Dimensional Analysis and Theory of Biological Similarity Present address : Instituto de Ciencias Medico-Biologicas, Departamento de Fisiopato- logia, Universidad de Conception, Chile

PHYSIOLOGICAL REVIEWS Vol. 55, No. 4, October 1975

Printed in U.S.A.

Dimensional Analysis and Theory of Biological Similarity

B. GtiNTHER*

Departamento de Medicina Experimental, Facultad de Medicina, Universidad de Chile, Santiago, Chile

I. Introduction. ...................................................... 659 II. Brief History. ...................................................... 660

III. Modern Theories of Biological Similarity. .............................. 661 IV. Different Similarity Criteria. ......................................... 662

A. Geometric similarity. ............................................. 662 B. Physical and biological similarities. ................................. 664

V. Organisms as Mixed Regimes, ........................................ 671 VI. Allometric Equations : Theoretical Foundation. ......................... 673

VII. Empirical Allome tric Equations. ...................................... 675 A. Morphometric studies. ............................................ 675 B. Physiometric studies. ............................................. 677

VI I I. Empirical Allometric Equations, Theoretical Reduced Exponents, and Optimality Principles. ........................................... 687

IX. Possible Relationships Between Two Allometric Equations. ............... 688 X. Dimensionless and Invariant Numbers. ................................ 689

A. Allometric canceling technique. .................................... 690 B. Matrix method .................................................. 691 C. Invariant number of mammalian circulatory system. .................. 691 D. Invariant number of respiratory system of mammals. ................. 692 E. Invariant number of cardiorespiratory system of mammals. ............ 692

XI. Iconic Treatment of Physiological Problems. ........................... 693 A. Block diagram of systemic circulation of mammals. ................... 693 B. Block diagram of gas transport system: blood, circulatory, and respiratory

systems. ....................................................... 694 C. Signal-flow diagram of cardiovascular system of mammals. ............. 695

XII. Summary. ......................................................... 695

I. INTRODUCTION

Structures and functions of all living beings, irrespective of their size, can be studied by means of some basic physical methods, viz., dimensional analysis and theories of similarity.

Unicellular and multicellular organisms have almost the same protoplasmic composition (proteins, lipids, carbohydrates, minerals, and water) and are subjected to similar physical and chemical influences from the environment. Conse- quently the organismic machinery can be considered from both a structural and a functional point of view as a harmonious assembly of parts, yielding at all instances

* Present address : Instituto de Ciencias Medico-Biologicas, Departamento de Fisiopato- logia, Universidad de Conception, Chile.

659

660 B. GijNTHER Volume 55

throughout the life of each organism a perfectly integrated whole, or in the words of Claude Bernard (3) an “ensemble harmonique.”

The theory of models, widely used in physics and technology, can also be successfully applied in biology for obtaining numerical solutions for any biological function definable in terms of the three-dimensional A4LT system (M = mass, L = length, T = time).

Furthermore, from only a few postulates it has been possible to derive a con- sistent tlieory of biological similarities, if organisms are considered as “mixed regimes” obeying simultaneously various similarity criteria, as for instance : I) rules that have been established for mechanical or dynamic processes and are valid for ItK Whulc uqanism; L”) kinematic events concerning organ functions; or 3) hydrodynamic or transport processes, which should be applied when dealing with events at cellular or molecular levels.

Finally, a set of dimensionless and invariant numbers has been obtained algebraically that is valid for multivariable morphophysiological systems. These nllmeriral relationships are independent of the size of the organisms considered.

Therefore behind the complexity and astonishing variety of forms and functions quantitative criteria of similarity for all living beings have been disclosed.

II. BRIEF HISTORY

The rules of geometric similarity, in particular those applying to polygons, have been known since Euclid’s time (300 B.C.). Later, Archimedes (287-2 12 B.C.) established that for “similar” geometrical bodies the corresponding surfaces increase as the square and the volumes increase as the cube of linear dimensions.

More than 300 years ago Galileo (1564-1642) not only discussed geometrical similarity of terrestrial and aquatic animals of different sizes, but also introduced dynamical aspects, particularly regarding the strength of bones. Galileo concluded that neither terrestrial animals nor trees can grow beyond a certain size. He advanced two solutions for the physical problems involved : I) a change of the relative proportions of an animal, and 2) harder and stronger materials are needed as the size of the animal increases. By way of illustration on the subject, Galileo pointed out that: “A dog could probably carry two or three such dogs upon his back; but I believe that a horse could not carry even one of his own size.”

