The Physics of Running Power

Physics of Running Power 

There is a certain theorm we were taught in school that goes something like this : The kinetic energy of a system of particles is the kinetic energy associated with the movement of the center of mass and the kinetic energy associated to the movement of the particles relative to the center of mass.  

This is called Koenig's theorm.

In humans (bipeds), as the center of mass is propelled, fore and hind limbs are alternatively in contact with the ground, while the upper limbs oscillate freely both during the stance and the swing phase. This results in a linked multi-segment system.

Koenig's theorm can be applied to this system to model the mechanical work done in running. 

A. External Work 

The human runner consists of a central trunk and n number of rigid segments each of mass m. The total mass M of the runner is considered to be lumped at center of gravity. 

The potential energy while running is represented by M.g.H, where H is the vertical height of center of gravity from ground.  

The kinetic energy of M is 1/2.M.Vcg^2 where Vcg is the velocity of center of gravity. 

The total external work Wext comprises of the sum of kinetic and potential energy. 

B. Internal Work

The kinetic energy of ith segment relative to body center of gravity is 1/2.mi.Vr,i^2 where Vr,i is the linear velocity of that segment relative to body center of gravity. 

The rotational kinetic energy of the ith segment relative to body center of gravity is 1/2.Ki.ωi^2 where Ki is the radius of gyration of the ith segment around it’s own centre of mass and ωi is the angular velocity of that segment.

The total internal work Wint is the summation of every segment's linear kinetic and rotational kinetic energies. 

The computational scheme of calculating internal work assumes that energy transfers take place between segments of the same limb but not between limbs or between trunk and limb. 

The total mechanical work done for running is then the simultaneous summation of total external work done and total internal work done for a particular instant.  

I've represented this in a rather quirky picture with the physical equations underneath. 

Fig 1 : Illustration showing the physical equations in external work and internal work as contributions to total work. Kleg here is a lumped stiffness constant for the leg. The yellow dot represents the center of mass and the vertical amplitude of it's movement represents vertical oscillation. 

Power is the rate of doing work. 

For example, if the runner in the picture commits 5 Joules of total mechanical work per kilogram every second, power = 5 Watts/kg (1 W = 1 Joule/second). 

C. How Mechanical Energy Changes With Running Motion

A cycle of running motion from touchdown to touchdown of the same leg is called the stride. Total mechanical work done can be resolved over many strides to see how it varies. 

Some data from empirical testing is shown in Fig. 2 to get an understanding of change in work done. For an idea of magnitude of work, a scale of 100 Joules is shown on the right.  

Observe the troughs and peaks in mechanical energy. The largest oscillations in energy come from the lower leg comprising of the thigh and the foot. Cavagna has written that the lower limb is itself responsible for about 80-90% of internal work. 

The physical understanding here is that every time the leg is on ground around mid-stance phase, potential energy is at it's minima, therefore mechanical energy of center of mass also attains minima (red lines). 

The maxima in mechanical energy of center of mass is attained at the peak of flight phase after a maximum in trailing foot pushoff work and when potential energy is at it's maxima (green lines). 

Fig 2 : Variation in mechanical energy of given sites in the human body as a function of running phase

Duty factor is the percentage of the total time between strides (or steps) that a single foot is on the ground. Values for duty factor can vary from 50 to 90 percent, but are typically in the 60 to 80 percent range. 

As duty factor increases, an individual spends more time with his feet on ground and this has implications for the maxima in mechanical energy, or maxima in power. 

In other words, we might consider that it increases the time spent around the minima of mechanical energy, which thereby might decrease the overall mechanical energy of the center of gravity and overall running velocity.  

But given the same forward speed, the only way to decrease duty factor is to increase leg turnover rate, or cadence. 

Somewhere between too high a cadence and too low a cadence, most good runners will strike a balance to optimize ground speed, time spent on the ground and total mechanical work done. 

Fig 3 : Variation of Wint, Wext and total work done at 3 different running speeds as a function of running phase

D. Average Work and Average Power

The "average work done" for a duration of say 10 minutes means resolving these peaks and valleys of an up and down work signal into it's average. 

