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Human Bioenergetic Models

My understanding of what I believe to be current bioenergetic models in Oct. 2010

Motivation for Modeling

There are many reasons for creating a model of the human bioenergetic system.  It allows the performance of athletes to be quantified; this allows progression of training to be tracked, prediction performance under different conditions/distances, and comparisons between athletes.  For my purposes, I am using it to determine optimal pacing strategies.  

History

 Monod and Sherrer were the first authors to try and characterize the hyperbolic relationship between time and power output, in the 1960's.  Their model is referred to as the Critical Power model; it has a constant level of aerobic power output (the "critical power"), and a finite amount of anaerobic energy (the "anaerobic work capacity").  These can be found my having an athlete undergo multiple maximal, constant-power tests of different lengths.  The work done is described by  W = cp * t + awc; by taking two data points both unknowns can be solved.  Figures 1 & 2 below show the hyperbolic curve.  
 
Zoomed out Hyperbolic Curve
Figure 1. Long Duration Power/Time Relationship
 
Zoomed in Hyperbolic Curve
Figure 2. Short Duration Power/Time Relationship
 
 
These plots make some limitations of this model immediately evident.  The largest problem is at low times where the power output goes to infinity.  This is clearly unrealistic, due to limitations of muscle tissues which have a maximum force that can be generated.  Less egregious is the long term behavior where it suggests that the critical power can be held forever.  I believe this to be less of an issue as this model will most likely not be applied to very long duration efforts.  Due to these limitations, the constant power output, and additional issues more complex models been developed.  
 

Current Models

 I am choosing to classify the bulk of available models into two categories: "Curve Fitting" and "Rate/Capacity" types.  The "Curve Fitting" models have come about from observing the variable speed of an athlete, or power output, or energy consumption, etc. and finding a set of equations which provide similar qualitative behavior.  The "Rate/Capacity" models are mainly extensions of the critical power model; they add additional rate limiters, introduce other energy sources, etc.  They also suffer from some of the same limitations of the original critical power model.  

"Curve Fitting"

Ward-Smith describes in his 1999 paper a model which I have classified as a curve fitting type.  Ward-Smith hypothesizes that three forms (direct ATP, creatine phosphate, and glycolysis) of anaerobic power all follow a similar behavior: starting from rest there is a rapid increase to a peak followed by a gentle decline.  Each form is then represented by a dimensionless variable which describes the ratio of that forms power output to some reference power output.  That dimensionless variable is defined as the product of two functions; one representing the initial rise, the other the gentle fall and each one has its own time constant.  
The next step is fitting these parameters; using a combination experimental data and previous literature, all the relevant parameters are described.  
Harman in 2002 provides another example; his equations are based on Ward-Smith's work, but apply it to the 400-meter track event and include various mechanical phenomena.  Figure 3 below shows the behavior of these models.  
Figure1 in Harman 2002

Figure 3. Behavior of curve-fitting model (from Harman 2002) 

There are many more curve-fitting models; some are much simpler with power limits described piecewise linearly, others use different parameters which are not based off of the energetic system.  These two seem to be the best in category.  

All of these curve-fitting models have a few issues though; while the original critical power model described constant power output, and these do not, these simply follow a function's profile; there is no input/output relationship.  No effects of an athlete choosing to apply a different power are allowed for.  This makes it very difficult to develop an optimal control strategy from one of these models.  

 

"Rate/Capacity" 

 Morton in his 2006 review paper goes into great detail in describing this class of models.  He visualizes most of them as hydraulic models.  There are a number of extensions to the critical power model, as well as more complicated 3-component models.  Figure 4 below shows one of the extended models.  

Figure 3 in Morton 2006

 Figure 4. Critical Power Model with Aerobic Lag (from Morton 2006)

 There also are critical power models which link the maximum available power to the reserve left in the anaerobic tank (closer to how an actual hydraulic system would behave).  

One of the more complicated "rate/capacity" models is the Morton-Margaria model, in which there are 3 tanks: 1 unlimited aerobic, 1 anaerobic glycolysis, and 1 anaerobic cp/atp tank.  Figure 5 below shows this.  

Figure 5 in Morton 2006

Figure 5. Morton Margaria Hydraulic Model (from Morton 2006)

As shown in the picture, there are many parameters associated with this model.  While it can provide a qualitatively good match to the measured data (in terms of respiration/metabolic measurements) it is very difficult to correctly identify all the parameters.  No one seems to have done a parameter matching study so far.  

 

Conclusion 

What these models show is that it is possible to match athletic performance to a set of equations.  However, when dealing with a more complex performance, the associated parameters start to lose physical meaning, making comparisons between athletes harder.  None of the "curve fitting" models allow for development of optimal control strategies, and the more complex hydraulic models have not had all of their parameters identified.  For these reasons I have chosen to use the critical power model with an aerobic delay; the limited output rate version of this model could also be tested.  
 

References

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