Borelli (1608-l 679), the leader of the iatrophysical school of medicine, established that the “force” of a muscle varies in proportion to its cross-sectional area, whereas the “energy” that the muscle is able to deliver apparently varies in proportion to its volume. He also calculated that two animals of similar build, but of different size, can jump the same actual height and that the maximal velocity they can reach is constant.

Newton (1642-1727) established the general rules for “mechanical” similarity, since his physical similarity is the starting point for all modern theories on biological similarity. However, the symbols of dimensional analysis used in physics as well as in biology were introduced by Maxwell (183 l-1879).

Spencer (1820-l 903) studied the growth-limiting factors and concluded that

October 1975 THEORY OF BIOLOGICAL SIMILARITY 661

assimilation processes occur at certain surfaces, so that as growth proceeds the ‘%urface-volume ratio” becomes progressively smaller, making any further increase in size impossible.

Von Hoesslin (38), based on the dimensional analysis of various functions, concluded that the “power” developed by similar animals of different size varies in

proportion to a characteristic surface and that the “work” performed is correlated to the corresponding volumes. He also established that all animals belonging to the same model are capable of attaining the same maximal speed and that the frequency of a particular movement is correlated to the size of the animal.

In 19 17 Thompson (75) introduced dimensional analysis and similarity criteria into general biology. He carefully studied the locomotion of terrestrial animals and the flight of birds and insects from a physical point of view. The scale effects and the principle of similarity are discussed in his fascinating book On Growth and Form, which has since become a classic.

Putter (6 1) published in 1917-l 920 several papers on the problem of physiological similarity, in his Studien iiber physiologische k’hnlichkeit. He made extensive use of empirical exponential equations for describing quantitative aspects of growth and of many other functions.

III. MODERN THEORIES OF BIOLOGICAL SIMILARITY

A preliminary note on the subject was submitted on January 10, 1927 to the French Academy of Sciences by Lambert and Teissier (47), and that same year they published the first extensive study entitled Thtbie de la similitude biologique (48)) which is the starting point for all subsequent developments.

It might be worthwhile to transcribe the main a priori ideas of these authors: “que certains lois physiologiques obtenues experimentalement ne sont que I’expression des PropritWs mlcaniques ou lnergetiques tr2s gt?Xrales et peuvent A?re e’tablies par voie dlductive

* a partzr des d&nitions des diverses grandeurs en jeu (longueur, masse, tern@, force, travail, puissance . . .).”

The French authors employed Newton’s dimensional analysis and the rules holding for any mechanical similarity in which X corresponds to the length ratio (L I/&) between prototype1 and model, 7 to the time ratio (Tl/ Tz), p to the mass ratio (Ml/M 2), and <p to the force ratio (Fl/‘Fz).

They further assumed: 1) the constancy of “density” of homologous organs and organisms and 2) that the time ratio (7) is equivalent to the length ratio (X).

It follows that :

This equation means that the “efficiencies” (Ki) of homologous transformations are the same in homologous organisms.

From the dimensional analysis of any function definable in terms of the mass

(ML length (0, and time (T) system, it is possible to deduce the corresponding

l Prototype means a large- or small-scale organism to which a model is geometrically or physically related.

662 B. GtiNTHER Volume 5.5

power function of the length ratio (A), as shown in a reproduction of the original table :

Longueur ............... h = L Masse.. ................ p=M=‘h3 Temps.. ................ z = T--h Surface.. ............... s =Le=h2 Fdquence ............... f = T-l = h-4 Vitesse ................. v =LT-4 =l Accklhation. ............ Y =LT--2=1-” Force .................. y = MLT-2 = P Travail. ................ hergie

) ................. ) W= ML” T-2 = ‘h3

Puissance. .............. E = ML2T-3 = P D&bit.. ................. 6 =MT4 =P

Lambert and Teissier (48) emphasized the purely formal nature of the theory of biological similarity, because neither the mechanisms involved nor the causal relationships among variables are part of the theory. They discussed not only the theoretical foundations, but also the corresponding experimental findings, i.e., basal metabolic rate, heart frequency, respiratory rate, gestational period, growth processes, etc.