The mechanical work done curve can be transformed to a mechanical power curve by turning it into a rate per second. The average mechanical power is about resolving this curve of peaks and troughs to an average value representing that curve. 

A math trick is to remember that the average value of any function can be represented in integral form. Integration can be electronically implemented. To clean up the resulting curves for presentation, signals can be sent through filters and/or computationally 'smoothed' over a desired interval of time. 


E. Mechanical Efficiency 

Human beings have a maximal efficiency of converting chemical energy in food to contractile muscle work of about 25%. Let's call this contractile efficiency as Contr_Eff.

In human locomotion, mechanical efficiency can be expressed in terms of how much total work you put out relative to the net metabolic cost of running. This is Run_Eff. Walking efficiency can be labelled Walk_Eff.

The energy cost of running is expressed as ml O2 consumed per kg per m. This can be converted to a metabolic power (units of J per kg per m) using the conversion of volumetric oxygen to joules. The net metabolic cost is nothing but the cost of running minus the cost of stationary standing.

The ratio of total mechanical work per kg per m (or total power) and the net metabolic cost of running is defined as the mechanical efficiency : 

Presumably, for well trained runners, efficiency is better than that for average runners. Such has been said about East African runners.

Researchers like Cavagna have seen that there are 4 trends to efficiency when they looked at motion on level surfaces :

1. Run_Eff > Contr_Eff. Apprently, this is due to the storage and use of elastic energy through the action of recoil in the lower legs between each cycle of running. The wider the separation between both, the better is the uptake of recoil elements in running.

2.  Run_Eff > Walk_Eff. This is presumably because potential and kinetic energy are out of phase in walking (rolling egg), but nicely synchronized and in-phase during running (think of a pogo stick).

3.  Run_Eff increases linearly with speed, starting at 45% and maximising somewhere between 70-80%.

4.  Walk_Eff maximises at intermediate speeds with values of 35-40%. It then falls off with further increase in speeds. This is interesting and possibly explains why the human considers running instead of walking when speed is past a certain threshold.

Fig 4 : Variation of mechanical efficiency, internal (Wint), external (Wext) and total work (Wtot) and net metabolic energy cost  (En exp) with speed in running and walking regimes. Source : Cavagna (1976).

F. Contribution of Internal Work to Total Mechanical Work

As shown in Fig 4, things really depend on the magnitude of running speed.

Researchers like Cavagna have shown that when the log of internal power is plotted against a log of speed, the resulting linear line approaches a slope of 2. Which simply means that internal power, as a crude approximation, may vary as a square of running speed.

For example, in the graph I have stuck below in Fig.5, one sees that beyond a speed of 17 kph, internal work starts to become a greater percentage of total work done and exceeds external work. 

The magnitudes of numbers are interesting for perspective. Below 17 kph, external mechanical work (Wext) varies from a high of nearly 1.5 to a low of 1.1 J/kg/m. Internal mechanical work (Win) varies from a low of 0.5 J/kg/m to a high of 1.1 J/kg/m. 

Above 17 kph, Wext varies from a high of 1.1 J/kg/m to a low of 1 J/kg/m while Wint increases from 1.1 J/kg/m to 1.6 J/kg/m.

Fig 5 : Plot of changes in Wint (black) and Wext (red) as fractions of Wtot (green) over a continuum of running speeds

For this example, if speed were a modest 7:00 min/mile (13.8 kph), this means that Wint is around 0.7 J/kg/m and Wext is around 1.3 J/kg/m. In other words, the Wint and Wext % of total mechanical power is 32% and 68% respectively. 

Presumably this means that short distance, maximal intensity runners might benefit in knowing the magnitude of internal work done, or internal power. It also means a 10K runner running at 7:00 min/mile spends 30% of his total power internally. That's a sizeable chunk of running workload. 

G. What Do Running Powermeters Measure and Not Measure? 

What they Measure : 

Running powermeters such as the Stryd and Runscribe are inherently 9-axis IMUs. By combining it with a barometric sensor, you get the ability to measure acceleration (X,Y,Z) and orientation but also altitude by measuring the atmospheric pressure and using the difference between that and sea level atmospheric pressure. This is called pressure altitude. 

