Friday, April 6, 2018

The Running Locomotor : Cost of Transport and Work Efficiency

The technically minded performance runner would want to be deeply interested in the inner workings of the human locomotor, and the numerical possibilities associated with the business of running and running performance. Through a series of articles, I hope to probe into and gain a deeper understanding of these possibilities. As Prof. di Prampero wrote in the Journal of Sports Medicine in 1986, man is the only machine to be able to move about and also understand how he does it at the same time. 

One can draw interesting parallels between the power production processes at the cellular level in the human body and the 4 stroke combustion engine. 

In the latter, a governing thermodynamic cycle requires that a mass flow of combustibles flow into a chamber as a "batch" process. A mix of gasoline and air is introduced into the combustion chamber, said mix is compressed to high pressures, said mix is then ignited by a spark plug consequently intiating the power stroke which delivers useful mechanical power to a flywheel. In the final stroke, exhaust products are expelled out of the combustion chamber. 

Muscle is a chemical engine in the human locomotor. Electron microscopy has revealed, quite beautifully, that there exists a molecular "power-stroke" that is ultimately responsible for muscle contractions. 

Very simplistically, the requirement to deliver contractile force causes a "spark" from an innervating motor unit strong enough to activate clusters of muscle fibers according to the size of the demand.

Substrates chemically combine in coupled reactions and the energy used to make ATP. All human movement is paid in ATP. At the muscle sacromere level, the hydrolysis of a molecule of ATP hydrolysis leads to the cross-bridging of the protein myosin over another protein actin causing contraction of a sacromere.

great video shows this elaborate molecular "power-stroke" in actin-myosin overlap. For academic purposes, one can read about fascinating molecular motors. Research on motility within muscle has spanned several decades and we still only continue to learn about the molecular agents responsible for muscle contraction.

I. Cost of Transport 

The cost of transport becomes a decision maker for vehicle purchase. For example, a 40mpg family sedan will consume approximately 7 liters of fuel per 100km. A figure like this is considered 'good' by today's standards and gets a strong weighting factor in purchase.

In the human locomotor, oxygen uptake reflects the quantity of ATP used when aerobic metabolism can provide all of the energy at a given steady state running speed.

Given the conditions that running is steady state and no accumulation of lactic acid takes place, the oxygen cost of sub-maximal running (ml O2/kg/min) above resting value is known to be a linear function of running speed. This oxygen cost, when expressed on a per minute basis, becomes the "metabolic power".

Metabolic power divided by speed of movement yields cost of transport.

COT = Metabolic Power Demand ÷ Running Speed

where units are :
Cost of Transport, COT  = mlO2/kg/m
Metabolic Power Demand (net or gross) = ml/kg/min
Running Speed = m/min

Note : COT is also called  Cost of Running or simply Energy to Run (ECOR, Cr, Er etc) in some works.  If expressed as an energy cost (J/kg/m), the volume of oxygen uptake has to be converted to it's energy equivalent.

Net metabolic power demand and running speed assumes linearity to a good degree; the slope becomes COT.  As science consuming readers, we might be able to hold confidence in the linearity between metabolic power demand and running speed upto maximum metabolic power because a large collection of published studies show this correlation (Fig 1).

The linear relationship essentially means that COT is independant of running speed. That is, regardless of the speed of running, the runner's energy expenditure per unit distance is constant.

Fig 1 : Data accumulated from 10 studies (n=130) for adults performing treadmill running (8-20 kph speeds) show the linear relationship between oxygen cost (ml/kg/min) and running speed (kmph). In this dataset, the average regression line approximates. Oxygen cost (ml/kg/min) = 2.203 + (3.163 x kph). If VO2 = A + B x Speed, A = 2.203 +/- 8.285 and B = 3.163 +/- 0.474. Males (71.5%), females (28.5%), trained (50%), untrained (1.5%) and unknown training status (18.5%). Reference [3]. 

Fig 2 shows COT values for several runners from a popular marathon in Geneva published in [7]. It is interesting to note that for the same sub-maximal running speeds, COT differs among the runners sometimes upto 20%!

Therefore, calculating COT yields an excellent barometer by which to judge different runners just as fuel consumption guides vehicle purchase.

Runners with a low COT have greater margin to push speed leading to superior performance to cover a given distance. Following that thought, we might consider that the hypothetical runner with a superior COT would be the one to break the marathon sub 2 hour barrier.

Fig 2 :  Energy cost of running (COT) at constant speed on flat terrain as a function of speed. Filled symbols refer to the two less economical and open symbols to the two most economical among 36 subjects taking part in the “Marathon International de Genève”. Reference [7].

II. Influencing Factors of COT

A) Substrate Use 

It is essential to know in what proportions the human locomotor uses fats and carbohydrates to fuel exercise in order to derive the energy equivalents of their associated consumptions. Metabolic substrate use is dependant on intensity of run and variable of interest is decide fat-carb use and aerobic and anaerobic regime of operation is the respiratory exchange ratio.

For example, if the intensity is low enough in constant speed running and respiratory exchange ratio (RER) is 0.7, the human locomotor is known to operate in a predominantly aerobic fashion oxidising fats (palmitate). Based on this knowledge, a calorific equivalent of 19.6 J/mlO2 is used for this operational regime.

However, when exercise intensity increases and RER approaches 1, fraction of glucose (carbs) aerobically metabolised must be account for.  Glucose yields 21.1 kJ/mLO2 and is therefore volumetrically 7% more energetic than fats. (diesel automobile enthusiasts will fondly remark that diesel fuel is volumetrically more efficient than petrol and that we ought to use diesel more!).