Subsequent developments (21, 23-28, 67-69) are discussed in this review.

IV. DIFFERENT SIMILARITY CRITERIA

A. Geometric Similarity

Two systems are said to be geometrically similar when parts of the model (m) have the same shape as the corresponding parts of the prototype. Hence, two points (PI and P2) that correspond to each other are homologous points. In consequence, in two solids (& and S,) the three-dimensional coordinates of two homologous points (PI and &) are xl, yl , and ~1 for PI, and x2, ~2, and ~2 for P2.

2’1 and P2 are homologous if the following condition holds:

Xl y1 a -c-c-= x (I) x2 y2 z2

where X is a constant ratio. In biology, complete geometrical similarity between prototype and model

animals is not observed frequently. For instance, an elephant is not a geometrically enlarged rat nor is a dog a miniaturized horse.

It is common to observe that xl/x2 = Xl and y1/y2 = Z~/ZZ = X2 ; or we may even deal with completely distorted geometrical relationships, where x 1/x2 = Xl , whereas yl/y2 = X2 and <l/z2 = Xa . These models have been extensively described by Thompson (75) in his remarkable study.

October I975 THEORY OF BIOLOGICAL SIMILARITY 663

t I i L2 I

I I-. L, L2

1

-a l-4

x=2

FIG. 1. Geometric representation of length ratio (X), area ratio (X2), and volume ratio

04.

It is possible to extend geometrical similarity criteria to the relationship between homologous areas belonging to both prototype and model :

(2)

For two solids, whose respective volumes are Vi and Vz, the similarity conditions can be defined as:

In conclusion, geometric similarity can be characterized by the constancy of the length ratios (A), the area ratios (X2), and the volume ratios (X3), as shown in Fig. 1.

In this paper body weight (IV) is used extensively as a reference system. If we assume that the density (p) of prototype (6) and model (m) is practically the same, because both organisms have a similar body composition, and assume further that gravitational acceleration (g) is practically constant on earth, we may write the following ratios :

w PC MP- VP- x3

Wm Mm Vm (4)

where M = mass. For geometrically similar animals Kleiber (46) advocated the following length

ratio :

664 B. GUNTHER

LP L, -=- Wp1’3 Wm1’3

Then, for an area ratio (A) Kleiber obtained:

AP An -=- w 213

P w 2/3 m

Finally, for the volume ratio (V) we have:

v, vrn -=- wp wn

Volume 55

(5)

(6)

(7)

B. Physical and Biological Simikarities

It must be emphasized that in the biological world physical similarity, which we shall discuss now, is assured by the systematic departure from geometric similarity (19). Furthermore, the organism changes geometrically, SO as to remain the same physiologically, and a constant efficiency of many body functions can only be achieved by systematic size-correlated alterations of physical shape (9).

Among the more important reference works on dimensional analysis and physical similarity are treatises of Birkhoff (7), Bridgman (8), Duncan (1 Z), Focken (13), Ipsen (41), Johnstone and Thring (42), Langhaar (49)) Levi-Civita and Amaldi (53), and Sedov (66).

Four kinds of criteria can be applied to define a physical similarity: I> dynamic, 2) kinematic, 3) hydrodynamic, and 4) thermic. Therefore : I) two systems are said to be dynamically similar if homologous parts of the systems experience similar net forces; 2) kinematic similarity exists when the motions of two systems are similar: that is, if homologous particles lie at homologous points at homologous times; 3) for hydrodynamic similarity Ipsen (41) has established that “In similar flows, involving Reynolds number, the ratio of inertial to viscous forces must be the same at corresponding points” ; and a) thermic similarity exists between prototype and model when temperature distribution and heat flow at homologous areas are both similar.

In this review extensive use is made of the three-dimensional system introduced in physics by Maxwell, based on the concepts of mass (M), length (L), and time ( T).