Among the electrically talented, these chips are also called TenDOFs or '10-degree of freedoms' which is a fusion of 3 chipsets and a barometer which communicate to each other through sensor fusion algorithms (like a Kalman filter). The function of each of the chips are summed up below :

3 DOF Accelerometer : Senses acceleration in 3 directions - X, Y, Z

3 DOF Gyroscope : Senses angular velocity in 3 directions - Roll, pitch, yaw

3 DOF Magnetometer : Senses true orientation in 3 directions (compass)

Averaging the data that comes from the 3 chipsets is said to produce a better estimate of motion than that obtained using accelerometer data alone.

This is a great page to learn about how accelerometers work. If you want to get a practical idea of how an accelerometer works, you can play around with this app Google built. Basically it uses the sensors in your phone to give you an X and Y axis acceleration while running, but you're on your own about what to do with it.

Runscribe also has a RawData tool to inspect the raw file for the original signals. This forms part of their Science Package for researchers

Devices like Stryd are coded with a sleep mode when not active to save battery power. They 'wake up' when motion is sensed and start collecting run data only when a certain threshold in motion is passed.

The hard part is figuring out the coding. Because the raw data from 10dof's can be noisy, they have to be filtered to present meaningful motion data. 

Fig 6: An image explaining the components on a GY-80 10DOF chipset.

Since an accelerometer can integrate acceleration to get velocity and double integrate to get position, algorithms involving changes in velocity and position are easily implemented. 

Focusing on kinematics means these devices do not use any hardware to actually measure force and therefore save on lot of cost and footprint size. 

Fig 7: An image showing the assembly view of a Stryd powermeter

Unlike cycling powermeters, running powermeters have zero strain gages, therefore there is no direct measurement of force. The device simply uses a model that uses measured parameters to approximate running power. 

Fig 8 : A close look at the electronics inside a Stryd powermeter

Now precisely what algorithm the Stryd uses to measure power is not known. On their website, in a little blurb within the FAQ section, Stryd claims that by "approximating" the time-course of ground reaction force in the horizontal and vertical direction and multiplying it with velocity components integrated from acceleration, they can calculate power. This is shown in a screen capture from one of Stryd's Youtube videos.

Fig 9 : An image from Stryd's Youtube video featuring Dr. Andrew Coggan which shows the horizontal and vertical ground reaction force on the left and the accelerometer derived 'shape and form' of those forces on the right. Since the time Stryd dropped a chest mounted sensor for a footpod, they have claimed that the reproduction of ground reaction forces have become better (link in my post) 

  
The methodological debate here is how the model is approximating the ground reaction forces and with what level of accuracy does it capture that data for level and gradient running and for soft vs hard surfaces. We will not know the answer to that. That is the secret sauce after all.

I was told that values of external power had been calibrated against force plate treadmills in the laboratory within a range of 10 cm of shoe mounting height. We are told by others that the data is reasonably accurate. 

Stryd resolves power into a horizontal power and a form power, the latter which represents the cost of perpendicular bouncing in place. Stryd is marketing this as 'wasted' effort. The sum of horizontal and vertical powers become total power. 

Other power models such as that used by Garmin take a different approach and they swear by the accuracy of their models.

If two powermeters yield different values of calculated power, we can assume that majority of the differences stem from precisely what algorithm is being used. The rest of the differences are probably due to how the signals are filtered, processed and implemented in code and smoothed before they are presented to the user. 

I've conducted experiments with the Stryd during a laboratory VO2max test and there was reasonable correlation between power and metabolic cost. I do not know if similar correlational power would exist in outdoor running with weather and running surface factored in. Stryd has conducted an outdoor VO2 test using a metabolic cart loading onto a pickup and claim that power tracks running economy.  

Because external power is relatively more "stable" than heart rate and pace, it becomes a "useful" perhaps 'objective' parameter to design stress scores and performance management charts around. As a messenger of training intensity and training load, it is useful. 

There are a few key things to question when using low cost accelerometry to study human motion, particularly the messy problem of running.

