### The Physics recap

In an earlier blog post I examined the influence of altitude on the physics of cycling’s world hour record, and showed how the reduction in air density as altitude increases means one can travel faster for the same power output, or put another way, the power demand reduces at any given speed as altitude increases.

That resulted in this chart, which shows the relationship in power to drag ratio (W/m^2) for speeds ranging from 47km/h up to Chris Boardman's 56.375km/h record. I've slightly amended the chart to extend up to altitude of 3,000 metres. Click on the pic to see a larger version.

Each slightly curved coloured line represents a speed as marked, and from that you can see how the W/m^2 required reduces with increasing attitude. The chart clearly suggests there is an advantage to performing such record attempts at higher altitudes, but it's never that simple of course.

### And the Physiological impact...

As we climb to higher altitudes and air density drops, the "thinner" air also means a reduction in the partial pressure of oxygen, which negatively impacts the power output we can sustain via aerobic metabolism. That loss of power can be as much as 20% or more depending on how high we go, and our individual response to altitude.

So the gain in speed from the physics side of the equation is somewhat negated by the reduction in physiological capacity. But by how much, and what might be the optimal or "sweet spot" altitude for a cyclist seeking to set a new record?

The physics side of the equation is easier to predict than the physiological, since the physics applies equally to all, however individual physiological response to altitude is quite variable, both from person to person, but also depending on how well a rider has acclimated to altitude. There are even differences in how altitude affects elite versus non-elite riders.

There have been a few published papers examining the impact of altitude on aerobic athletic performance and from these formulas to estimate the loss of power as a function of altitude have been developed. There was one from the 1989 paper by Peronnet et al, two from the 1999 paper by Bassett et al, one each for acclimated and non-acclimated athletes. Adding to those, I have generated a fourth formula, based on the 2007 study by Clark et al. The relevant papers are:

So the gain in speed from the physics side of the equation is somewhat negated by the reduction in physiological capacity. But by how much, and what might be the optimal or "sweet spot" altitude for a cyclist seeking to set a new record?

The physics side of the equation is easier to predict than the physiological, since the physics applies equally to all, however individual physiological response to altitude is quite variable, both from person to person, but also depending on how well a rider has acclimated to altitude. There are even differences in how altitude affects elite versus non-elite riders.

There have been a few published papers examining the impact of altitude on aerobic athletic performance and from these formulas to estimate the loss of power as a function of altitude have been developed. There was one from the 1989 paper by Peronnet et al, two from the 1999 paper by Bassett et al, one each for acclimated and non-acclimated athletes. Adding to those, I have generated a fourth formula, based on the 2007 study by Clark et al. The relevant papers are:

Péronnet F, Bouissou P, Perrault H, Ricci J.:

A comparison of cyclists' time records according to altitude and materials used.

Bassett DR Jr, Kyle CR, Passfield L, Broker JP, Burke ER.:

Comparing cycling world hour records, 1967-1996: modelling with empirical data.

Clark SA, Bourdon PC, Schmidt W, Singh B, Cable G, Onus KJ, Woolford SM, Stanef T, Gore CJ, Aughey RJ.:

The effect of acute simulated moderate altitude on power, performance and pacing strategies in well-trained cyclists.

Peronnet et al used empirical data from actual world cycling hour records to estimate the impact of altitude on an elite cyclist's power output. The assumptions used in estimating altitude induced power loss may have some error; in particular due to methods used to estimate the power for each rider as neither the power nor coefficient of aerodynamic drag was actually measured.

According to the old Wattage forum FAQ item by Dr David Bassett, Jr, the two Bassett et al formula were derived from earlier papers examining altitude impact on aerobic performance of four groups of highly trained or elite runners. So while these formulas were not derived from cyclists we can still generalise from those to the loss of aerobic capacity for cyclists.

Finally, the study by Clark et al measured the impact on peak oxygen utilisation (VO2), gross efficiency and cycling power output on ten well trained but non-altitude acclimated cyclists and triathletes by testing riders at simulated altitudes of 200, 1200, 2200 and 3200 metres. They examined a number of factors, including maximal 5-minute power output, VO2 and gross efficiency relative to performance at 200 metres, as well as sub-maximal VO2 and gross efficiency.

I used these data to generate a formula similar to those from Peronnet et al and Bassett et al. Of course there is an assumption of an equivalent reduction in 1-hour power as for 5-minute power. Clark et al noted slightly greater reductions in VO2 peak than for 5-minute maximal power, and no change in gross efficiency at 5-min max power with altitude. So there is some anaerobic metabolic contribution presumably making up the difference. There was some loss of sub-maximal efficiency noted at a simulated 3200 metres.

I chose in this instance to use the reduction in 5-minute power rather than fall in VO2 peak as the base data for the formula, and applied an adjustment to offset the formula for sea-level equivalency to bring it into line with the formula by Peronnet et al and Bassett et al. Of course when you look at the reported data there are of course sizeable variations within the test group at each simulated altitude, so the formula is based on group averages for each simulated altitude.

Here are the formulas:

The effect of acute simulated moderate altitude on power, performance and pacing strategies in well-trained cyclists.