Under the simplifying assumption of zero anaerobic contribution, an average value of 20.9 J/mlO2 is used in literature to account for both fats and glucose oxidation, although this average value corresponds to a RER = 0.96.

Beyond a respiratory exchange ratio = 1, the human locomotor is anaerobic and equivalency value of 20.9 J/mlO2 without inclusion of the energy contribution of lactate introduces an error into the calculations. Therefore, improper assumptions about substrate use can lead to error-prone estimates of energy production depending on training status of the runner.

Fig 3 : The red lines indicate the corrected VO2 equivalent of running as a function of running intensity in sloped and level running conditions when blood lactate contribution is accounted for.  Black lines neglect this contrbution. In this particular study on trained runners, the difference of neglecting lacate contribution amounted to a mean value of 0.02 mlO2/kg/m for level running and 0.03 mlO2/kg/m for sloped running. Reference [11].

Aerobic Regime

The shape of the aerobic COT in relation to running intensity has been reported to mildly curvilinear that tends to flatten out with intensity. The net oxygen consumption in the following relation is dependant on subject and assumes that during locomotion, the resting metabolism remains unchanged.

Aerobic COT = [Net Oxygen Consumption x Calorific Equivalent] ÷ Speed

where units are :
Aerobic COT = J/kg/m
Net Oxygen Consumption  = ml/kg/min [Reported to be between 3.5-5 ml O2/kg/min]
Calorific Equivalent = J/ml
Caloric Equivalent of Aerobic Metabolism (Fat & Carb) = 20.9 J/mlO2  (average value)
Speed = m/s

Fig 4 : Non-linear increase of aerobic COT in several non-athletic male subjects (n=29) while running indoors. Reference [4].   

Anaerobic Regime

When the energy demand exceeds the locomotor's aerobic capacity, the fraction of energy production from anaerobic sources come into the picture. A byproduct of anaerobic metabolism is lactate, therefore measurements of blood lactate ([bLA]) in standardized laboratory protocols constitute a valid cardiorespiratory assessment of exercise intensity. Not accounting for [bLA]'s energy contribution (what literature calls "oxygen debt") may have varying degrees of error based on the subject measured on and the exercise intensities (Fig 3).

The precise shape of the anaerobic COT in relation to running intensity has been reported to be sharply curvilinear. The net increase in blood lactate (net bLA) is multiplied by an equiavalent of 60 J/kg/mM or 3 mlO2/kg/mM to determine the net energetic value of lactate. When divided by the overall distance covered, one gets the net anaerobic COT.

AnaerobicLa COT = [Net bLA x O2 equivalent x Caloric Equivalent of Carb. Oxid.] ÷ Distance

where units are :
Anaerobic La COT = Anerobic Lactate COT, ml/kg/m
Net bLA rise  = mM/l
O2 Equivalent = ml/mM/kg. This is between 2.7 and 3.3 mlO2/mM/kg.
Caloric Equivalent of Carb. Oxidatation (glucose) = 21.131 J/mlO2
Distance = m (running time x speed)

Fig 5 : Non-linear increase of anaerobic COT in several non-athletic male subjects (n=29) while running indoors. Reference [4]. 
A third contribution to energy supply comes from anaerobic alactic stores, or the cleavage of phospocreatine PCr but this is only prominent in short distances under maximal running conditions. For example, in the 400m sprint, 10-12% of total energy has been reported to come from this contribution. However, in long distance running, this contribution maybe conveniently neglected.

AaerobicaLa COT = [PCr x O2 equivalent x Caloric Equivalent of PCr] ÷ Distance

The metabolically derived COT, COTm is a summation of anerobic and aerobic contributions. 

COTm (J/kg/m) = Aerobic COT + Anaerobic La COT 

COTm from a large number of studies done on athletic subjects approach the value of 0.9 kcal/kg/km or 3.7 J/kg/m in indoor conditions without environmental influences.  Under the same conditions, the non-linear shapes of the aerobic and anaerobic COT combine to produce a net linear shape in COTm as shown in Fig 1.

B) Environmental Conditions : Accelerated Running

The above discussion is valid for indoor settings. In an outdoor running environment, the influence of air resistance starts to play a substantial role in fast running. Furthermore, accelerated running out of block starts such as track running incurs a kinetic cost of accelerating the body from zero to final speed in the acceleration phase.

The energy cost of overcoming wind resistance is particularly appreciable beyond 5 m/s. For a man of 1.75m and 70kg in mass, wind resistance only accounts for 6.5% of the total cost although it can and have been known to affect speeds significantly in short distance track races.

Even under still wind conditions, runners "create" their own wind by virtue of moving speed. Speeds approaching the sub-2 hour marathon barrier (5.8 m/s) under still wind conditions will require +8% higher energy compared to running with no air resistance (Pugh, 1970).

C) Environmental Conditions : Slope of Terrain

A strong environmental condition known to affect COT is the slope of the running surface. Fig 5 distills the work of some prominent researchers on slope effects. Recall that Minetti's regression 5th order equations for slope effects on running cost are also reflected in the GOVSS power calculation algorithm.

So we see that upto a slope of 2%, COT is linear. Running on a slopes of 3-5% will require upwards of  10J/kg/m! Therefore, higher work loads can be accumulated under hill running in a given amount of time compared to flat running and this has implications for training. On the other hand, in a race or long hiking situation on very steep terrain, the runner is faced with how to minimize energy costs of travel. There is advantages in traversing up a zigzag path to artifically flatten the slope.

Fig 6 : COT along the direction of motion as a function of the incline of the terrain. COT is independant of speed and only depends on slope. Reference [7].

D) Environmental Conditions : Heat and Humidity

The running machine faces a substantial reduction in work capacity in hot and humid climates. The reasons are seen below.