Any variable (Q) can be defined in the model as a power function of the following kind :

Q 1 = Mla*L+ T,r (8)

For the prototype the same variable can be defined as:

Q 2 = M2*-L2@- T27 (9)

It is now possible to establish a quantitative relationship between both functions by means of Newton’s reduction coefficient (x) :

QJ x=- Q2 (N


Finally, the mass ratio A&/A& = p, the length ratio Ll/Lz = X, and the time ratio Tr/Tz = r are introduced into equation 10, yielding:

The basic assumptions of the four independent physical similarity criteria are shown in Table 1, and from the dimensional analysis and the corresponding scale equations it is possible to obtain the general similarity equations.

1. Mechanical O?- dynamic similarity criteria

The mechanical similarity scale equations lead to p = X3 and 7 = P2, and if these values are introduced into equation I1 we obtain:

or x = ~3~~p.py

(12)

x= X3”+ Is+ WY w

and if instead of the length ratio (X = LI/L~) we use the mass ratio (p = M l/Mz) or the weight ratio (w = WI/W,), equation 13 becomes

which also appears in the last column of Table 1. For all other physical similarity criteria the algebraic procedure is identical and is summarized in Table 1.

Table 2 contains 28 physical entities of physiological interest, their corresponding definitions, physical dimensions (MLT system), and the procedure for cal- culating the exponent of the weight ratio (u). In the last column, the reduced exponent (6) is shown, in order to be able to compare the theoretically calculated exponents to the empirically obtained values, in accordance with the well-known allometric equation2 set up by Huxley (40), saying that:

y = a*Wb (15)

where y = any function that can be defined in accordance with the MLT system, a= empirical parameter, W = body weight, and b = reduced exponent.

The discrepancy between the theoretically predicted reduced exponents (eq. 14 and Table 2) and the experimental findings prompted Lambert and Teissier (47) to abandon the mechanical similarity criteria to seek a better theoretical foundation.

2. Kinematic or biological similarity

By assuming the constancy of body composition and the equivalence between the length and the time ratios, Lambert and Teissier (47, 48) proposed the following general equation :

X = @! + l/3/3 f l/37 (16)

2 Allometry has been derived from the Greek : iiXXos = different; &pou = measure.

TABL

E 1.

Fou

r ph

ysica

l sim

ilarit

y cr

iteria

of

biol

ogica

l im

porta

nce

in ac

cord

ance

with

m

ass

(AI),

len

gth

(L),

time

(T),

and

tem

pera

ture

(to

> sy

stem

and

as f

unct

ions

of

bod

y we

ight

ra

tio

(w>

Item

Si

mila

rity

Crite

ria

Post

ulat

es

Dim

ensio

nal

Anal

ysis

Scal

e Eq

uatio

ns

Gene

ral

Sim

ilarit

y Eq

uatio

ns

A Me

chan

ical

or dy

nami

c I)

Body

de

nsity

=

cons

t. 2)

Acce

lerati

on

of gra

v- vit

y =

cons

t.

B Ki

nema

tic

or bio

logica

l 1)

Body

de

nsity

=

cons

t. 2)

Leng

th

ratio

(L

l/Lz>

eq

ual

to tim

e ra

tio

WI/T

21

C Hy

drod

ynam

ic or

trans

- 1)

Reyn

olds

numb

er

= po

rt co

nst.

.u=

veloc

ity,

2) Ki

nema

tic

visco

sity

D =

diame

ter

= co

nst.

D Th

erma

l I)

Body

de

nsity

=

cons

t. [H

] =

ML2e

T-

2 2)

Spec

ific

heat

= co

nst.

8 =

t0Jt

02

3) He

at flo

w pe

r ar

ea

time

= co

nst.

M/V

= 1.0

LT

-2 =

1.0

M/V

= 1.0

wJ

52

= TO

-2

Re

u* D

=-

V

Cl y

= L2

’ T-1

M/V

= 1.0

H/

Met”

= 1.0

H/

AeT

= 1.0

px3

= 1.0

X

= W

a +

l/3/9

+

116~

x*7-

2 =

1.0

X

= W

a +

1138

+

li3y

.P

x2’ T

--l =

1.0

X

= c3

a +

l/3/3

+

2i3y

p*r3

=

1 .o

X =

(&p

+ l/3

@

+ 1i

3y

+ O

S

~2',=

-2*@

--1

= 1.

0

&h'T

-3

= 1.

