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

2. Importance of new power-duration model

3. Performance Manager Chart in WKO4

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

5. 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 )

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

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

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

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

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

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

12. 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™

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

14. Distribution of normalized residuals: WKO4 model

15. Normalized residuals vs. duration: WKO4 model

16. 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.

17. 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.

18. Model fit parameters in WKO4

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

20. External validation: WKO4 model

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

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

23. Normalized residuals vs. duration: WKO4 model

24. 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).

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 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.

26. 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.

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:

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:

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:

30. 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.

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

32. Pmax vs. 1 s power

33. Model is sensitive to changes across years
: trained for mass start road races : trained for 3 km pursuit : JRA 4-6x/wk : trained for 40 km TT (at altitude). 2010: JRA 3x/wk (plus weights) : JRA 6-7x/wk.

34. Model is sensitive to changes across years

35. Model is sensitive to changes w/in a year

36. Power-duration history chart in WKO4

37. Application of the model to running

38. Application of the model to swimming

39. Application of the model to skating

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

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.