Considering the running locomotor and the ambient surroundings (ground + air) as a thermodynamic system, heat production is a function of energy cost, speed and weight :

Heat Production = COTm x Speed x Weight

where units are :
Heat production = Watts
COTm = Joules/kg/m
Speed = m/s
Weight = kg

On the other hand, heat dissipation is a function of surface area (or mathematically the square root of body surface area). Heat dissipated by means of conduction, radiation and evaporation added to the storage of heat within the body must balance heat production.

Heat Production = Heat Lost in (Conduction + Evaporation + Radiation) + Heat Stored in Body

The technical issues that lead to an impact on running speed are the following :

1. The running locomotor's aerobic capacity or VO2max could shrink, hence there is a derate in aerobic potential.
2.  The running locomotor faces a cardiovascular drift running in the heat.
3.  Heat production is constrained by speed and weight
4.  Heat dissipation is constrained by body surface area, temperature and relative humidity

Fig 7 : Heat production (W, Y-axis) as a function of running velocity (X-axis) and COT. Iso-temperature lines are shown in bold. Reference [12]. 

Ultimately, what this entails is a substaintial % decrease in sustainable speed in hot, humid environment dictated by the need to be able to cool the body. This leads to the following realities :

1. Distance runners are smaller than middle-distance runners to limit heat production, because weight has a 2-fold effect on heat production compared to heat dissipation.
2. Long distance running speed is temperature derated in hot climate because the running locomotor seeks to maintain heat balance without letting core body temperature rise to dangerous levels. For example, marathons in temperatures of 20± 25°C are 6%±10% slower than marathons in temperatures of 10±12°C.
3. Increasing age possibly has a multiplicative effect on COT degradation as well as the effect of the ability to shed weight. 

Noakes published an interesting graph indicating speed cutoffs to maintain estimated heat balance. Observe that for heavier runners, the speed derate are higher.  These are only indications, rather than absolute values as they mentioned in their paper.

Fig 8 : Illustration of the derate in running speeds where heat production and maximum heat dissipation are in balance. Illustation provided are indications, not absolute values. Reference [13]. 

E) Environmental Conditions : Altitude

It is known that COT falls with rising altitude. Overground sea-level oxygen cost of running has been reported by Daniels to be 4.5% greater than that measured at an altitude of 2,300m.

This has been attributed to the a) greater reliance on carbohydrate at high altitude for the same absolute running speed, which serves to explain the lower metabolic cost since the oxygen uptake for metabolising carbohydrate is lower than that for fats and b) lower work of ventilation due to lesser resistance to breathing [8]. However, since carbohydrate stores are low and due to low partial pressures of oxygen at great heights, these advantages are negated and the human runner has to compromise on work intensity to survive over long high altitude distances.

F) Other Contributers to COT : Training Status, Mass and Size

Training status has the ability to affect the Overall COT. Though literature is filled with estimates ranging from 6-24% reduction, an estimate of 8% can be expected in beginner runners on a 10 week training program, anywhere between 2-7% in endurance runners and about 7.5% after 9 weeks on an explosive training regimen. However, all of these estimates are subject to the specific protocol administered and calculations used.

Humans adapt with running training. They lose fat mass, build muscle and may alter their biomechanics in a way that elicits more tendon contribution in energy storage. Certainly the fat mass loss with training is something all runners are familiar with. Loss of fat around distal areas of the limb possibly lead to higher reductions in COT. The reduction of every 100 grams of mass from around the feet can lead to nearly 1% reduction in COT. This reduction is fairly consistent across a range of running speeds.

Several researchers noted that size and stature invariably affect the oxgen cost of running, with larger individuals having a lower energy cost and younger children haivng a higher energy cost. An analysis of studies report a gross estimate of 2% increase in the gross energy cost of running from ages 18 to 8 years.

A Size-Independant Cost of Transport by dividing COT by the product of mass and height did not solve the interdependancies of mass to oxygen consumption. Alternative hypothesis suggest that the larger the body dimension, the larger the amount of energy stored and released through the stretch-shortening cycle of the leg extensor muscle (see below).

Under the dictation of some of the above influencers, a correction to the laboratory estimate COTm can be written as :

COT = COTm + Correction Due to Combination of (Wind + Altitude + Slope) Effects

III. Predicting Time For Covering Distances

The overall cost of transport is a powerful metric. Knowing COT and the maximum metabolic power in proper units helps assess what is the maximum possible speed the human locomotor can achieve.

Maximum Running Speed (m/s) = Maximum Metabolic Power ÷ Overall COT

where units are :
Maximum Running Speed = m/s
Maximum Metabolic Power = W/kg or J/kg/s
Overall COT = J/kg/m

Since operating at the maximum metabolic power results in fatigue in the locomotor within approximately 7 minutes (runner dependant), the following equation allows the prediction of time to cover short distances upto 3000m :

Best Short Distance Running Time (s)  = Distance ÷ Maximum Running Speed

For longer distances requiring more than 420 seconds of running time, it is impossible for the locomotor to sustain maximum metabolic power without fatigue. In this running regime, only a fraction of maximum metabolic power can be sustained and therefore, the endurance time is approximated by :

Best Endurance Speed (m/s) = Highest Fraction of Maximum Metabolic Power ÷ Overall COT

Best Long Distance Running Time (s) = Distance ÷ Best Endurance Speed

So herein lies a secret to running. For a given metabolic power, best endurance speed is achieved by being able to race at a higher fraction of that metabolic power. But since metabolic power is itself under several environmental influences such as heat and altitude, there are uncertainties to such simple predictions.

