Sunday, December 29, 2013

Power Up a Tower: Part 3

In Power up a Tower: Part 1, I showed the relationship between work (J) and power (watts). I also introduced the power to mass ratio metric (watts/kg) which can be used to measure the performance of different climbers climbing the same building. Finally, I showed an example of how the power to mass ratio of climbers depends on the height of a climb.

In Power up a Tower: Part 2, I built upon the principals of work & power presented in part 1 and I showed the relationship between power output and race length. Using this relationship, I was able develop a method to predict racer’s finish time based on prior results in a different building (i.e. at different height and race duration).

In this post, I will focus on a slightly more practical topic for the average fitness enthusiast: How many calories are burned while climbing stairs?*

*Dedicated to Jason Larson who inadvertently sparked my interest on this topic.

To answer this question, you first need to understand that overall calorie burn can be broken up into two separate pieces:
1) Calories burned due skeletal muscle exertion (e.g. using your muscles when climbing stairs.
2) Calories burned due to other processes occurring within the body (e.g. process of digestion, the functioning of your brain and other organs, etc.).

As I explain each piece, I’ll use my latest Sears Tower climb as an illustrative example. Here are a few pieces of information you’ll need to know about my latest climb.
Climber: Alex Workman
Mass: 80 kg
Mass of shoes & clothes: 1 kg
Percent Body Fat: 11%
Height of Climb: 412 meters
Duration of Climb: 15 minutes*

*Actually, it only took me 14 minutes and 53 seconds, but for illustrative purposes, 15 minutes is close enough.

Calories Burned via Bodily Processes:
The 1st piece (skeletal muscle exertion) is the primary calorie burner during vigorous exercise, but the 2nd piece (bodily processes) cannot be completely ignored. After all, the majority of your overall daily calorie expenditure is consumed by your organs while you go about your day-to-day activities (sleeping, eating, sitting, reading, etc.). So what exactly are your caloric needs for your internal processes? It turns out that this topic is pretty complicated, since there are thousands of chemical interactions happening concurrently inside your body every moment of your life. Fortunately, you can approximate this value by using your Resting Metabolic Rate (RMR). RMR is defined as is the amount of energy expended daily by humans (and other animals) at rest. One method of calculating your RMR is the Cunningham Formula:

P = 500 + (22 x LBM)
where P is your daily caloric needs
where LBM is your Lean Body Mass in kg.

Lean body mass is used in the formula because fat cells have negligible caloric needs to exist, whereas muscle tissue and other organs require energy to function, even when resting. There are several different ways to measure your LBM directly, but as a rule of thumb, you can reference the table below (courtesy of Wikipedia) and make an educated guess.

Body Fat Percentage
Essential fat

Using this table, you can estimate your Lean Body Mass:
LBM = your mass  – (your mass x Body Fat Percentage)

Example 1: How many Calories did I burn during the Sears Tower climb due to bodily processes?
Answer: My LBM = 80 kg – (80 kg x 11%) = 71.2 kg.
Using Cunningham’s formula, my daily Caloric needs =  500 + (22 x 71.2 kg) = 2066.4 Calories per day
Since the duration of the climb was only 15 minutes (a quarter of an hour) I burned 2066.4/24 hours x ¼ hour = 22 Calories (rounded to nearest Calorie)

Calorie Burned via Exercise:
Now that we have a method for determining the calories burned due to the natural processes within your body, let’s take a look at calories burned while stair climbing. Using the information presented in Part 1 of this series, the total work output needed to climb a stairwell is:

Work = Mass x 9.81 m/s2 x Distance

Work is in joules
Mass is in kg (total mass including body mass and mass of clothing & accessories worn)
Distance is in meters (i.e. the total height of the stairwell)

Since 1 joule = 0.239 calories, and 1000 calories = 1 Calorie (aka the kilocalorie which is the dietary Calorie we’re most familiar with) the Calories of work  for a stair climb is:

Calories of Work Output = (Mass x Distance)/426.5

Mass is in kg.
Distance is in meters
Units for the 426.5 factor is (kg x m)/Calorie

Example 2: How many Calories of work output were needed to climb the Sears Tower?
Answer: My total mass is 81 kg, which includes both my body mass (80kg) and the mass of my clothes & shoes (1 kg). Since the vertical height of the Sears Tower climb is 412 meters, the Calories of work output = (81 kg x 412 m)/426.5 = 78 Calories (rounded)

Looking at the above example, you might think that something seems to be missing. How is it possible that I only needed 78 Calories of work output (the equivalent rate of about 313 Calories per hour) during my 15 minute stair climb?

The truth is, studies have shown that your skeletal muscles are only about 18%-25% efficient at converting energy into work*. The remaining 75%-82% of energy spent by your body is given off as heat**. Although the studies I’ve read on this topic were focused mainly on cycling and rowing, I believe that it also would apply to stair climbing since it is a similarly efficient exercise.

*Compare that with a modern combined-cycle gas turbine/steam turbine power plant that is over 60% efficient!
**Now you know why you sweat so much when climbing stairs.

