perspectives

A two-reservoir human body1 Seymour

S. Alpert,

energy

ABSTRACT

A macroscopic

and

food

model

changes

intake

is that

undergoing

it can

level

weight

in individuals

over-eating

may

application

suggested

diet

activity

that

of by single

changes

are

fit by

on the

initially

model

reservoir

to

experimental

concepts.

The

American

Journal

ofClinical

Downloaded from https://academic.oup.com/ajcn/article-abstract/32/8/1710/4692351 by guest on 15 February 2018

data Am.

Nutrition

Experimental

and

the

efficiencies

displacement this

parameter.

are

32:

metab-

A feature on The

of the

considered. fluids

hypothesis

Nuir.

basal data

variation

ofaqueous

J. Clin.

exercise, of energy.

of a single

weight

utilization

to a volume

weight,

conservation

fluids.

adjustment body

body

of the

in aqueous the

food

relating

principle

equilibrium

A macroscopic two-reservoir energy model of the human body has been developed which relates body-weight, exercise, basal metabolism, and food intake. A feature of this model is that it is capable of accounting for changes in aqueous fluids. The value of a successful energy model is manifold. In addition to providing a predictive capability, it can be an aid in the defmition of human abnormality and variability. A model also suggests those parameters that should be carefully measured in future cxperiments. The validity of the model must be judged both by its ability to agree with observed experimental data and by the reasonableness of its underlying simplifying assumptions. The model presented herein additionably resolves some of the observed paradoxes suggested by single reservoir concepts. The problem of human variability was mitially avoided by considering data averaged for several experimental subjects. We have relied heavily on the experimental work of Keys et al. (I) who, in 1944, studied 32 young male volunteers undergoing a semistarvation diet of long duration. Considering the period of observation (#% 1 year), the effort and care in making the many experimental measurements, and the imposition into the life-style 1710

model the

for

of different

lead the

from

account

a semistarvation in the

two-reservoir

is developed

body by

of the

PhD.

olism, this

model

in nutrition

The

of

subjects effects

of

equilibrium hypothesis

by fat is demonstrated

obviates 1710-1718,

paradoxical

results

1979.

of the subjects, it is unlikely that an experiment ofthis magnitude and completeness will soon be repeated. For purposes ofbrevity this work will be referred to as the Minnesota Experiment. In comparing our model to experimental data, we necessarily have had to estimate the value of poorly known parameters; hence, our resulting calculations provide numerical values that may contain inaccuracies on the order of 10 to 20%. Since we must use defmitc numerical values to deduce functional relationships, it should not be inferred, however, that an extremely high level of computational accuracy exists. It is in the spirit of attempting to deduce a reasonable, valid, and functional model of the body’s energy relationships that this work is reported.

Theory The principle of the conservation of energy is the underlying basis of our theoretical approach. Although we basically consider only energy quantities, it is generally

assumed

‘From the The University

that

the

principles

of good

nutrition

Department of Physics and ofNew Mexico, Albuquerque,

apply;

Astronomy, New Mex-

ico 87131.

32: AUGUST

1979,

pp.

17 10-17

18. Printed

in U.S.A.

TWO-RESERVOIR that is, the fuel intake represents necessary nutrients and minerals.

The two reservoirs

ENERGY

a suitably

that we consider

mixed

diet

for energy

MODEL of

storage

purposes are the fat store and the remainder of the human body which we consider to be capable of storing and providing energy. We follow the convention suggested by Garrow (2) in using the word “fat” to mean the chemical composition of glyceryl esters and fatty acids as opposed to the term “adipose tissue” which is known to contain aqueous fluids and small amounts of protein in addition to the fat. We use the term “fat-free” to refer to the remainder of the human body not including fat. In the terminology of this paper, the fat-free store includes muscular protein, interstitial fluids, bone mineral, blood plasma and cells, and other unspecified tissues. In our usage, the protein and fluid content of adipose tissue are also part of the fat-free reservoir. The distinction between fat and fat-free weight is significant since densimetric measurements yield direct information on precisely these quantities.

The specific

assumptions

that are fundamental

to the

development of our model are listed below. I) The rate of energy utilization associated with activity is directly proportional to the body weight. This assumption is commonly made by those experimentally engaged in the measurement ofthe energy cost of various activities (3-5), and is considered to be valid as long as the activity includes body motion. There are some activities, however, that require little body movement such as typewriting or bicycle ergometer riding; the energy cost these activities is not directly related to body weight (6). Even the most passive typist, however, must maintain a sitting posture; it can be deduced from data presented by Durnin and Passmore (7) that nonmobile sitting has energy costs that vary proportionally to body weight. We believe that generally most daily activity, when averaged will be directly related to body weight. 2) The fat store, in the energy deficit situation, provides at most a supply of energy the rate of which is proportional to the size of the fat store. There is little theoretical justification for this assumption other than it is one of the simplest ones that can be made. It can be concluded from this assumption, as will be shown later, of

that

the

fat

store

decreases

logarithmically

with

time;

this

situation is in fact observed in the Minnesota Experiment and, hence, the assumption is experimentally justified. 3) The basal metabolic rate (BMR) is only a function of the fat-free weight and does not depend on the size of the fat store. This assumption is based on the experimental work of MacMillan et al. (8) who studied the resting oxygen consumption rate of 24 male and 25 female Edinburgh medical students and also densimetrically measured their fat-free weights. The ratio of the oxygen consumption rate to the fat-free weight was very closely

the same

for both

the men

and

the women

suggesting

that the different size of the fat store in the two sexes was unimportant. This seems reasonable from our definition of the fat store which excludes protein and aqueous components; these excluded substances are presumed to be in close association with the blood supply and in need of oxygen for maintenance. 4) There is a multiplicative food utilization factor that

relates

which all of

the

intake

is used for the existing

energy

activity, food

to

basal

energy

that

heating,

amount

of

or storage.