1. Inter-device reproducability : Given a bunch of running powermeters from the same OEM, to what degree will each device converge upon a similar value given a controlled running task?

2. Intra-device reproducability : Given several identical running tasks over a period of time, to what degree will a given device reproduce metrics over that time?

3. Validity : Given a "measurement" from a running powermeter, to what degree are the results meaningful , i.e what co-relation do they have in relation to real physiological demands of running?

Criterion validity measures how well the data corresponds to gold standards of measuring the same thing. Convergent validity is the extent to which the measurements made by the sensor are associated with those made with other assessment methods that intend to measure the same or similar aspects.


Where to Apply the Sensors : 

There are devices that measure running power hitting the market that are affixed to different positions on the body. Some, like Garmin's pods, are affixed to the shorts. Some, like the Runscribe and Stryd, are mounted on the shoes, either at the heels or on the laces. An earlier version of the Stryd, called Pioneer, came as a chest mounted strap.

There is a debate on what is the best position. For consistent measurement, mounting at the shorts  and the chest is said to be problematic since the pod also exhibits motion when the fat layer and short waistline moves relative to the core of the body.

However, using the same argument, the Stryd foodpod exhibits relative motion on certain shoes as the foot-mount bracket "slips" on the laces. I suppose any mounting position, if properly controlled, is valid. But it does seem that the foot-mounting positions seems to be the least worst among all three positions.

What They Don't Measure : 

Estimating the true workload of running is a tricky business due to a couple of reasons.

1) First, in level running at constant speed, there is a substantial recovery of elastic energy at each stride, that brings about a corresponding reduction in the mechanical work performed by the active muscles.  In other words, when your muscles shorten to propel you forwards, some of the energy it uses has been stored from the previous stretch cycle.

On non-level running surfaces, the portion of negative eccentric work starts to become a significant factor on appreciably steep downhill slopes, where the leg muscles are working to both brake and stabilise the human runner from toppling forward. In this regime, use of elastic recoil maybe lesser than on level running.

For these reasons, it is inappropriate and erronous to assume a running efficiency value equal to that of purely isotonic work (i. e. on the order of 25 %).

Powermeters can't tell you want is truly going on with the elastic recovery portion of running, however some algorithms like Leg Spring Stiffness (LSS) maybe a step in this direction. One also has to realize that LSS maybe subject to various interpretations depending on the mathematical implementation. See this post for an overview.

2) Secondly, powermeters do not sense internal power.  In literature, researchers have summed up the kinetic energy needed to move all important limbs in running, multiplied them by two for contra-laterality and divided by the stride time (2 steps) to express that in terms of power (W).

A problem with the above calculation is that it might over-estimate the amount of actual muscular power. Consider the case where if the energy can be "transferred" from one limb to the other due to speed and momentum without any muscular contraction actually happening, the total work you calculate is greater than that actually being used.

Sensing internal work is most likely hardware and computational intensive.  Studies show that for the same running speed, different methods of calculating internal power yield around a 1000% difference between highest and lowest values. 

3) Running powemeters do not factor in wind and therefore, any extra workload to move against stiff aerodynamic resistance is unaccounted for.

For example, in metabolic terms, a +4.5 mph headwind translates to a +5% increase in VO2 according to data from Dr. Jack Daniels. For maintaining the same pace, the running powermeter will "lag" behind metabolic intensity because wind resistance is not factored for.

Since fast runners "create" their own wind even in calm conditions, not being able to assess a true intensity of working in the fluid medium could be an issue, particularly in places with high air densities and air pressures. This might present a problem to a runner with an inflexible pacing plan.

4) Similarly, external power does not factor in a temperature. It is left to the runner to calibrate a power based pacing strategy against the ambient temperature and humidity.

While power is said to be "objective", it does not in any way diminish the need for the runner to calibrate against perceived effort and possibly, even heart rate. A sensible approach is one that is holistic, especially if runner in question is someone known to push their body to the extremes. 

H. The Implications of Not Knowing a "Total" Running Power

1) One does not know the true mechanical workload of a run. 

