±3.4%1.2

±0.5%

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Model fit parameters in WKO4

Andrew R. Coggan, Ph.D., Peaks Coaching Group, and/or TrainingPeaks are the owner(s) and/or licensee(s) of all intellectual property rights to this presentation and the material it contains. These works are protected by copyright laws and treaties around the world. All such rights are reserved. 19

External validation: WKO4 model

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External validation: WKO4 model

Andrew R. Coggan, Ph.D., Peaks Coaching Group, and/or TrainingPeaks are the owner(s) and/or licensee(s) of all intellectual property rights to this presentation and the material it contains. These works are protected by copyright laws and treaties around the world. All such rights are reserved. 21

WKO4 vs. other models

“It ain’t braggin’ if you can back it up”

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Limitations of other models: residuals

CP1-10 CP3-30 CP 3-parameter

AIS Ward-Smith Pinot and Grappe

Andrew R. Coggan, Ph.D., Peaks Coaching Group, and/or TrainingPeaks are the owner(s) and/or licensee(s) of all intellectual property rights to this presentation and the material it contains. These works are protected by copyright laws and treaties around the world. All such rights are reserved. 23

Normalized residuals vs. duration: WKO4 model

Andrew R. Coggan, Ph.D., Peaks Coaching Group, and/or TrainingPeaks are the owner(s) and/or licensee(s) of all intellectual property rights to this presentation and the material it contains. These works are protected by copyright laws and treaties around the world. All such rights are reserved. 24

Comparison of root mean squared errors

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%

Ɵ=√𝑣𝑎𝑟𝑖𝑎𝑛𝑐𝑒+𝑏𝑖𝑎𝑠

Andrew R. Coggan, Ph.D., Peaks Coaching Group, and/or TrainingPeaks are the owner(s) and/or licensee(s) of all intellectual property rights to this presentation and the material it contains. These works are protected by copyright laws and treaties around the world. All such rights are reserved. 25

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 0.87 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 n/a n/a n/a

WKO4 model 0.93 21 ±1.5 ±0.5

Andrew R. Coggan, Ph.D., Peaks Coaching Group, and/or TrainingPeaks are the owner(s) and/or licensee(s) of all intellectual property rights to this presentation and the material it contains. These works are protected by copyright laws and treaties around the world. All such rights are reserved. 26

Comparison of parameter estimates

Model Pmax (W) W’ or FRC (kJ) CP or FTP (W)CP1-10 ∞ 13.9±4.4 303±56CP3-30 ∞ 19.2±6.1 285±52

CP 3-parameter 1219±309 22.4±8.1 277±53AIS ∞ 12.0±1.3 276±45

Ward-Smith 1145±318 n/a 360±64Pinot and Grappe 1438±413 n/a n/a

WKO4 model 1132±273 19.3±6.8 286±51

Andrew R. Coggan, Ph.D., Peaks Coaching Group, and/or TrainingPeaks are the owner(s) and/or licensee(s) of all intellectual property rights to this presentation and the material it contains. These works are protected by copyright laws and treaties around the world. All such rights are reserved. 27

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.

Andrew R. Coggan, Ph.D., Peaks Coaching Group, and/or TrainingPeaks are the owner(s) and/or licensee(s) of all intellectual property rights to this presentation and the material it contains. These works are protected by copyright laws and treaties around the world. All such rights are reserved. 28

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.

Andrew R. Coggan, Ph.D., Peaks Coaching Group, and/or TrainingPeaks are the owner(s) and/or licensee(s) of all intellectual property rights to this presentation and the material it contains. These works are protected by copyright laws and treaties around the world. All such rights are reserved. 29

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.

Andrew R. Coggan, Ph.D., Peaks Coaching Group, and/or TrainingPeaks are the owner(s) and/or licensee(s) of all intellectual property rights to this presentation and the material it contains. These works are protected by copyright laws and treaties around the world. All such rights are reserved. 30

Comparison of parameter estimates

Model Pmax (W) W’ or FRC (kJ) CP or FTP (W)CP1-10 ∞ 13.9±4.4 303±56CP3-30 ∞ 19.2±6.1 285±52

CP 3-parameter 1219±309 22.4±8.1 277±53AIS ∞ 12.0±1.3 276±45

Ward-Smith 1145±318 n/a 360±64Pinot and Grappe 1438±413 n/a n/a

WKO4 model 1132±273 19.3±6.8 286±51

Andrew R. Coggan, Ph.D., Peaks Coaching Group, and/or TrainingPeaks are the owner(s) and/or licensee(s) of all intellectual property rights to this presentation and the material it contains. These works are protected by copyright laws and treaties around the world. All such rights are reserved. 31

FRC vs. W’

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Pmax vs. 1 s power

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Model is sensitive to changes across years

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Model is sensitive to changes across years

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Model is sensitive to changes w/in a year

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Power-duration history chart in WKO4

Andrew R. Coggan, Ph.D., Peaks Coaching Group, and/or TrainingPeaks are the owner(s) and/or licensee(s) of all intellectual property rights to this presentation and the material it contains. These works are protected by copyright laws and treaties around the world. All such rights are reserved. 37

Application of the model to running

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Application of the model to swimming

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Application of the model to skating

10 100 10000

2

4

6

8

10

12

14

16

18 2013 WR in speed skating (men)

Time (s)

Aver

age

pace

(m/s

) Functional reserve distance = 208 mFunctional threshold pace = 13.0 m/sAverage prediction error = 3.3%

Andrew R. Coggan, Ph.D., Peaks Coaching Group, and/or TrainingPeaks are the owner(s) and/or licensee(s) of all intellectual property rights to this presentation and the material it contains. These works are protected by copyright laws and treaties around the world. All such rights are reserved. 40

The new power-duration model in WKO4

• The most advanced mathematical model of the exercise intensity-duration relationship ever developed.

Andrew R. Coggan, Ph.D., Peaks Coaching Group, and/or TrainingPeaks are the owner(s) and/or licensee(s) of all intellectual property rights to this presentation and the material it contains. These works are protected by copyright laws and treaties around the world. All such rights are reserved. 41

Next week: the final installment!

Current (i.e., WKO4) and possible future applications of the new power-duration model will be presented and discussed.

Andrew R. Coggan, Ph.D., Peaks Coaching Group, and/or TrainingPeaks are the owner(s) and/or licensee(s) of all intellectual property rights to this presentation and the material it contains. These works are protected by copyright laws and treaties around the world. All such rights are reserved. 42

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