IV. Towards Optimizing Cost of Transport

Fascinatingly, both mechanical engine and human locomotory movements exhibit a "non-linear" shape to cost of operation.

Consider that the fuel combusting mechanical engine has a speed and torque dependant optimum fuel efficiency. Driving a car "too slow" or "too fast" introduces a bigger penalty on brake fuel consumption than a more optimum cruise speed somewhere in between.  Therefore, an "island" that contains the optimum fuel consumption is by design placed at mid-engine speeds and high torque.

Fig 9 : Optimum island for specific brake fuel consumption in an engine. Reference [1]

Similarly, when COT data is collected for several runners, an optima for low COT shows up at a specific value of running speed. For example, in the data below, minimum COT appears to be around 11.1 kph. Therefore, just like the mechanical engine, there is an optimal movement load and speed for lowest costs. 

Fig 10 : Optimal speed to minimize COT in 9 trained runners. Reference [2]

At a kinematic level, speed is determined by the product of stride frequency and stride rate. The running locomotor attains a minima in COT at an optimum stride frequency that varies from individual to individual.

A set of data from 12 subjects in Fig 8 show a U shaped profile in COT with respect to stride frequency. A similar behavior is also seen in cyclists, where the optimum pedaling frequency for low metabolic cost is around 60rpm. Yet, cyclists impose a self selected 90rpm possibly to reduce torque demand and muscular effort.

Fig 11 : Relation between cost of transport (COT) and stride frequency for 12 physically fit and experienced runners. Reference [5].

Scientists tell us that metabolic cost is primarily linked to the cost of producing muscular force. So could optimal movement speed be governed by the force and speed of muscular firing?

For example, it is known muscles are governed fundamentally by force-length and force-velocity relationships. At a whole body level, it is also known that human runners incur the least operational cost at an optimally selected stride frequency. These relationships are fundamentally non-linear in nature and subject to inter-individual differences.

Secondly, operating the locomotor with a shorter ground contact time involves fast fiber contractions (faster muscle firing) leading to higher energy costs. This explanation has served very well in understanding why smaller animals have higher metabolic costs and COTs.

The practical takeaway from this discussion is that a self imposed step frequency may or may not necessarily correspond to the optimum required to achieve minimum metabolic cost and minimum COT. There is some trainability value in this aspect.

V. Apparent Efficiency of Running

In addition to the costs of transport, knowing efficiency as it relates to the maximum extractable work for a given metabolic input is also required to analyze the performance of any engine.

What's the efficiency of running?

At this juncture, we need to define the apparent efficiency of running.  The apparent efficiency can be seen as the end result of all possible losses and energy saving mechanisms during complete cycles of running motion.

Apparent Efficiency = Total Mechanical Work Done / Metabolic Cost

where total mechanical work done = external work + internal work

Apparent efficiency can be either expressed as "gross" or "net" depending on whether the energy cost of vital functions that are not directly related to exercise (e.g. the O2 consumption of the brain, of the gut, kidneys and internal organs, as well as the minor fraction due other organs' metabolism) is included in the metabolic cost (the denominator).

Apparent Efficiency (Gross) = Total Mechanical Work Done / Gross Metabolic Cost

Apparent Efficiency (Net) = Total Mechanical Work Done / Net Metabolic Cost

Where Net Metabolic Cost = Observed Metabolic Cost - Resting Metabolic Cost
Resting Metabolic Cost = Approx. 300 mlO2/min (≈ 1.5 kcal/min or ≈ 100 W)  for an adult man of about 70 kg body mass and is essentially unaffected by the exercise.

Muscular contractions require splitting of ATP, the energy currency in the body. To synthesize ATP can take several substrate routes but if we assume exercise to be fundamentally aerobic, then the amount of oxygen processed in unit time becomes a proxy for the power of cellular energy production.

The process leading to the splitting of ATP in the isolated muscle comprises two steps and each of these steps have an associated efficiency.

1. ATP-synthesis/energy liberation from decomposition of nutritients : Phosphorylative coupling
2. Energy liberation during ATP-splitting/ATP Hydrolysis : Mechanical coupling

Overall muscle contraction efficiency =  Phosphorylative coupling efficiency x Mechanical coupling Efficiency

A range of reported values for phosphorylative coupling efficiency and mechanical coupling have been reported in literature (Fig. 12) Certaintly, it appears that aerobic work is more efficient overall with the ATP resynthesis efficiency being as high 64% which when multiplied with a modest 40% for ATP hydrolysis efficiency yields an overall efficiency = 25.6%. On the other hand, anaerobic muscle efficiency maybe somewhat lower at 21.5% as reported by Margaria. 

Fig 12 : Components in ATP turnover efficiency in human muscle. Reference [6].

However, running is not purely contraction movement, rather a mixture of positive work (the push-off) where ATP is split to apply force against the terrain and negative eccentric work (the landing) that dissipates energy while the terrain applies force on the body. It is then the apparent efficiency of positive-negative work that deserves attention. 

Researchers discovered several decades ago that the summation of the theoretical oxygen requirement to power the different parts of the body during endurance running is over-estimated by approximately 50% when they compared theory to empirical data from level running experiments. In other words, the apparent work efficiency of whole body running was greater than the 25% efficiency for isolated muscle contraction.

How much greater? 40% or more for level running (Fig 13, 14).

Fig 13 : Values of mechanical efficiency from several studies. 

Scientists agree that a reason for high apparent efficiency has partly to do with the fact that during the landing phase of running, passive, elastic elements that are connected to muscle bellies in the human body absorb some of the elongation of the muscle, store and release energy into the next phase of the cycle. 

