I’ve raised the issue before with Andy Coggan. He has not given proper attribution to Peronnet and Thibault as the source model for his own model.
The Peronnet and Tibault model is a reasonable model that is an extension of Ward-Smith which a modification of LLoyd which is an extension of Hill.
P(t) = AWC / t + MAP
AWC is anaerobic work capacity
MAP is maximal aerobic power
P(t) = AWC / t * (1-exp(-t/tau)) + MAP
tau is the time constant of the exponential function. In plain terms what this function says is that AWC is not instantly available but increase exponentially with time so that the maximum power is constrained.
P(t) = AWC / t * (1-exp(-t/tau)) + MAP * (1-exp(-t/tau2))
tau2 is again a time constant of the exponential function. In plain terms Ward-Smith again said that MAP is not instantly available but instead must increase with oxygen kinetics. If you do the math:
tau = AWC/Pmax (almost, there is technically a tiny MAP contribution since the exponential function does not start from zero so that starting value should be subtracted from Pmax).
P(t) = AWC / t * (1-exp(-t/tau)) + MAP * (1-exp(-t/tau2)); t </= Tmap
P(t) = AWC / t * (1-exp(-t/tau)) + MAP * (1-exp(-t/tau2)) – a*Ln(t/Tmap): t > Tmap
a is the slope of the decline in MAP, which decreases log-linearly starting at t > Tmap. Tmap is the longest duration that MAP can be sustained.
P(t) = FRC / t * (1-exp(-t/tau)) + FTP * (1-exp(-t/tau2)); t </= TTE
P(t) = FRC / t * (1-exp(-t/tau)) + FTP * (1-exp(-t/tau2))) – a*Ln(t/TTE): t > TTE
As noted above tau can be substituted with FRC/Pmax. The practical difference between the two models is that P&T specified Tmap as a fixed parameter = 420s while it is a fitted parameter in WKO. (As an aside, to be consistent with oxygen kinetics, which are indeed described by an exponential rise in response to high intensity exercise, the model should actually used the integral of the function above if the modeler was working from first principles as claimed. Unfortunately, when the integral function is used the model does not perform well )
To illustrate that the models are in fact mathematically the same here is an overlay of P&T on top of the output from WKO:
To show that there are actually two models here, I offset P&T slightly:
As an example of how this model can get janky with very low AWC and long tau2 (causing the anaerobic power to fall off before the aerobic power ramps up):
Looks a bit ridiculous with model output (red) actually dipping, which is clearly not physiological as MMP is by definition monotonically decreasing. Likewise, the sharp inflection at Tmap indicates very sub-maximal long duration data.
To be clear, I am not intentionally putting bad numbers in to the model to make it look bad, rather I am just reproducing the curves off a WKO5 help page:
Here are some links to a spreadsheet with the P&T model and a ppt with these last two images overlaid.
Note, there is nothing wrong with using source models and prior work. It is what we did to arrive at a not very different end, source functions cited and all.