Crap-shoot modeling

I can be a bit obsessed at times. Especially when I want to understand something intriguing like physiology and my lactate measurement results. So yesterday I read an interesting article by Moxnes and Sandbakk:

Moxnes and Sandbakk,  Theoretical Biology and Medical Modelling 2012, 9:7

Basically, they construct a mathematical model of the following energy pathways:

Figure 1: Main pathways of energy metabolism with indication of the three metabolic energy delivery systems. From Chapter 11 from “World Book of Swimming: From Science to Performance”. 

Here is one of the plots where they compare their model with data measured on a cross-country skier (Norwegian authors!) on a cross-country threadmill:

Figure 2: Steady State concentrations of blood lactate as a function of the maximum aerobic power while threadmill roller skiing: model vs experimental data

This mathematical model is quite sophisticated and this physics guy will  need some more time to grasp all the details.

But physics guy turned manager sometimes has to prove to himself that he has not totally forgotten how to build a model. And to a physics guy the whole system looks simply like a lactate pump, pumping lactate into one container (the muscles) where it can accumulate. The muscle container is connected to a “blood” container by a diffusive wall, so lactate can move from one container to the other depending on the relative concentrations. Then there is another pump, which is the aerobic power pathway which extracts some of the lactate from the blood container. Add some realistic assumptions on how fast the different pumps switch on (the anaerobic, lactate producing one is faster in that respect than the aerobic, lactate consuming one). From Moxnes and Sandbakk I added that the efficiency of the aerobic (lactate disappearance) pump decreases as the aerobic power approaches the maximum aerobic power.

It didn’t take long to throw this into SciPy for some numerical calculations and now I could start comparing with my own experimental data. I could put in various power vs time profiles and thus model the entire test protocols including rests and the time lactate was measured after stopping the exercise.

First, there were a few parameters which I had to guess and could use to tune my model to fit the data. As the old saying goes, with four parameters I can fit an elephant, and with five I can make him wiggle his trunk.

Figure 3: Fitting an elephant with four (complex) parameters. After Mayer, Khaled, and Howard (Am. J. Phys. 78, 648 (2010).

So don’t be too surprised that I got my data to fit really well. It may look impressive to an outsider, but it really isn’t much more than a toy. (I may be a little too harsh to myself here. The model was quite insensitive to a few of the constants that I used. For example, for my maximum power I took 469W, the average I can old in a 1 minute test.

Figure 4: Comparison of model and experimental data from different measurement protocols

So now that I had my constants fitted to make the model behave nicely, I could make a few plots:

Figure 5: Blood Lactate concentration vs time at different power levels around “threshold”
Figure 6: “Long step test” results as simulated by my model (10 minute of exercise at power, a 2 minute break to measure, then on to the next power level)

If the model correct in predicting my lactate levels, you can see that my steady state power of 200W is definitely below the 2.0 mmol/L level that my on-line training friends are using. That may be a good level for them to use, but for me it is too high.

Another thing that is interesting is that things quickly go south for me when I go above 205 W. As I am planning to do a half marathon SB/PB attempt on Christmas, I used my model to look what it would predict for lactate concentration during the row.

Figure 7: Using the model to try to predict lactate for a half marathon ergometer row. Circles: Lactate concentration. Triangles: Erg power

The plot in figure 7 is a bit complex. Basically, I tried two strategies which both should bring me within a few seconds of my Personal Best of 1:22:47. Both are negative splitted, but while in the first (blue) strategy I start to increase the pace early on (after about 40% of the row is done, then at each halving of the remaining distance), the second (orange) strategy stays one second above the average pace for about an hour.

To me this seems quite relevant. Rowing for so long at threshold power, one has to be very careful to not go above it and end up with painful legs with another 40 minutes to go.

I would love to try my crap-shoot model on somebody else’s data set. Send me an email. I need a description of the measurement protocol (power vs time during exercise, rest duration (power during rest if active), and the blood lactate measured.


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