In other words, the human runner can activate "pre-stretch" in series connected elastic elements in the musculotendon unit just before touchdown, thereby storing energy which is then re-used for powering the next takeoff. The reduction in oxygen consumption is explained by the reduction in concentric response from the muscle and the lowered speed of contraction.

What this simply means is that work done by tendons does not have to be performed by muscles - therefore, tendons reduce muscle work, and therefore metabolic cost, during running.

This fact is empirically supported, with many studies showing that runners perform the work of running with an efficiency that exceeds that of isolated muscle (Cavagna et al., 1964; Heglund et al., 1982; Minetti et al., 1999). These observations support the idea that tendons do much of the work ‘for free’, thus increasing the apparent efficiency. This does not violate the principle of the second law of thermodynamics, as some people mistakenly claim.

VI. Spring Mechanics and Efficiency

Human locomotors naturally oscillate like a bouncing ball in order to run forward. As discussed in another post, human running can be approximated very well by a linear spring-mass model. However, because some energy is lost at each step due to friction and heat (attenuation), muscles need to constantly add some energy to the system to power forward movement.

Inspite of this little complication, the relation of COT to speed, stride frequency and the elastic behavior of the human running motion can all be fascinatingly tied up to support the metabolic cost of force production hypothesis.

The empirical finding on a treadmill was that humans chose a self-selected stride frequency corresponding to one which minimized metabolic energy expenditure,  maximises apparent work efficiency and which corresponds closely to the calculated natural frequency of the "spring" (assuming damped harmonic motion).

The beautiful plot in Fig. 9 reveals more details. At low, medium and high running speeds (5.3 kph - 11.1 kph), human runners' freely chose a running cadence that corresponded to the minimum metabolic cost and maximum apparent efficiency.

This was despite the fact that mechanical power was greater at low cadences due to higher vertical work against gravity and lower at higher cadences due to a minimzation in vertical work done. In other words, these studies suggest that the running locomotor is somewhat blind to mechanical power minimization and instead the goal is to optimize cadence around the point where work efficiency is highest and metabolic cost lowest.

Fig 14 : The ratio of imposed step frequency and freely chosen cadence approaches 1 at the point where metabolic cost is minimized and apparent efficiency is maximized.  Reference [9].

One also notices in Fig 9 that the apparent work efficiency increases as the speed increases, the magnitude is more than double (50%) of the metabolic efficiency of converting chemical energy to work in isolated human muscle (25%).


The human engine is likened to a mechanical engine, where molecular motors power the strokes responsible for movement while converting only a portion of the input energy to actual work. This post explored two key areas which influence human running performance - the cost of transport and the apparent efficiency.

Cost of transport is driven by metabolic substrate use and the effect of environment upon cost.

The learning process in endurance running is mostly about finding how to strike a balance between this need to achieve speed on one hand and minimizing cost of transport and maximizing efficiency on the other. It boils down to the cost of producing force at the molecular level and researchers are only beginning to understand that there maybe a tradeoff between force and efficiency, i.e some fundamental mechanistic limitations prevent muscles being both powerful and efficient at the same time [10].

Numerous technologies are available today to aid the human runner to find their "pace"; these however are simply aids and the best runners still appear to run mostly by "feel". We know very little about how the human locomotor judges and acts upon internal and external feedback signals by way of sensory control systems but the fact is that it happens. So it becomes essential to investigate what those effort governance theories are and what the supporting observations might be. This aspect will be tackled in a future post. 


[1] Brake Specific Fuel Consumption (BSFC) – (2018).
[2] Mayhew, J. L. (1977). Oxygen cost and energy expenditure of running in trained runners. British Journal of Sports Medicine, 11(3), 116–121.
[3] Leger, L., & Mercier, D. (1984). Gross energy cost of horizontal treadmill and track running. Sports medicine, 1(4), 270-277.
[4] Beneke R, Leithäuser RM. Energy Cost of Running Related to Running Intensity and Peak Oxygen Uptake. Dtsch Z Sportmed. 2017; 68: 196-202.
[5] Lieberman DE, Warrener AG, Wang J, Castillo ER. Effects of stride frequency and foot position at landing on braking force, hip torque, impact peak force and the metabolic cost of running in humans. Journal of Experimental Biology [Internet]. 2015;218 :3406-3414.
[6] Scott, Christopher B. A Primer for the Exercise and Nutrition Sciences: Thermodynamics, Bioenergetics, Metabolism. Humana Press, 2010.
[7] di Prampero. Mechanical efficiency, work and heat output in running uphill or downhill. Annales Kinesiologiae. 2011.
[8] Fletcher JR and MacIntosh BR (2017) Running Economy from a Muscle Energetics Perspective. Front. Physiol. 8:433. doi: 10.3389/fphys.2017.00433
[9] Cavagna, G. The Resonant Step Frequency in Human Running. Pflügers Archiv. 1997; 6:678–684
[10] Barclay, C. (2017). The basis of differences in thermodynamic efficiency among skeletal muscles. Clinical and Experimental Pharmacology and Physiology, 44(12):1279-1286
[11] REIS, Victor Machado, OLIVEIRA, Diogo Roberto, CARNEIRO, André Luiz, FERNANDES, Hélder Miguel, & SCOTT, Christopher. (2016). Inclusion of blood lactate O2 equivalent in the VO2-intensity regression at level and 10.5% grade running. Revista Brasileira de Educação Física e Esporte, 30(2), 255-261.
[12] Arsac, L. M., Deschodt-Arsac, V., & Lacour, J. (2013). Influence of individual energy cost on running capacity in warm, humid environments. European Journal of Applied Physiology, 113(10), 2587-2594. doi:10.1007/s00421-013-2696-6
[13] Dennis, S. C., & Noakes, T. D. (1999). Advantages of a smaller bodymass in humans when distance-running in warm, humid conditions. European Journal of Applied Physiology, 79(3), 280-284. doi:10.1007/s004210050507

Saturday, February 10, 2018

Technical Analysis of a 1:38:00 RAK Half Marathon

Going sub 1:40:00 in any half marathon is considered a decently competitive time achievement for age groupers.