Peronnet et al used empirical data from actual world cycling hour records to estimate the impact of altitude on an elite cyclist's power output. The assumptions used in estimating altitude induced power loss may have some error; in particular due to methods used to estimate the power for each rider as neither the power nor coefficient of aerodynamic drag was actually measured.

According to the old Wattage forum FAQ item by Dr David Bassett, Jr, the two Bassett et al formula were derived from earlier papers examining altitude impact on aerobic performance of four groups of highly trained or elite runners. So while these formulas were not derived from cyclists we can still generalise from those to the loss of aerobic capacity for cyclists.

Finally, the study by Clark et al measured the impact on peak oxygen utilisation (VO2), gross efficiency and cycling power output on ten well trained but non-altitude acclimated cyclists and triathletes by testing riders at simulated altitudes of 200, 1200, 2200 and 3200 metres. They examined a number of factors, including maximal 5-minute power output, VO2 and gross efficiency relative to performance at 200 metres, as well as sub-maximal VO2 and gross efficiency.

I used these data to generate a formula similar to those from Peronnet et al and Bassett et al. Of course there is an assumption of an equivalent reduction in 1-hour power as for 5-minute power. Clark et al noted slightly greater reductions in VO2 peak than for 5-minute maximal power, and no change in gross efficiency at 5-min max power with altitude. So there is some anaerobic metabolic contribution presumably making up the difference. There was some loss of sub-maximal efficiency noted at a simulated 3200 metres.

I chose in this instance to use the reduction in 5-minute power rather than fall in VO2 peak as the base data for the formula, and applied an adjustment to offset the formula for sea-level equivalency to bring it into line with the formula by Peronnet et al and Bassett et al. Of course when you look at the reported data there are of course sizeable variations within the test group at each simulated altitude, so the formula is based on group averages for each simulated altitude.

Here are the formulas:

**x = kilometres above sea level:**

**Peronnet et al:**

Proportion of sea level power = -0.003x

^{3}+ 0.0081x

^{2}- 0.0381x + 1

**Bassett et al**

**Altitude-acclimatised athletes**(several weeks at altitude):

Proportion of sea level power = -0.0112 x

^{2}– 0.0190x + 1

R

^{2}= 0.973

**Bassett et al**

**Non altitude-acclimatised athletes**(1-7 days at altitude):

Proportion of sea level power = 0.00178x

^{3}– 0.0143x

^{2}– 0.0407x + 1

R

^{2}= 0.974

**Simmons’ formula based on Clark et al:**

Proportion of sea level power = -0.0092x

^{2}– 0.0323x + 1

R

^{2}= 0.993

There is some variance between each formula's estimates, although the gap between the Non-acclimatised athlete estimates by Bassett et al and by Simmons based on Clarke et al is not all that large, ranging up to a ~2% variance.

Had I chosen to use the reduction in peak VO2 for 5-min max power, then I'd expect those two lines to be closer. In any case, these data by Clark et al reasonably match earlier reported findings of the impact of altitude on sustainable aerobic power. And once again - the individual response varies - these are simply averages based on the limited data available and for the cohorts tested. As always, YMMV.

The formula by Peronnet et al is the least aggressive at reducing the estimate of a cyclist’s power at higher altitudes, and that may be due to various not insignificant assumptions used in calculating each rider’s power outputs.

The formula by Peronnet et al is the least aggressive at reducing the estimate of a cyclist’s power at higher altitudes, and that may be due to various not insignificant assumptions used in calculating each rider’s power outputs.

### OK, so now we have estimates of both the physics upside and the physiological downside of altitude, What happens when we merge the two?

Well if I recreate the chart showing the physics, and overlay on that the curve showing power output as a function of altitude, this is what we get if we examine a rider capable of sustaining 51km/h at sea level:

Let me explain how to interpret the chart.

First of all, the vertical axis scale has been changed for clarity – the slightly curved coloured lines still represent the power to drag ratio required to attain a given speed at various altitudes.

So let's examine the case for a rider capable of sustaining 51km/h at sea level.

The thick orange line represents the power to drag required to sustain 51km/h. At sea level that's ~1,800 W/m^2 (Red circle 1). The exact value depends on a few other assumptions of course, so let's just use that as our "baseline" W/m^2 value.

Now if we apply the Bassett et al formula for power reduction for an altitude-acclimatised athlete, then their baseline sea level power (and with it their power to aero drag ratio) falls with increasing altitude. This drop in sustainable power with increasing altitude is indicated by the black dotted line.

We can see the power to drag ratio resulting from the physiological impact of altitude (the dotted black line) doesn't fall as quickly as the power to drag ratio required to sustain 51km/h (the thick orange line).

If you trace the black dotted line from left to right, we can see that at Red Circle 2, the power to drag ratio crosses the line marked 52km/h at an altitude of ~700 metres. Then as you trace the dotted line further to the right, we can see it cross the 53km/h line at ~1,500 metres. Tracing the line to the right hand edge of the chart out to 3,000 metres altitude, we can see it doesn't quite reach the 54km/h line, falling a little short at 53.9km/h. So for this altitude-acclimatised athlete, they can gain an extra 2km on their hour record simply by choosing to ride at an altitude of 1,500 metres.

