## June 22th 2015, Counting Calories

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Once a track has been recorded, we can calculate various track statistics about the track like average time and speed.

A particular interesting statistic is, of course, how many calories were burnt when walking or hiking. In principal we have the following two options:

- Calculating the calories involves measuring the consumed energy. The only direct way to do this is to measure the oxygen intake of a person during a particular activity. Then, the energy per liter oxygen is about 4.5 to 5.1 kcal, depending on the power level. The higher the power level, the more the body will produce energy in an anaerob fashion (by burning fat → 4.5 kcal per liter oxygen), else the body will produce energy in an aerob fashion (by burning carbohydrates → 5.1 kcal per liter oxygen).
- Another way would be to measure the power (Watts) required for an actual activity. Then we may divide the power by the efficiency rate by which muscles transform energy into motion and integrate over time to get a wattage. The physical unit of wattage is J (Joule). For each kJ we gain 0.239 kcal or 3.6 Wh.

The efficiency rate is about 0.22 for an untrained person and 0.25 for professionals. When biking we may measure the wattage with an ergometer. For regular biking tours this is no option, though. As an alternative we could model the physics behind biking. For more information about the involved biking physics and formulas see www.kreuzotter.de. Then we could calculate the wattage from the physical model. But this requires the exact knowledge of the physical parameters involved, which are often unknown.

For example, the necessary wattage for biking at a particular speed depends on two main factors: on the the air drag and the wheel friction. The air drag depends on the effective frontal area of a person, which in turn depends on the sitting angle and the height of a person. And the wheel friction depends on the tyre pressure and its profile and the trail consistency and texture. Both of which change often during a biking tour. So it is virtually impossible to make precise assumptions on the physical properties involved with calculating a wattage profile.

Since there is no way to measure the oxygen intake or wattage directly without an ergometer or oximeter, the exact direct calculation of the spent calories is NOT possible with a smartphone’s GPS.

However, the oxygen intake is related to other characteristics we can measure, so that it is possible to indirectly measure the calories. In such a case, the indirect measurement will suffer from both systematic errors and imprecise assumptions on a person’s metabolic rate, so that the figures will always be off to a certain extent. Treating those indirect figures as absolute numbers is therefore meaningless.

If the calories are intended to be an approximate estimate, for example to compare several runs with each other, then we may look at the following indirect approaches:

- The heart beat is directly related to the metabolic rate. So we can estimate the oxygen intake indirectly by measuring the pulse rate. The relation is non-linear, though. What is more, the relation varies from person to person and depends on such factors like age and the level of training. Without measuring the individual function of the oxygen intake on the actual pulse, we can only make average assumptions like:

- When hiking or walking, the consumed kilo calories approximately relate to the horizontal hiking distance in kilometers times the person’s weight in kilogram:

- An elevation change of $\Deltah$ meters directly relates to the potential energy $m\cdotg\cdot\Deltah$ necessary to lift a person vertically:

- For a resting person the BMR (basal metabolic rate, that is is calories burnt in 24h) can be approximated by the Harris-Benedict equation:

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