Consider that this year at the RAK Half, only 12% of the competing runners (both male and female) posted a gross time better than 1:40:00 and the rest 88% were slower. By net time, I assume this percentage will be still lower.

In this year's RAK Half, I bested the previous year's race time by nearly 14 minutes and certainly, I've never run a half this fast since college years, so therefore it is an all-time PB at this distance.

What does it take to run sub 1:40:00? I would like to share some of my own data to shed some technical light on the subject.

Note : Please click all images to zoom in.

Race Day Ambient Conditions

The morning of race saw a starting time temperature of 15 deg C, 88% humidity with 11 kph northerly winds (most likely measured at 10m height off the ground). Barometric pressure was 1014mbar. 

Between 7 and 9am, the temperature rose maybe 2 degree C at most.

These conditions yield a calculated air density of 1.2203 kg per cubic meter and a WBGT (wet bulb globe temperature) of around 16 deg C. 

Race Course

The AIMS certified course is 21.1 km long. The course is mostly very flat with 30-40m of total ascent making it suitable for a flat out race.

Incline data as a function of distance :

Weight Trend 

The trend of weight (without shoes etc) several weeks leading into the race is shown below. The weight trend hovers over 62-63 kg. At the time of the race, adding the mass of shoes and running attire to that figure would put me at a racing weight of approximately 63 kg.

HRV Trend 3 Weeks Before Race

A monitoring of daily heart rate variability 3 weeks from race day revealed that :

1) RMSSD fluctuated with highs reaching the weekend (Friday) of weeks 3 and 2 before the race. Hourly tapering of runs (last 2 weeks) showed a decrease in daily RMSSD.

2) Log transformed RMSSD normalized to R-R intervals, an indicator of fatigue, actually increased during the time 3 weeks before the race and declined during the last taper week.

Not reading too much into this but 1) and 2) might indicate a readiness to perform & heightened parasympathetic activity during tapering period. My weekly hours and average training paces for the last 3-4 weeks to the race are included in the 3rd plot below.

Certainly interesting and deserves more study.

HR Trimp Performance Chart Trend

The following image shows TRIMP performance chart (generated in Golden Cheetah) along a 2.5 month period from Nov 20 2017 to race day on Feb 9 2018  (I ran a 10K race in early November 2017 so I chose to start tracking PMC then). 

While absolute values are not important, the trend says I was more or less in a maintenance phase in the months of Nov and Dec by running an average of 3-4 km every day. In the month of January 2018, I picked that dosage upto >4 km every day. Therefore, I accumulated some residual fatigue indicated by the stress balance line (yellow) following which a taper period relaxed the stress balance to -4 just before race day. Overall fitness (blue line) increased gradually to a peak a week before race day. 

It is interesting to keep the -4 stress balance in context with the race performance achieved at the race. Atleast what the curve shows is that I went into the race slightly fatigued but not at a level that made me dysfunctional.  

Certainly, a performance management chart can be made using the language of external power and RSS, but to me, TRIMP and HR based PMC is much more trust-able when I want to assess heart stress.

Race Data

Aggregated run data is shown below from several devices, namely the Polar V800, Runscribe+ (beta) and Stryd powermeters.

Net Time : 1:38:00 (Data from GPS and 2x accelerometers)
Pace : 4:32 min/km | 7:18 min/mile | 13.23 kph | 8.22 mph | 3.67 m/s

Ave. External Running Power : 234 Watts  (to move center of mass)
Ave. External Power to weight ratio : 3.7 W/kg  
External Power Intensity : 95-98 % of Critical External Power 
Basis of Critical Power : Exponential fit over 90 Day power duration curve
Normalized External Race Power : 233 W

GOVSS Power : 373 W (a proxy for internal + external run power)

Ave. Heart Rate : 190 bpm
Racing HR as %  : 91% (Karvonen method)
HR Zone Distribution (Polar) 

Total Steps : 18,584 
Ave. Step Length : 1.16 m | 3.8 ft
Ave. Stride Length : 2.32 m | 7.6 ft
Ave. Step Rate : 191 steps a minute
Ave.  Stride Rate : 95 strides per minute 

Estimated Vertical Oscillation of Center of Mass : 0.061 m
Ave. Ground Contact Time : 0.215 seconds
Estimated Leg Spring Stiffness : 11 kN/m

Ave. Impact Shock : 12.2 G (correlates with vertical ground impact force)
Ave. Braking Shock : 10.6 G (correlates with horizontal braking forces)

Kinematic Variables & Their Distribution Over Time 

The following series of time series screenshots show box plot distributions of kinematic variables, something I really like. On the Y-axis is dependant variables of interest (such as ground contact time) and on the X-axis is time. 

Step Rate 
Half way point
Median : 189 spm
Overall, high cadence and very even throughout. 

Ground Contact Time
Half way point
Median : 0.218 s
Overall, low GCT and very even which speaks for the evenness in step rate and footstrike type.

Flight Ratio
Half way point 
Median : 30.9%
Overall, a slightly fluctuating proportion of flight time around the 30% mark. In all my past data at these paces, I have not seen numbers substantially higher than 30%.

