Since this was a longish run I had time to think about this and that. From my previous post I showed how you can estimate the calories you would burn per kilometer run per kilogram body weight. My ballpark figure was 1.17 kcal/(kg*km). The 'real' value -from an actual study- was 0.97 kcal/(kg*km) for a group of practiced runners. Note that the unit 'kcal' is the same as Calorie (the ones on food labels) and that 1 kcal = 4.184 kJ. I decided when my run was finished I'd use this empirical number to find how much fat you burn after running a given milage D (or vice versa).
To take a specific example, I want to know how far a 145 lb (66 kg) person would have to run to burn one kilogram (2.2 pounds) of body fat (To put that in perspective, if that same person had 8% body fat (what many a distance runner may have) they'd have 11.6 lbs (5.3 kg) total fat). Spoiler alert: the value for D is going to be surprisingly large.
The other number we need is the energy stored in fat and glycogen per kilo. From wikipedia,
Energy in fat = 9000 kcal/kg = C
Energy in sugars = 4000 kcal/kg = M
Again, energy required for running = 0.97 kcal/(kg km) = K
Now for a bit of calculus (!). Sorry, but since your bodyweight changes with distance I feel compelled to do this. Energy gained per unit weight lost:
dE/dm = C
Energy consumed per unit distance run:
dE/dD = -mK
Combine these two formulas and integrate over the mass and distance intervals m = [mi, mf] and D = [0, D], respectively:
Cdm = mKdD
ln(mf/mi) = -DK/C
or solving for D as fraction of total fat lost:
Dfat = (C/K)*ln(mi/mf)
Aside: if your body weight stays constant (via eating), then the calculus can be skipped and we get
D = E/(m*K) = E/(m*0.97)
where E is the energy content of the food taken in, regardless of fat/sugar/protein content and m is your constant bodyweight.
We have our formula(s)! But wait a second: There is also 1900 kcal of glycogen energy stored in our body too. That is, according to Canova's Marathon Training manual (p. 18), there is 1500 kcal in our muscles and 400 kcal in the liver. The mass of this energy is
1900 kcal /4000 kcal/kg = 0.475 kg ~ 1 pound
The distance our 145 lb runner would get with 1900 kcal is, after subbing in M for C,
Dglyc = (M/K)*ln(mi/mf)
= (4000/0.97)*ln(145/144)
= 29.8 km ~ 30 km
which matches most people's experience 'hitting the wall' before the 20 mile mark in a marathon (some fat is used prior to hitting the wall, so the 'wall' experience varies with each person and their training). For the total distance traveled we include this glycogen D value with the total:
Dtotal = Dglyc+ Dfat
= 30 km + Dfat
Hence Dfat is for 1 kilogram of fat burned (and after subtracting 0.5 kg of glycogen lost, so that mi = 65.5 kg and mf = 64.5 kg):
Dfat = (C/K)*ln(mi/mf)
= (9000/0.97)*ln(65.5/64.5)
= 142 km
Hence Dtotal = 172 km, well over 100 miles! I told you it was going to be a big number. You might suppose that I made a calculation error. I thought so too, which is why I used the formula first for glycogen-based running distance. It works.
Implications? Even an ultra-marathon like the 135 mile Badwater demands just over a kilo of fat-stored energy, so in theory you could run much farther. Accounting for all the the elevation changes in this mountain run you'd probably burn and few extra (thousand) Calories. We can even account for your basal metabolic rate, which we'll ballpark at 1500 kcal/day for this one-day race. Combining elevation and BMR, you'll lose maybe an extra 3000-4500 calories or 0.33-0.5 kg of extra fat, though this is just a guess.
These calculations explain how some individuals like Marshall Ulrich have managed to run the Badwater course multiple times, i.e. totaling some 500+ miles in ten days of near-constant running. Of course he had outside assistance in part because both water and electrolytes are depleted much faster than fat. Kidneys can process about 15 L of water per day (if they have to).
It explains how a Swedish man survived two months in his car in winter without food (but with water). Assume his metabolism was 1500 kcal/day. Then 60 days is 90,000 kcal of basic energy needs. That would equal to over 10 kg of fat. He apparently lost closer to 19 kg body weight, probably due to water losses and his body using a higher BMR to fight the freezing outside temperatures. A skinnier/leaner man would have died.
Even under normal conditions, weight loss is complex. The body has to balance its confidence in holding on to 'enough' fat (because it never knows for sure) but not so much as to burden exercise. This is why no exercise at all messes up the equation. Yet no amount of exercise eliminates all, or even most, of your body fat (compared to what you actually require in a race). This your body does despite the obvious benefits (i.e. making you run faster). Why? You body has no interest in winning a race, only in long-term survival. Hence you will always have far more fat than you need. From your body's POV you could always be part of the next Donner party. This is a non-negotiable aspect of your body. Some coaches and athletes may attempt calorie restriction to lose weight (I define this as eating fewer calories than you burn per day). These people are, in my opinion,