0

October 1975 THEORY OF BIOLOGICAL SIMILARITY

TABLE 2. Dimensional analysis, physical entities, and calculation of reduced exponents corresponding to mechanical similarity criteria (eq. la>

667

Item Physical Entity Description Dimensions Theoretical

Reduced M T Exponent a s” Y UO

1 Acceleration

8

9

10

11

I2

13

I4

15

16

17

18 Momentum

19 Moment of

21 Power

22 Pressure ; stress

23 Resistance to flow

24 Surface tension

25 Torque

26 Velocity

27 Viscosity (dynamic)

Action

Area

Conductance

Compliance

Density

Elastic modulus

Energy ; work

Flow

Force

Frequency

Heat quantity

Impulse

Inertia

Length

Mass

Mass generation

Period

inertia

Volume

Rate of change of velocity

Work X time

Length squared

Flow per unit pressure

Distensibility

Mass per unit volume

Stress

Force X distance

Volume per unit time

Mass X acceleration

Inverse of time

Calories

Force X time

Resistance to change of

state

Fundamental unit of space

Quantity of matter

Production of a sub-

stance

Mass X velocity

Effectiveness of mass in

rotation

Time of one complete revolution

Work per unit time

Force per unit area

Pressure difference per unit flow

Force per unit length

Effectiveness to produce rotation

Distance per unit time

Resistance to change of form

Length cubed

0

1

0

-1

-1

1

1

1

0

1

0

1

1

1

0

1

1

1

1

0

1

1

1

1

1

0

1

0

1

2

2

4

4

-3

-1

2

3

1

0

2 1

0

1

0

0

1

2

0

2

-1

-4

0

2

1

-1

3

-2

-1

0

1

2

0

-2

-2

-1

-2

-1

-2

-1

0

0

0

-1

-1

0

1

-3

-2

-1

-2

-2

-1

-1

0

0.00

1.50

0.66

0.50

0.66

0.00

0.33

1.33

0.83

1 .oo

-0.16

1.33

1.16

1 .oo

0.33

1 .oo

0.83

1.16

1.66

0.16

1.16

0.33

-0.50

0.66

1.33

0.16

0.50

1 .oo

Interestingly, the reduced exponents (b) calculated according to equation 16 are in close agreement with the experimental findings. Nevertheless, the predictive value of the above-mentioned theory could be improved further by introducing the operational-time concept [Giinther and Martinoya (33)], thus finally yielding the

668 B. GUNTHER Volume 55

following equation :

x = oar + l/3/3 + 1/3Q--O.O65~ WI

where - 0.065 = empirical correctioli factor of the operational-time exponent I’, whose numerical value is I = 1 when a clock is actually used during the experi- ment; I? = - 1 when we are dealing with an experimental frequency, and I? = 0 when no clock is used during the experiments.

The numerical correction factor (- 0.065) is empirical since it represents the difference between theoretical and experimental values of two functions that have been statistically well established in physiology, namely, basal metabolic rate and heart rate of mammals. For the metabolic rate we have in theory (eq. IS) an exponent b = 0.66(6) and in actual practice b = 0.734, yielding a difference of 0.067; for heart frequency the difference is obtained from -0.33(3) and -0.27, which yields 0.063. In order to improve the predictive capacity of the original theory of similarity advocated by Lambert and Teissier (47, 48)) we have proposed for this correction factor of the operational time a mean value of -0.065.

In Table 3, dimensional analysis, calculation of the reduced exponent (b), and introduction of the operational-time concept for 16 different functions of biological interest are shown.

3. Hydrodynamic and transport sim ilar it ies

Hydrodynamic similarity is based on the constancy of the Reynolds’ number (Re) and of the kinematic viscosity (v) in both prototype and model (see Table 1, item C).

If the corresponding scale equation is introduced into Newton’s reduction coefficient (eq. 11) we obtain:

Table 4 shows the dimensional analysis of 16 variables and their corresponding reduced exponent (6) calculated according to equation 16.

The transport similarity then can be derived from the general equation:

dA - 2z dT

-KS d(fw) dx

where A = any quantity that is transported, T = time, K = diffusion constant, S = area through which the transport process takes place and that is perpendicular to the transport process, x = direction of transport, and V = volume.

