Monday, September 25, 2017

Berlin Marathon 2017 : Estimating Eliud Kipchoge's Running Power



Eluid Kipchoge pounded out the tarmac at the Berlin Marathon today with the 5th fastest marathon time of 2:03:32. 

Berlin was wet and windy this year and Kipchoge had no benefit of drafting in the latter stages. We know that cost of running against the wind becomes significant after 20kph. 

The extra power to run against the wind and maintain pace on wet roads may partly explain the huge difference in finishing time between Nike's Sub2 attempt at the Monza track and today's IAAF sanctioned marathon. Conditions also meant Kipchoge didn't even beat his own personal best of 2:03:05 today at the marathon. 

The average pace from the determined veteran especially at the middle of the race today was astonishing, so I decided to take a stab at estimating running power from pace data. 

Many thanks to Dr. Pietro Prampero from beautiful Italy for helping out with some of the math surrounding metabolic power. 

Earlier in the day, Ross Tucker posted Kipchoge's 5km splits overlaid on other key pieces of information on his Twitterfeed. These are possibly unofficial, but to dive into the numbers, they'll do fine for now. Many thanks to him as well.

Fig 1 : 5K split data for Kipchoge at Berlin Marathon 2017. Courtesy Ross Tucker, Science of Sport.



Method 1 : Estimating Kipchoge's Marathon Power from Riegel Profile

Fig 2 : Riegel slope for Kipchoge's best times. Resulting fatigue factor = 1.0236.

Kipchoge's best times at the 10K, Half Marathon and Marathon distances are 00:28:11, 00:59:25 and 2:03:05 respectively. Using those times to construct his Riegel fatigue factor on the Ln Speed vs Ln Distance graph gives a result of 1.02369 (Fig 2). Contrast this with the general men's road racing Riegel fatigue factor of 1.0497.

Using this fatigue factor to predict Kipchoge's absolute best time from the half marathon time of 00:59:25 results in 2:00:47!  This is just 22 seconds more than the actual time he ran at Monza during Nike Breaking2's 'deeply pampered' attempt. So we know this is somewhat possible but somewhat also impossible on an IAAF course.

Some amateur predictions of best power to weight ratios in the different classes of runners was posted on Stryd's facebook page in September. The numbers of best in class power to weight ratios were obtained using formulae popularized in the Secret of Running book. A screenshot of the exact posting is below for reference in Fig 3 (click to zoom in). 

If we assume that these best in class ratios are for external power only, then we can assume that Kipchoge's 10K time of 00:28:11 might be very close to 100% of his critical power. If Kipchoge's weight is 56kg, this results in a 10K power of 386 W for 6.9W/kg, equalling the table of best in class ratios.

If that's the case, the recommended pacing from Stryd's guideline for the marathon is 89.9% of 100% CP which is the 10K power. Using 10K power of 386W, this results in a marathon external power of 347 W.

Is this predicted by Riegel?  Using the uncorrected Riegel exponent of -0.0236 applied to an assumed half marathon power of 364 W (6.5W/kg) results in a Riegel predicted marathon power of 352 W.


Fig 3 : Power to Weight ratios for different classes, posted by Michael Arend on Stryd's Facebook forum.


Method 2 : Estimating Kipchoge's Running Power from Metabolic Power

For outdoor running, the product of the mass and distance normalized energy cost of outdoor running (Crout), and the forward ground speed (v) yields the net metabolic power necessary to move at the speed in question.

Gross Metabolic Power, W/kg =  Crout   x  v   ..... 1)    

where units are :
Crout in  J/kg.m
v in m/s

The metabolic power is required to reconstitute the ATP utilised for work performance, regardless of the actual oxygen consumption which may be equal, greater, or smaller than the metabolic power itself.

During constant speed running on flat compact terrain the net energy cost of running (above resting, Cr) is independent of speed and amounts on an average to 4  J/kg.m [Lacour and Bourdin, Eur. J. Appl. Physiol., 2015].  This is strictly true for treadmill running in which case the energy for overcoming the air resistance is nil.