True workload is true total workload. Unless you're a kangaroo, humans commit internal work to run. How an algorithmic estimation fares against true intensity among different runners, of different age groups, of different geographical backgrounds, on different terrain - all carry an uncertainty to it. At the heart of why you would want to measure the intensity of an exercise is the ability to get valid information.

2) For faster runners, not being able to assess Wint means not knowing a sizeable proportion of total workload that may contribute (or deduct) from movement efficiency. 

Maximal running elicits high amounts of joint torque and power in fractions of seconds. Short distance track runners who use a substantial portion of limb power to propel forwards are probably better off with traditional or slightly more advanced techniques of training to extract maximum potential. 

3) Metrics using external mechanical power may not actually explain performance differences among runners and may not be helpful to address biomechanical issues.

The complexity with running lies in the fact that for the same speed, there are wide variations in economy among runners. We still don't exactly know what makes East African runners so good at what they do but various theories have been provided, one of them being running economy.

Traditionally, economy has been measured in metabolic terms. For example, if you need 200 ml O2/kg/min to run at 7:00/mile, that's your running economy at that speed.

With the coming of power sensors, a commercial market of sorts has opened up to introduce new ways to interpret this data. There's a plethora of metrics thats pouring out from these efforts.

It is my humble opinion that some caution must be exercised when basing value judgements using efficiency metrics calculated using external power.

For example, Stryd's analysts write that running efficiency is 20-25% and that 40% efficiency is "inhuman" without distinguishing between a total running output and external running output. Expressing clarity in what goes into the effiiency calculation avoids confusion. As we have seen before, mechanical efficiency can be substantially different based on speed, running surface and elastic energy recoil. 

In another example, Andrew Coggan and TrainingPeaks have introduced a "novel" metric called Running Effectiveness. This is a complex metric that is based on a ratio of speed (in m/s) over external power to weight ratio (W/kg). 

Notwithstanding the need to get several things correct in this ratio and filter for course and weather decoupling to get a sensibly stable number, it must be borne in mind that the power in the denominator is still an "external power" only. 

Therefore, trying to use Running Effectiveness for basing value judgements about runners maybe methodologically flawed. Depending on speed, a major chunk of total power - internal power - is not factored in.

Similarly, the inverse of the above metric is packaged into a metric called Energy Cost of Running (ECOR) by the authors of the book Secret of Running. In the book, they implore runners to monitor this metric and try to reduce energy cost.

Personally, from more than 10 months of running data, I do not have confidence that any decrease I'm seeing in ECOR isn't simply a function of normal variability in the data. Therefore, the analyst must be aware that a change that is less than or within the tolerance attributable to device variability is not conclusively a positive change purely from training.

The general advice to reduce cost of running is absolutely well taken, but hang on. Again, ECOR is calculated using only external power and presents a possibly limited picture of true cost. If you accomplish reducing ECOR in your runs, so what? Is an improvement in ECOR actually tied to something happening within the body?

Therefore, I do not think it is helpful to use a surrogate cost of running based on external power for cross-comparisons when you do not address and control for a major chunk of running biomechanics which is the movement of the limbs.

As Donald Rumsfeld once remarked, there's known unknowns and unknown unknowns. Internal power is a known unknown. The unknown unknown is what fraction of total power the internal power really is and how that varies among people. A reduction in ECOR or an increase in RE may not disclose the entire picture.

Carrying an evidence based approach in the application of such metrics is advisable. For example, a validation study could be conducted on an appropriate sample of runners to assess the correlational power of metrics like Running Effectiveness in relation to being able to explain actual performance variations among runners.


ALSO READ :

Technical Review of the Stryd Power Model


REFERENCES

External, internal and total work in human locomotion.
P. A. Willems, G. A. Cavagna, N. C. Heglund
J Exp Biol. 1995 Feb; 198(Pt 2): 379–393.

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

Dubai Desert Road Run 10K Race 1 : The Metrics

Today's Dubai Desert Road Run 10K packed a quality field of 392 with some of the fastest male and female runners in Dubai showing up. As many observed, this became a fast race for a season opener and there was plenty of visible excitement in the air to complement the energy.