If you trace the black dotted line from left to right, we can see that at Red Circle 2, the power to drag ratio crosses the line marked 52km/h at an altitude of ~700 metres. Then as you trace the dotted line further to the right, we can see it cross the 53km/h line at ~1,500 metres. Tracing the line to the right hand edge of the chart out to 3,000 metres altitude, we can see it doesn't quite reach the 54km/h line, falling a little short at 53.9km/h. So for this altitude-acclimatised athlete, they can gain an extra 2km on their hour record simply by choosing to ride at an altitude of 1,500 metres.

### OK, so what happens if the athlete is not acclimatised to altitude?

This time the non altitude-acclimatised power line is indicated by the lower black dashed line. It starts at 1,800 W/m^2 at sea level indicated at Blue circle 1, but as we trace that line to the right, it falls away more quickly than for the altitude-acclimatised athlete, crossing the 52km/h line at ~1,000 metres altitude (Blue circle 2) and not reaching the 53km/h line by the time the athlete is at 3,000 metres, where in this case the athlete would be estimated to achieve a speed of ~ 52.9km/h (Blue circle 3).

So while the acclimated athlete can improve their speed by 1km/h by going from sea level to 700 metres, and increase speed by 2km/h by going up to 1,500 metres, to achieve the same speed gains the non-acclimated athlete would need to ride at an altitude of 1,000 metres and would not be able to attain a 2km/h speed gain even at 3,000 metres.

We can see that as the altitude increases, the extra speed gains begin to diminish, and there are risks in going too high, especially if you are not acclimated, or experience an above average decline in power with altitude.

Conversely, if you are well acclimated and/or have a below average decline in power with altitude, then there are benefits in going higher if maximising speed is your primary objective.

Any rider considering an hour record would do well to consider the opportunity presented by tracks located at altitude. Of course costs, logistics, regulations all factor into the choice of venue, and how much time a rider may need to acclimate to altitude, and their individual response to altitude.

Any rider considering an hour record would do well to consider the opportunity presented by tracks located at altitude. Of course costs, logistics, regulations all factor into the choice of venue, and how much time a rider may need to acclimate to altitude, and their individual response to altitude.

If a sea level based rider were considering a fly-in / fly-out attempt without much acclimation time, then I'd suggest choosing a good track that is not too high, as the risks of a larger than expected power decline increase significantly, and the potential speed gains diminish as well increasing complexity of execution as nailing pacing gets trickier. Of course the more experience a rider has with altitude and its impact on their performance, the more confident they can be with predicting an ideal location.

### So what tracks are there at altitude?

Indoor laminated wooden 250m tracks at altitude include:

- Aguascalientes, Mexico: 1,887 metres above sea level
- Guadalajara, Mexico: 1,550 metres above sea level
- Aigle, Switzerland: 415 metres above sea level
- Astana, Kazakstan: 349 metres above sea level
- Grenchen, Switzerland: 340 metres above sea level

There are track at much higher altitudes, but they are 333 metre outdoor tracks with concrete surfaces:

- La Paz, Bolivia: 3,340m
- Cochabamba, Bolivia: 2,571m
- Arequipa, Peri: 2,295m
- Mexico City, Mexico: 2,260m

### So what's actually possible by the bigger guns of the sport. e.g. Wiggins, Martin, Bobridge and company?

I'll save that analysis for a future post, as well as a look at generating a formula to estimate the range of potential speed gains as a function of altitude, given an estimated sea level performance.

## 4 comments:

Nice work! But if I believe your formulas both acclimatized and nonacclimatized athletes are optimized at the same, close to 3000 meter altitude: that is where, for each, the physiological and wind drag curves become parallel, which marks the point at which the marginal losses equal the marginal gains.

Yeah, that's probably about right, at least theoretically. I haven't plotted the "mathematically optimal altitude", that's for another day perhaps, but since the marginal gains reduce and the risks increase with ever higher altitudes, and we can of course only choose the velodromes that are available, I figure it's somewhat moot. And who knows what anyone's individual power curve looks like, these are sample population averages.

Just train smart and go to Mexico. :)

One of the other factors to consider is the altitude domain of validity of the power reduction models used. e.g. the Bassett et al non altitude-acclimated model is a cubic equation, which at altitudes above 3,000m begins to curve back towards less negative gradients, while the Bassett et al altitude acclimated model is quadratic, and gradient continues to become more negative with increasing altitude. I'd be hesitant to suggest the models are valid at very high altitudes.

The power reduction formula I created based on the Clark et al data is quadratic, and that data is derived from testing cycling power up to a simulated altitude of 3,200m, so we at least have some bounds for its validity, although the individual differences are sizeable.

I did a quick and dirty plot extension, and the Bassett et al acclimatised athlete based on sea level speed of 51km/h reaches a theoretical maximum speed of a bit under 54km/h at ~3,300 metres.

I'm having a hard time believing anyone would really perform that well at such altitude.

Anyone up for a trip to La Paz? :)

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