Sunday, February 3, 2019

The Diet Equation

Having perhaps added a bit of weight during the recent holidays, we turn our attention to the fundamentals of weight gain, weight loss, and dieting.

To start, if we ignore the burning of our calories by our metabolism, we would have the simple weight-gain (fat-gain) equation

W’ = C’ / (3600 cal/lb)

W’ is the average weight change rate in pounds per day (lb/day).
C’ is the average daily calorie intake rate in calories per day (cal/day).

If we ingested an additional 80 calories per day (an extra slice of bread), we would expect a rate of weight gain (ignoring energy use) of

W’= (80 cal/day) / (3600 cal/lb) = 0.022 lb/day = 0.36 oz/day,

too small to notice. 

A large daily diet of C’=1800 cal/day would produce, without any energy use,

W’ = (1800 cal/day)/(3600 cal/lb) = 0.5 lb/day, 

a half-pound gain per day! That cannot be the total story.

We use calories by our basic metabolism and our activities.

We’ll first look at our basic metabolic rate factor, r, the calorie burn rate (calories per pound per day) while sleeping or if quadriplegic, and include this in our formula:

W’ (lb/day) = [C’ (cal/day) – r (cal/lb-day) W (lb)] / (3600 cal/lb)

Or, without the units displayed, the Diet Equation

W’ = [C’ – r W] / 3600                                                        [1]

W is weight (long-run average weight).

A basic resting metabolic rate factor is r = 10 (cal/lb-day), and we combine it with the rest of Equation [1] to get

W’ = [C’ – 10 W] / 3600

For our example, a person weighing 120 pounds to start and taking in 1800 cal/day, would have (restoring units and using the calories-to-fat conversion factor of 3600 cal/lb)

W’ = [1800 – 10 (120)] / 3600 = 0.17 lb/day,

a slower rate of weight gain.

To get the rate of weight gain to be zero, W’=0, we need

C’ = r W

which occurs when C’ is lowered or when r is raised, producing weight 

W = C’ / r                                                                [2]

in the long run.

We can adjust our long-run average weight by changing C’, the average daily calorie intake, or by raising r, the metabolic rate, by becoming more active.

The additional long-run-average weight due to an average additional 80 calories (a slice of bread extra each day) is the change in C’ / r, or

+80 (cal/day) / 10 (cal/lb-day) = +8 pounds.

This is shocking! We added 8 lb from just an extra slice of bread a day.

How long will it take to reach this level? Even longer than if there were no metabolism going on, which would simply be the number of days to accumulate an extra 8 x 3600 cal:

8 x 3600 cal / 80 cal/day = 360 days,

such a sneaky, slow gain we would not easily notice it.

Including our metabolism, cutting back by a slice of bread a day (80 cal/day) would produce a slow, long-term-average weight loss of 8 pounds, even for this low-metabolism r = 10 case. Patience pays, people!

P.S.  If you want to get a more exact estimate of your BMR (Basic Metabolic Rate), use the following site, which gives BMR = r W by height, weight, age, and gender:

Another calculator allows you to include these factors and activity level:

They have a maintenance level r=10 and a moderately active level r=13 among the examples I tried.

If you hope that walking will help, you are right, but it takes walking a mile for a 180-pound person to expend 100 calories (and proportionally more or fewer calories for people of other weights). 

(c) Douglas Winslow Cooper, Ph.D. 02/03/2019

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