Overall Standings

The top 100 in standings show who's done their homework over the summer. I'm somewhere in there. It seems to me that older individuals are getting faster while (we) the whatsapp generation continue to tumble down in fitness. Need to reduce thumbing up and down stupid messages and run more! 

Fig 1 : Official results, top 100 at Desert Road Run 10K Race 1

I didn't exactly throw a kitchen sink at the race. Starting a bit on the conservative side and tossing a negative split cost me a bit of ground to cover in the later part. However, for the same course and same race for more or less similar ambient temperature profile, I broke a 3 min PR from 2013 which is interesting. As you can tell, I do not do a lot of these 10K's.

The following data table are my numbers from the race. Cost of running and running economy are surrogate values calculated purely from external mechanical power and running speed. I also choose to leave critical power and/or FTP from the data table. The -ve splitting today meant dabbling with something like a -3% to +20% CP distribution beginning to end, capping with a final kick at + 60% CP.

Fig 2 : Run data for Desert Road Run 10K Race 1

A Look at Riegel Fatigue Factor

Readers will note that I extensively explored Riegel fatigue factors for both world class male and female racing in this post and this post. Many readers on Facebook expressed the concern for applicability of the exponents to everyday age-groupers.

Very conveniently, I ran a constant pace 3K (00:12:12) and a 5K TT (00:23:40) within 1 month's gap of each well before today's race. The first and second TT were in Abu Dhabi, by the Corniche, an obviously humid place to run due to the effect of water. Today's race in Dubai was in-land, a tad bit on the cooler side and much less humid.

The slope of Ln Speed to Ln Distance, when corrected to the Riegel fatigue factor becomes 1.0995 with a 51.7% regression fit. This exponent predicts a 10K time of 00:51:00 from previous 5K time. Actual finish time however was 00:46:18.

Fig 3: Riegel exponent for race and pre-10K time trials executed in the months of August and September 2017

Which means nearly 5 minutes is unexplained by Riegel and that's inexcusable for my finishing time and placing. The poor linear fit of the slope is most likely due to insufficient data or variabilities in course, temperature and pacing strategy.

I will build this up as I accumulate more seasonal data for other distances and hopefully I can get a more composite picture of what the final fatigue factor is.

Well done to all runners today. See you soon.   -Ron

New Fatigue Factors for Running : Men's Road and Track Racing

Zersenay Tadese during the unbroken world record half marathon at Lisbon in 2010.

Readers will recall that in a recent post, I did an exercise to calculate Riegel fatigue factors from updated world record performance times for women's road and track racing. I then ran some numbers on a near-world class local female runner aged almost 40 to predict her half marathon time from her most recent 10K time. The Riegel model predicted the closest times to her actual finishing time compared to several other models widely available on the internet. 

In this post, I attempt the same thing for men's road and track racing.

I. Fatigue Factors for Men's Road Racing

I constructed a table with the latest road racing world record times for men. The following ln-ln plot of time to distance results in a slope of -0.0497 (R²  = 0.8101). When corrected into the fatigue factor form, it results in a number 1.0497. This is a bit different to Riegel's estimation of 1.07732 back in the day, a percentage decrease of 2.5%. 

Fig 1 :  ln-ln graph of men's world class road racing bests

Fig 2 : Table showing the data behind the updated Riegel model for world class men's road racing.

Apart from Stanley Biwott's 1:27:13 30K, no major recent records have been set in other distances. This shows the strength of Kimetto and Tedese's placings in the long and mid-distance categories. These records haven't been broken for several years although EliUd Kipchoge seems very hungry in 2017. A world record attempt at the marathon may mean taking out the half marathon world record in the process. 

II. Fatigue Factors for Men's Track Racing

The following ln-ln plot of time to distance for men's track results in a slope of -0.0777 (R²= 0.9901). When corrected into the fatigue factor form, it results in a number 1.0777. 

There are two interesting aspects to this number. First, the linear fit is very close to actual results. Secondly, this number is very close to Riegel's estimation of 1.07732 back in the late 1970's.

Fig 3 :  ln-ln graph of men's world class track racing bests

Fig 4 : Table showing the data behind the updated Riegel model for world class men's track racing.