Stride Length 
Half way point 
Median : 8.4 ft (2.56 m)

Stride Length (Left/Right Distribution)
A bit doubtful on the data but interestingly, it's showing that a decrease in SL in one of the feet is complemented by an increase in the other foot. The dark blue curve must be for the right foot. I'd have to continue to monitor this in past and future data to understand if this is a real variation between left and right sides or just noise.

Footstrike Type
Half way point 
Median : 9 (Between midfoot and forefoot)
This data comes from accelerometers strapped to the heel but eitherway, the indication is not far from what I thought it'd be. 

Impact Force
Half way point
Median : 12.4 G
This is not an indication of actual force but certainly a proxy for negative vertical acceleration. 5-15 Gs is a normal range. 

Braking Force
Half way point 
Median : 10.9 G
This is not an indication of actual force but certainly a proxy for negative horizontal acceleration. 4-13 G's is a normal range.

Run Power (GOVSS)
Half way point 
Median : 421.6 W
GOVSS power involves a computation of internal and external power to run and uses an efficiency correction upon metabolic demand. In other words, this plot gives an indication of total metabolic demand with time.

Pace and Power Splits

Maximum variation in pace = 21 seconds/km.


Readers might recall my post on my dismal performance in the same race in 2017. In that year, I dragged myself across the finish line in 1:52:00 and went back home pissed and determined to get better next year. 

In that post, I revealed pace and power histograms and some other interesting metrics. This season, a self-coached and methodical running streak from September 2017 resulted in a strong performance and a 12th place in my age category. 

I still think the most basic of all tools - a training log and a simple stopwatch - should inform most runners how structured their plans are, if they are making improvements and how much rest they are getting in between. 

The plethora of data metrics from inertial measurement units, heart rate monitors and GPS devices are nice to have and for the added tradeoff in analysis time, you get some marginal improvements in information which may or may not suit everyone.

Yet, we should not lose sight of the forest for the trees. Summing up some greater generalities about achieving sub 1:40:00, I have the following points :

1) Specificity of training : Commit a purpose to most runs, if not all runs.

2) Reverse your gear : Work backwards from shorter distances. You must break barriers in short distance (aerobic) racing to get faster at longer distances. Remember our friend, Riegel? A 44:00 10K will lobby harder for your sub 1:40:00 half campaign than one slower. 

3) Run slow to run fast :  Increase volume of low intensity runs and optimize volume of fast runs. 

4) Be a good guest : Our friend is Mr. Improvement, we'd like him in our house. Complement training sessions with adequate rest in between; an extensive endurance run may take 8-12 hours for supercompensation timing while an intensive anaerobic training session might require 40-60 hours for the same.  Doing stupid things when these bodily changes are taking place will shut the front door on Mr. Improvement. Corny, but this is fact. 

5) The journey counts : Take a year to work towards the half marathon goal of 1:40:00. Run with friends, run often, have fun.

Saturday, January 27, 2018

Ground Contact Variables Affect External Running Power Derived From Accelerometry

In my previous post, I reviewed Stryd's running power model. While looking into Stryd's whitepaper and reading several other journal papers, I suspected that if running powermeter algorithms employed summation of energy changes at center of mass, then could variation in detected ground contact variables explain some of the striking variations in reported power between competing accelerometer platforms in the market?

Ground contact variables are things like ground contact force, ground contact time, stance time, step time, step rate, vertical oscillation and so on. Minimizing errors between detected variables and laboratory equipment may minimize variations in computed power, however we are yet to understand how these accelerometers work in outside running relative to variety of footstrikes and terrain types and how those errors vary relative to actual variables.

To understand what the effect of ground contact variables is on computed running power, I played around with some hard numbers and did a sensitivity analysis using a power model I built.

First, I got in touch with Professor Alberto Minetti to get some raw force plate data for running. Professor Minetti is widely regarded as an expert in the biomechanics of running and is an honorary research professor at Accademia Nazionale dei Lincei in Rome, Italy. He also leads the Laboratory of Physiomechanics of Locomotion at the Department of Pathophysiology and Transplantation at the University of Milan.

Prof. Minetti shared force plate data for a shod front foot striker running at 4 m/s, which is 14.4 kph or 6:42 min/mile. 

I then developed a model to estimate the rate of changes in external work done using the EESA algorithm (see previous post). The model applies the same computation algorithm for external power as described in several sources in literature. 

I. Effect of Variation in Detected Ground Contact Time On Running Power

Stryd's whitepaper showed that the error in ground contact time between force plate and their footpod IMU is 2.83%. The running speed and footstrike type within the data was not discussed. These images from my previous post are reproduced here. 

Figure 1 : Modeled vs actual vertical force-time signatures in a rear-footed runner for an unspecified running speed. Base image courtesy of Stryd. Markups by the me. Observe that the Stryd thinks the footstrike is front footed when the force-time signature from the forceplate shows a rear foot strike.

Figure 2 : Plot showing goodness-of-fit of modeled GCT to force plate measurements. The average error is stated to be 2.83%. The number of runners, running speeds, shoes worn, footstrike mechanics and slope on the treadmill are all unknown which raises the question of how the error varies as a function of each of those factors.  Image courtesy of Stryd.

Now to the data from Prof. Minetti and my model :

Time signatures of the forces in the front foot striking runner are shown in Figure 3, where Fx, Fy and Fz are antero-posterial (A-P, or horizontal), vertical and medio-lateral forces respectively. Mechanical energy changes for all steps in the data and per step are shown in Figures 4 & 5 respectively. 

Figure 3 : Force-time signatures for a 67kg front foot striking shod runner running at 4 m/s. Force plate data of Minetti, Milan (Italy). Force plate acquisition frequency = 1000 Hz.

Figure 4 : Mechanical energy changes for a 67kg front foot striking shod runner running at 4 m/s for 3 steps.