The last piece of the puzzle is that work performed to climb a stairwell excludes the work performed to swing the arms &  legs and to turn around the corners in the stairwell. Fortunately a paper published by Minetti et. al. (Skyscraper running: physiological and biomechanical profile of a novel sport activity) offers some insight. In this paper, the authors experimentally derived that only about 80% of the total power output is used to climb vertically. Another 5% of the total output is used to swing your limbs and another 15% is used while turning around corners. However, I don’t fully agree with all of these findings. My biggest concern was that the authors assumed that that the power needed to turn around a landing was equivalent to the power needed to run in a 2 meter (about 6’7”) diameter circle. From experience, I know that I make much tighter turns in the stairwell *and* I’m using my arms to keep my center of mass as close to the hand rail as possible. Really I’m just pivoting around the rail (on my inside foot) rather than running around a circle. As such, I believe that the author’s model was quite conservative. Although I have neither experimental evidence nor a kinematic model to predict the power needed to turn around a stairwell landing, I estimate that:
  • 90% of work output is used to climb vertically
  • 5% of work output is used to move your limbs
  • 5% of work output is used to turn around the landings

Including both muscle efficiency and excess power wasted in the stairwell (on turns & extraneous motion) we can use the following formula to calculate the total number of Calories used to climb a stairwell:

Calories used to Climb a Stairwell  = [(Mass x Distance)/384]/Aeff
Mass is in kg.
Distance is in meters (vertical height of climb)
Units for the 384 factor is (kg x m)/Calorie*
Aeff is the muscle efficiency factor, a unit less number somewhere between 0.18 and 0.25
*note: this is 90% of the 426.5 factor used in the 2nd example

Example 3: How many Calories did I use to climb the Sears Tower?
Answer: We’ve already shown I need 78 Calories of work output to climb the Sears Tower, but this only represents 90% of the total work output exerted by my body.
Therefore overall output of my body = 78 Calories/90% = 87 Calories (rounded).
Since my muscles are pretty inefficient at converting energy into work, I burned far more than just 87 Calories - mostly in the form of heat. Assuming I’m about 22% efficient (estimate), I burned about 87 Calories/22% = 393 Calories during the climb, which is a rate of about 1576 Calories per hour.*

*Note: this does not include the 22 additional Calories burned via my body’s internal processes (which is an order of magnitude smaller than the Calories burned due to climbing).

Have you ever used a piece of exercise equipment and noticed the term “MET” on the electronic display? Well, MET is an acronym for “Metabolic Equivalent of Task”. One MET is defined as 1 Calorie/(kg x hour), and it was originally considered to be the resting metabolic rate of an “average” person. The reason why exercise equipment manufacturers include MET is because it provides a measurement of the intensity of a workout. For example, walking very slowly has a MET of 2.3. This means you are burning 2.3 times as many calories (via exercise) than your body naturally burns while at rest. Here is a short list of common activities and their MET values (courtesy of Wikipedia):

  • Watching TV: MET = 1
  • Walking briskly (3.4 mph): MET = 3.6
  • Jogging: MET = 7
  • Push-ups/Pull-ups: MET = 8
  • Running: MET = 10+ (depending on speed)

Since this article is primarily about stair climbing, lets calculate my MET for climbing up the Sears Tower.

Example 4: What was my MET for climbing the Sears Tower?
Answer: Since we’ve already calculated my RMR in Example 1 and my Calorie expenditure due to climbing in Example 3, the ratio should give an approximate value of MET. In my case that was 393 Calories burned due to climbing per 22 Calories due to my body’s internal processes. This give an approximate value of 
METest= 393/22 = 17.9.
However, my RMR is not exactly the same as the an average person’s. So, taking the definition of 1 MET = 1 Calorie/(kg x hour), it looks like my actual MET is 
METact= 393/(80kg x .25 hours) = 19.7

Taking the ratio of these two MET calculations (i.e. METact/METest) it appears my RMR is about 10% more than the average person.*

*Which makes sense ‘cause I’m totally jacked.

Further Questions:
As you can see from the examples above, stair climbing is a great way to burn calories. However, there are several areas that require further exploration:
1) It has been shown that a trained athlete requires less energy to perform the same amount of work, but that is mainly because a trained athlete recruits fewer muscles to perform a given amount of work. Unfortunately, there is a lot less information about how muscle efficiency (i.e. translating energy into mechanical work vs. heat) varies from person to person. Can muscle efficiency also be increased with training? Is there a biochemical upper limit?
2) How much extraneous work is really spent swinging the arms and legs and turning around the stairwell turns? Minetti et. al. has piqued my interest on this topic. In addition, I’d also like to know the effect of using the rails vs. not using the rails when climbing. I know from experience that using the rails (i.e. using the arms) helps to “save” my legs from anaerobic fatigue, but at the same time, using the rails is not as mechanically efficient as using the legs. I’d like to quantify this trade off. 
3) RMR is a very good measurement of the metabolic needs of your internal processes when the body is at rest, but does it change during exercise? For example, I know that intense exercise makes difficult to think, so I must assume that less energy is used by the brain during exercise. Does the same hold true for other organs?