No recent upheavals have happened in men's track racing, with the last record in the 25k and 30K distance being established in 2011. El Guerrouj, Bekele and Gebrselassie's records meanwhile have stood the test of time proving the remarkable forms they were in when they competed.

III. Application to a World Class Local Male Runner Y

I applied the Riegel fatigue factor for young male world class road racing to a 30 year old long distance runner Y to see how off the predictions are when using his 10K time to predict half marathon time. Both the races were held in the U.A.E and just a couple of months apart. 

Runner Y is a local running celebrity of sorts, having established record times in everything from your weekend park run predictor to the RAK half marathon. 

The math behind the following predictions were already explained in the past post. 

Riegel fatigue factor = 1.0497
Individual : Y, local male sponsored athlete
Age Bracket : 30-35
Actual 10K time (T1) = 31:40
Predicted 21.1K time (T2) = 1:09:21
Actual 21.1K time  = 1:06:10

Outcome : The prediction was under-estimated by 3 minutes 11 seconds. Runner Y finished in 1st place at the half marathon. The 2nd placed male runner finished in 1:15:06. For that difference in outcome between first and second place, the Riegel method still predicts the podium. So in a corrupted yet logical way, it appears that the difference of 00:03:11 is something we can live with. 

IV. Comparison of Riegel Predictions to Other Time Prediction Models for Runner Y

The following table compares Y's actual performance with old and new Riegel models and several other running predictors easily available on the internet. Please keep in mind that the case study was to predict half marathon time using 10K time.

Fig 5 : Comparison of updated Riegel prediction for men's road racing against 10 other models widely available on the internet. The error in prediction in also calculated and shown.

As shown in the table, the updated Riegel method performs the best prediction among several other models. In the women's case as well, it performed the best half marathon prediction. 

Please note that the Runner's World predictor tool did not work. The times entered were outside the "limits" of the tool, which perhaps is something deliberately programmed by them to prevent application for elite athletes. 

V. Conclusion

In this writeup, I updated the fatigue factor or what people term the Riegel exponent to include road racing and track racing data for men. The updated fatigue factor for world class road racing = 1.0497 and that for world class track racing = 1.0777. 

As opposed to women's racing, what is remarkable is that several of the men's records in both track and road have stood the test of time. Very few records have been broken in recent years. This further re-inforces the fact that there is more news to cover in women's racing, it's alive and kicking and these are exciting times. 

Men have some catching up to do. 

I used the updated Riegel model for men's road racing to predict the half marathon performance of a 30-35 age bracket world class road runner living in the U.A.E. The Riegel model appears to give the best prediction when compared to 10 other models. Further, the updated Riegel fatigue factor gives a better prediction in the case study that the number from Riegel's original paper. 

When considering both the men's and women's prediction I wrote about, the Riegel method appears to be a simple and reliable method that anyone could easily program. The fatigue factor can also be tweaked to fit the data of specific runners.

Since world records are being broken year after year, it seems justified that running prediction calculators for world class runners should update their fatigue factors as new data comes in.

Coaches are advised to guide with application and interpretation issues for recreational runners especially those above the age categories that the model was made from. As shown in this post, prediction tools are only a guideline. 

Therefore, two words of caution :

A. The understanding is that specific data for a runner takes precedence over estimations done using data for world class athletes, the latter which can detrimentally over-estimate race time if mis-applied.

B. Specific data involving similarities in race course and weather conditions takes precedence over distributed data containing a variety of different races courses and weather. In general, it is better to avoid exceptionally tough courses and/or conditions involving generous amounts of assistors, such as a tailwind or rabbits, from the dataset.

References

Riegel, Peter S. “Athletic Records and Human Endurance: A Time-vs.-Distance Equation Describing World-Record Performances May Be Used to Compare the Relative Endurance Capabilities of Various Groups of People.” American Scientist, vol. 69, no. 3, 1981, pp. 285–290. Link to paper.

Prediction Models :

Runner's World Predictor : Link.

Daniel's Equivalence Predictor : Link.

McMillian Race Predictor : Link.

Godwin Race Predictor : Link.

Purdy, Cameron & VO2 Max Predictors : Link.