Figure 5 : Mechanical energy changes for a 67kg front foot striking shod runner running at 4 m/s for a single step.

External power (Pext), vertical power (Pv), horizontal power (Px) and lateral power (Pz) are shown in the tabulated data in Figure 6 for a -5 to +5% variation in detected ground contact times. 

Figure 6 : External running power and it's 3D components per step for a rear footed shod runner running at 4 m/s. Values were computed using the EESA algorithm (or Cavagna method) as described in literature for a range of ground contact times from -5% to +5% relative to the highlighted base value. Computer model was developed by Ron George. 

As shown in Figure 6, a variation of -5 to +5% in detected ground contact time has an impact of -5 to +5% in estimated external running power, all other factors kept the same. 

Here, you can see power extending from 365 Watts to 331 Watts because of the error in vertical force-time signatures. 

A computational difference between platforms for ground contact time can arise from 3 
factors :

1. Errors in estimated ground force-time signatures from accelerometry.
2. Inability to differentiate between rear foot strike and front foot strike and suble variations in between.
3. Variations in the minimum force threshold set in the power algorithm while computing ground contact time. In literature, the threshold value from a minimum of 10 N to as large as 50 N, which translates to an appreciable difference in estimated stance times for a given running speed. 

This is just one source of variation possible between different accelerometers in reported power. Another example of variation is from differences in estimated vertical oscillation or vertical translation of the center of mass. 

II. Effect of Variation in Detected Vertical Oscillation On Running Power

The Stryd whitepaper revealed that an error of 3.18% existed between force plate vertical oscillation and that derived from the footpod. This is shown in Figure 7. 

Figure 7 : Estmated vs actual vertical oscillation in the Stryd footpod. Stated average error with respect to force plate = 3.18%. Base image courtesy of Stryd. Markups by me.

What does an error of 3.18% in vertical oscillation mean? It simply means that there is a difference in the vertical landing force that the footpod detects relative to force plate data. Since vertical velocity and vertical oscillation are single and double integrals of detected vertical accelarations respecitvely,  any error in the acceleration signal translates into errors in the vertical oscillation.

Now to the data from Prof. Minetti and my model :

External power (Pext), vertical power (Pv), horizontal power (Px) and lateral power (Pz) are shown in the tabulated data in Figure 8 for a -5 to +5% variation in accelerometer derived vertical oscillations.

Figure 8 : External running power and it's 3D components per step for a front footed shod runner running at 4 m/s. Values were computed using the EESA algorithm (or Cavagna method) as described in literature for a range of vertical oscillations from -5% to +5% relative to the highlighted base value. Computer model was developed by Ron George. 

As shown in Figure 8, a variation of -5 to +5% in derived vertical oscillation has an impact of -0.3 to +1% in estimated external running power respectively, all other factors kept the same. 

Here, you can see power extending from 347 Watts to 351 Watts because of the error in vertical force-time signatures. 

A computational difference between platforms for vertical oscillation can arise from errors in the vertical acceleration and vertical force-time signatures computed by the device. 

III. Effect of Variation in Horizontal Speed On Running Power

The Stryd whitepaper revealed that an error of 5% existed between force plate horizontal speed and that derived from the footpod. This is shown in Figure 9. 

Figure 9 :  Stryd modeled forward speed change compared to force plate measures. Indicated accuracy = 95%. Grade of running surface unknown, but presumably level. Image courtesy of Stryd.

Now to the data from Prof. Minetti and my model :

External power (Pext), vertical power (Pv), horizontal power (Px) and lateral power (Pz) are shown in the tabulated data in Figure 10 for a -5 to +5% variation in accelerometer derived horizontal speed. 

Figue 10 : External running power and it's 3D components per step for a front footed shod runner running at 4 m/s. Values were computed using the EESA algorithm (or Cavagna method) as described in literature for a range of horizontal speeds from -5% to +5% relative to the highlighted base value. Computer model was developed by Ron George.

As shown in Figure 10, a variation of -5 to +5% in derived horizontal speed has an impact of -7 to +16% in estimated external running power respectively, all other factors kept the same. 

Here, you can see power extending from 324 Watts to 402 Watts because of the error in horizontal force-time signatures. 

This analysis is a little incomplete because if horizontal speed changes, so can the vertical force-time signature. Therefore, there are multiplicative effects on computed power from the coupling of horizontal speed and vertical force.

A computational difference between platforms for horizontal speed can arise from errors in the horizontal acceleration and horizontal force-time signatures computed by the device. 


A running powermeter that utilizes ground contact variables in it's calculation of running power can spell out a range of values for power depending on variation in those ground contact inputs. 

The impact of error in three of these ground contact variables, chiefly vertical oscillation, ground contact time and horizontal speed, were explored in this article by independantly varying force time signatures and inspecting their impact on computed power. 

Other sources of error exist. For example, if an accelerometer and the algorithm used cannot distinguish between a front footed strike and a rear footed one, the underlying force impulse characteristic is misjudged. Misjudging impact force-time signature impacts the computed potental energy change and hence potential work.  

Based on the analysis for one front foot runner at 4 m/s, one observes that some of the striking differences in reported power between accelerometer platforms like Stryd and Garmin may lie in the variations in derived ground contact variables. Out of the three explored variables, errors in vertical and horizontal force time signatures can make an appreciable impact on the theoretically computed power (See Figures 6 & 10). 

It is my hope that this writeup gives runners and coaches a qualitative and quantitative feel for the impact of accelerometer error upon the new metric of running power. Basing training prescription on faulty devices and secondary & tertiary metrics derived from them can carry a risk. Not being aware of such risk within new technology has implications for both undertraining and over-reaching.