The dimensional analysis of equation 19 leads to :

[K] = L2. T-’

and the corresponding scale equation reads as follows :

1.0 = X2*7+ or 7 = X2

These results are independent of the dimensions and the nature of the trans-

October 1975 THEORY OF BIOLOGICAL SIMILARITY

TABLE 3. I)imensional analysis of 16 diferent j‘knctions, determination of operational-time factor, and calculation of reduced exponent (b)

669

Item Functions

Physical Dimensions Reduced Opera- Exponent t ional- Numerical Theoretical

M T ~‘+--~g ;” Time Value of Reduced Factor -0.065 I Exponent

a Y + f/3Y 07

w

-____- -~ - -

2

3

4

5

6

7

8

9

10 II

12

13 14

I5

16

Area

Compliance

Energy ; work

Force

Frequency

Length

Mass

Mass flow

Period

Power

Pressure ; stress

Resistance to flow

Tension

Velocity

Volume

Volume flow

-

1

2

4

2

1

0

1

0

0

0

2 -1

-4

0

1

3

3

0

2

-2

-2

-1

0

0

-1

1

-3

-2 -1

-2 -1

0

-1

0.66 0 0 0.66

1 .oo 0 0 1 .oo

1 .oo 0 0 1 .oo

0.66 0 0 0.66

-0.33 -1 0.065 -0.27

0.33 0 0 0.33

1 .oo 0 0 1 .oo

0.66 -1 0.065 0.73

0.33 1 -0.065 0.27

0.66 -1 0.065 0.73

0.00 0 0 0.00

-0.66 1 -0.065 -0.73

0.33 0 0 0.33

0.00 -1 0.065 0.07

1 .oo 0 0 1 .oo

0.66 -1 0.065 0.73

ported item A. For this reason the general transport equation (eq. 19) is valid for either mass transport or for the transfer of momentum (24).

4. Thermal similarity

Two attempts have been made to apply similarity criteria to heat-generation and heat-loss processes in homeotherms (3 1, 52). In the first attempt, five fundamental variables were employed [<mass (M), length (L), time (T), heat (H), and temperature (t)], whereas in the second only four variables were necessary (M, L, T, t) and only three postulates had to be established in order to get the corresponding reduction coefficients (see Table 1, item D).

\

Since p = X3 is valid for the first postulate, we can write the third postulate as x37-3 = 1.0, and consequently we obtain, surprisingly enough, Lambert and Teissier’s (48) original second postulate, namely r = X (Table 1, item B). Finally, we have to introduce this scale equation (7 = A) into the second scale equation in order to get 6 = X0.

Thus, the reduction coefficient (x) can be defined as:

x = &3(3cr + B + y + OS> WI

Table 5 shows the dimensional analysis of 19 thermal functions and their reduced exponents (b) calculated according to equation 20. We may add that the


TABLE 4. Dimensional analysis and calculation of theoretical reduced exponent (b) in accordance with transport or hydrodynamic similarity

Item Functions Physical Dimensions

Theoretical Reduced M T Exponent (b), Eq. 18 a s” Y

I

2

3 4 5 6 7 8 9

10 11 I2 13 I4 15 16

Area

Density

Diffusivity

Energy ; work

Force

Frequency

Length

Momentum

Power

Pressure

Surface tension

Velocity

Viscosity (dynamic)

Viscosity (kinematic)

Volume

Volume flow

0

0

1

0

0

2 0 -3 0

2 -1

2 -2 1 -2

0 -1

1 0

1 -1

2 -3 -1 -2

0 -2 1 -1

-1 -1

2 -1

3 0 3 -1

0.66 0.00 0.00 0.33 0.00

-0.66 0.33 0.66

-0.33 -0.66 -0.33 -0.33

0.00 0.00 1 .oo

0.33

introduction of the notion of operational time into equation 20 might perhaps improve the predictive value of the theory.

Unfortunately, only a few references on thermal characteristics could be found in the literature concerning interspecies measurements. Morrison and Ryser (58), in a systematic study on body temperature of mammals as a function of body weight, have concluded that body temperature is independent of body size, even though body weights covered a range of 1: 106. This experimental finding is in accordance with the theoretically predicted exponent (b = 0.00) for body temperature (Table 5, item 8).