When running on terrain in the absence of wind, the overall Crout, including the energy expenditure against the air resistance is larger than that applying to treadmill running (Cr_Tr) by an amount proportional to the square of the air velocity (which in this case is equal to the ground speed, v):

Crout  =  Cr_Tr  +  k’.v²  ..... 2) 

The values of the constant k’ (J.s2 .m-3.kg-1) reported in the literature range from ≈ 0.012 [Pugh, J. Physiol., 1971, di Prampero J. Sport Med.,1986] to ≈ 0.018 [Tam et al., Eur. J. Appl. Physiol., 2012].

Based on average Berlin marathon speed for Kipchoge and an assumed average k’ = 0.015,  Crout =  4.49 J/kg/m. This cost of running is assumed to stay constant over the duration of the marathon, but in reality, it may even increase.  

Equation 2) also shows that the effects of the air resistance are not as high as one would expect; indeed, up to a speed ≤ 20 km/h, the energy expenditure against the wind accounts for ≤ 9 – 14 % of the total cost.

To establish a time of 2:03:32, Kipchoge ran at an average pace of 5.7 m/s. If we assume this to have been his maximum aerobic speed, vaer,max,  his maximum aerobic metabolic power can be estimated from the relation :

Maximum Gross Metabolic Power, W/kg = (vaer,max  x Crout ) / F    ...... 3)

where F is the fraction of maximum metabolic power.

Elite marathoners are known to utilise 75-85% of their aerobic maximums for the marathon. An assumed F = 0.85 means Kipchoge's maximum gross metabolic power = 30 W/kg.

Assuming 20.1 Joules of energy per ml of O2 , Kipchoge's VO2max = 89.7 ml O2/kg/min.

Also, if we assume a modest conversion efficiency of 23%, then :

Kipchoge's Mechanical Running Power = 23% of 30 W/kg = 6.9 W/kg

A lower fraction F will mean higher mechanical power demand to run at this speed, so for the same metabolic efficiency, it benefits Kipchoge to operate at a high fraction of his aerobic potential. This shows the importance to elite marathoners of increasing F and decreasing Crout to it's minimum possible. 

How much optimization is possible becomes the eternal question for discussion. 




Method 3 : Estimating Kipchoge's Running Power from Regression Curves

The Berlin Marathon course is more or less flat, with some 35m of elevation gain in total. Since power is more or less directly proportional to speed on flat terrain, I needed a simple regression equation expressing power as a function of pace for running.

Diving into old research papers, Cavagna et. al's 1967 work on external work in level running is not a bad place to start.

Cavagna wrote then that in running, the potential and kinetic energy of the body do not interchange but are simultaneously taken up and released by the muscles with a rate increasing markedly with the speed.

If that is correct, then total external mechanical work done during running must be in phase. In other words, external work done is approximately equal to sum of work done in forward motion and work done against gravity. 

Wext ~ Work done to lift center of mass + Work done to move forward

From experiements done on experienced males performing constant pace runs in a heavily instrumented indoor corridor, Cavagna et.al plotted a graph of cal/kg/min (external power) vs average forward speed and found the following relationships (filled dots represent running) : 

Fig 3 : Empirical data for pace vs external mechanical power for experienced runners. Cavagna (1976).  

With a little bit of graphics trick, the equation of the regression line was extracted as :

Cal/kg/min = (4.8223 x km/hr speed) + 8.9124
R² = 0.9997

Based on splits and an easy regression formula from Cavagna's data, the table of estimated power/weight values for 42.195 kms converted to external mechanical power, Watts/kg are shown below :


Fig 4 : Table of calculated power to weight ratio's (external) per given amount of distance at the Berlin Marathon 2017. 


The power to weight numbers are truly astonishing as to seem almost unlikely, since it is based on a trend line. Therefore, I call it a 'high' estimate in the column.

From Cavagna's own graph, you can see a variance of 0.7 W/kg for the same running speed of 20 kph. So at the low end, you end up with a mechanical power to weight ratio of 6.8 W/kg. This more or less matches the estimate using Method 2. 

I think the best we can do right now is express mechanical power within a range, the large uncertainty is justified due to insufficient empirical data. However, it is a first good guess.