is available

for

energy

Not

these

Downloaded from https://academic.oup.com/ajcn/article-abstract/32/8/1710/4692351 by guest on 15 February 2018

OF

HUMAN

1711

BODY

modes of utilization. Durnin and Passmore (7) point out that a small fraction of food energy is passed by the body. Additionally it is well known (9) that there is a thermic effect associated with food processing, digestion,

and

absorption;

thermic such activity, effect factor 5)

effect does energy is not or storage. are specific to we will later That amount

the

fractional

energy

involved

in the

not represent a waste of energy but accounted for in terms of the BMR, The passed energy and the thermic food type; hence, the food utilization use will be an averaged one. of energy that is not associated with

activity, related

basal heating, or storage and that is not directly to the amount of ingested energy will be considered to be waste energy. In normal situations this waste energy will be a small quantity and is assumed to be constant. Primarily it consists of the thermal energy of voided excreta and perspiration. The larger amount of energy associated with the heat content of exhaled air is already implicity accounted for in the activity energy

and

in the BMR.

6) In the deficit situation, the fat-free reservoir can provide stored energy in proportion to the amount of protein catabolized. The conversion factor of stored energy per unit of fat-free weight is not generally a constant value but is related to both the amount of protein catabolized and to the associated aqueous fluid change. 7) The carbohydrate store, while highly important as an immediate source of activity energy, is not significant in size or amount of stored energy. The carbohydrate reserve is known to be small (10). Since it can be rapidly depleted it will have little effect in a long term study. We will consider the carbohydrate store to be a part of the fat-free reservoir. From

the

principle

of

the

conservation

of energy

making use of some of the assumptions stated can write the following differential equation: a df/dt

+

/1

above,

and

we

dl/dt -#{128}P--BMR(l)-W-#{244}(f+l)

(1)

where a is the energy conversion factor associated with the fat store f, $ is the energy conversion factor associated with the fat-free weight 1, #{128}is the efficiency of food utilization, P is the daily energy value of food intake, BMR(/) is the functional dependence of the basal metabolic rate on the fat-free weight, W is the small amount of waste energy assumed to be time-independent, 6 is the activity coefficient which relates energy expenditure to the body weight (f + 1), and t is the time. It should be kept in mind that both fi and S in equation (I) may themselves be functions of time. Equation (1) relates two dependent variables, f and 1, to a single independent variable, t. We can go no further in a solution of equation (1) unless another equation relating the variables is available. In the severe deficit sutiation where the fat store provides energy at a rate proportional to its size,

we may

write: adf/dt

where

+

Sf- -af

(2)

a iS a constant. We have included a minus sign on the right hand side of equation (2) since this allows a to be positive in the energy deficit situation. The quantity c;f represents the net rate of energy supplied to the fatfree body over and above that amount expended on the fat store itself. If the activity coefficient 6 is relatively

1712

ALPERT

constant,

we

can

solve

(2) deriving

equation

the expres-

sion f

=

exp(-

fo

(a +

8)

t/a)

t3)

where f0 is the initial size of the fat store. We will later justify that a is much larger than 6 so that even if 6 does vary with time, equation (3) is still reasonably correct. Combining equations (1), (2), and (3) leads to the expression $dl/dt-eP-BMR(l)--#{246}l-W

(4) +

In the severe

(4) will

deficit

case,

(c,+

(-

8)

side of in the fat-free weight. We will refer to this case as the nonprotein sparing situation since the deficit energy will neces-

equation

energy

exp

010

be negative

the right

indicating

hand

a decrease

sarily be derived from protein sources. If, however, the BMR and/or the activity coefficient decreases suffici#{235}ntlywith time, the right hand side can become positive. It is unlikely that the fat-free weight will increase at

the expense We suggest

of the fat store in the fuel scarce situation. that before the right hand side of equation

(4) becomes positive it will pass through zero indicating that the fat-free weight is energetically stabilized. This marks the transition to the protein sparing case and equation (4) will no longer hold, the fat-free weight being constant. After protein sparing occurs, the fat store no longer has to supply energy at its maximal rate. The equation

for

the

fat store

a df/dt

of

+

will

then

eP,-

=

become

BMR(1)

-

61

W -

(5)

where

1 and BMR(1) are constant. In the protein sparing the fat store is capable of providing a net energy flow to the stabilized fat-free body at a rate given by the right side of equation (5) (with sign reversed). If at any time due to changing conditions, the following inequality holds

case,

ep then

the

BMR(l)-

nonprotein

W -

sparing

case

81