If Kipchoge's weight is assumed to be 56 kg, this results in the following range for power, accomodating all the numbers from Methods 1-3 :


Range, Mechanical Power (Kipchoge) = 350-420 Watts




Weaknesses of Various Estimation Methods

The estimations are not without issues : 

1) Riegel factor mis-predictions : The external mechanical power results from Method 1 are constructed using time prediction Riegel formulas applied to power. It is understood that this is external power as distinct from internal power. Riegel fatigue factors are specific to Kipchoge but the Riegel method has been known to sometimes over-predict and sometimes under-predict actual timings. One would assume the same inaccuracies to fall through to power. This is specifically because course and weather patterns are not accounted for in these simple formulae.

2) Validity of Riegel exponent for power : While power is proportional to pace, we don't know if the Riegel exponents can be applied to external mechanical power in the same way as it is done to time. There is no evidence that external power scales the same way for world class runners. Do good runners get better at running faster but while conserving mechanical power and increasing mechanical efficiency? Without answers to such questions, a blind application of Riegel exponents to scale external power to world class runners maybe erronous.

3) Power to weight ratios for world class runners : Any classification of external mechanical power as a function of body mass^1 for world class runners vs recreational runners done by individuals is unvalidated by science. To generate these tables, a Riegel exonent of -0.07 is also used by the creator of the tables which maybe erronous. The issue is that we are not sure if it's external mechanical power alone that separates the best from the rest or if other factors such as mechanical efficiency also help explain the differences. It goes back to all the complexities behind what makes good runners or really good runners the way they are.  We are also not sure if body mass^1 in the power to weight ratio number is correct. We might find that an allometric exponent applied to body mass in the ratio might have a higher correlation to metabolic cost. Physically, this might mean that world class runners, especially east Africans, who are smaller in stature, might exhibit higher mechanical efficiencies while running.

4) Assumptions of VO2 utilization and metabolic efficiency : Methods 2-3 make assumptions about fractional VO2 utilization and metabolic efficiency that seem almost arbitrary. There is no empirical data specific to Kipchoge to back this up. I'm also not sure if runners like Kipchoge can getaway with a lower VO2 utilization by relying heavily on carbohydrate rich drinks throughout the run. 85%, while in the range of some papers in literature, is on the high side. Is 85% of VO2 for a marathon sustainable in practice?

5) Savings due to pacers : The calculation neglects a shielding factor in the beginning of the race due to the formation of pace makers. Therefore, it speaks nothing about the savings from those initial stages of the race. Fluid phenomena can only be simulated using fluid codes that are expensive and take time. An example is Siemens' fluid simulation of Nike's Breaking2 attempt published on Linkedin.

6) Savings from shoes : Any effect the shoes had on his run is neglected. Is there some hidden juice in the spring carbon plate? Should it be deemed important, why could a debutant marathoner Guye Adola stick with Kipchoge 98% of the way without any publicized aids on his feet? That's up for debate. My argument is that any aid obtained from the shoes is negligible in the big scheme of things. 



Conclusion

From first order mathematical relations, I estimate that absolute external power Kipchoge used to run the Berlin Marathon in 2:03:32 is in the range 350-420 Watts. The mechanical power was established using three methods and several assumptions. The Riegel power prediction is the most conservative. The latter two methods predict numbers on the high side.  

Popular social media site Strava's analysis into Kipchoge's pacing structure shows that Kipchoge had just 8 seconds of total variation in pace throughout the duration of the marathon. That somewhat matches Ross's 5K split data as well. 

Since power is proportional to pace on flat terrain, we can "assume" a proportionally tight variation on actual external power. 

The post acknowledges that calculated external power values maybe on the high side for a thin, lightweight runner such as Kipchoge. Weaknessess of the estimations were spelled out in a separate section above. 

Kipchoge can run with lower power numbers by economizing on his cost of running and maximizing on the fraction of his aerobic potential.

Given that his competitors, themselves top names in the running business, couldn't seem to hang on neither today nor at Monza inspite of a systematically pampered course must tell something about Kipchoge's high fractional VO2 utilization and low cost of running. This remains to be verified. 


References

1. Cavagna, G. A., Thys, H., & Zamboni, A. (1976). The sources of external work in level walking and running. The Journal of Physiology, 262(3), 639–657.

2. Joyner, M. J., & Coyle, E. F. (2008). Endurance exercise performance: the physiology of champions. The Journal of Physiology, 586(Pt 1), 35–44.

3. Strava analysis of pace variance, Link :

4. Riegel Fatigue Factors for Men's (Updated) : Link

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