Thermal insulation (L2. To H-l* t or M-l l T3 l t) of the winter fur of arctic animals in relation to body weight is practically invariant from the size of a fox (5 kg) to the size of a moose (> 500 kg), as has been shown by Scholander et al. (65). The theoretical exponent is b = 0.00 in accordance with equation 20 of thermal similari tv.

Conversely, the weight-specific thermal conductance (cal l g-l l h-l l “C-l) in birds, mammals, and lizards has been studied by Eerried and Kessel (37). For intact birds these authors found that the thermal conductance (C) can be defined as follows :

log c = 0.66 - 0.52 log W (34

the body weight (W) being given in grams. In birds without feathers they found:

log c = 0.86 - 0.44 log W (22)


TABLE 5. Dimensional analysis of i’9 thermal functions according to a-dimensional systems and calculation of theoretical reduced exponent (b)

Item Functions Physical Dimensions Theoretical

Reduced *I4 T Exponent (b), a s” Y t3 Eq. 20

I Total heat content

2 Heat transfer coefficient

3 Insulation

4 Heat generation rate

5 Heat conductivity

6 Specific heat (unit mass)

7 Specific heat (unit volume)

8 Temperature

9 Temperature (time)

IO Thermal conductance

II Heat capacity (temperature)

I2 Internal energy (unit mass)

13 Heat production

14 Temperature gradient (length)

I5 Stefan-Boltzmann constant

16 Enthalpy (unit mass)

17 Entropy

18 Thermal expansion coefficient

I9 Thermal resistance

1

1

-1

1

1

0

1

0

0

1

1

0

1

0

1

0

1

0

-1

2

0

0

2

1

2

-1

0

0

2

2

2

0

-1

0

2

2

0

-2

-2

-3

3

-3

-3

-2

-2

0

-1

-3

-2

-2

-3

0

-3

-2

-2

0

3

0

-1

1

0

-1

-1

-1

1

1

-1

-1

0

0

1

-4

0

-1

-1

1

1 .oo

0.00

0.00

0.66

0.33

0.00

0.00

0.00

-0.33

0.66

1 .oo

0.00

0.00

-0.33

0.00

0.00

1 .oo

0.00

-0.66

and for birds without feathers and skin, the log-log plot yielded :

log C = 1.16 - 0.48 log W

In 24 mammals, the thermal conductance was described as:

log C = 0.691 - 0.505 log w

Finally, for varanid lizards the allometric equation reads as follows:

log c = 1.354 - 0.37 log w

which means that the thermal conductance of varanid lizards is 10 times higher than that of intact birds and mammals and 4 times higher than in defeathered birds, as found by those authors.

The calculated allometric exponent (6) for the weight-specific thermal conductance (M-lo T-VI- t-l or L2 l TG3. t-l) is b = - 0.33 according to thermal similarity (eq. 20) and b = -0.66 according to heat transport similarity (24). The experimental findings mentioned above are within this range.

V. ORGANISMS AS MIXED REGIMES

The complexity of the morphological and physiological organization of living beings cannot be adequately described by means of any single similarity rule. For


this reason, it is necessary to treat this kind of situation as “mixed regimes” in the technological sense [ Johnstone and Thring (42)].

The possible similarity range varies from the mechanical or dynamic similarity criterion (7 = X1j2) up to hydrodynamic or transport similarity (7 = X2), with the kinematic or main biological similarity (7 = X) placed halfway.

As a general rule, we may state that dynamic similarity is the main factor when we are analyzing mechanical problems, where gravitational forces pre- dominate-movements of the head, legs, and of the body in general, related to skeleton, joints, and tendons and caused by the contraction of striated musculature.

Kinematic or biological similarity criteria should be applied when we are dealing with the functions of organs (heart, lungs, liver, kidneys, etc.) and their association into systems (circulation, respiration, metabolism, absorption, excre- tion, etc.).

Finally, hydrodynamic or transport similarity is particularly valid at the cellular and molecular level; such as, for instance, the viscous properties of air and blood, the various diffusion processes at the air-blood barrier and at the capillary- cell exchange boundary, as well as heat transfer along the organism, from the mito- chondrial level up, the blood transport system, until the outer surface of the body is reached, where heat transfer is performed through radiation, convection, and evaporation.

In conclusion, no single similarity criterion can provide a satisfactory quantitative explanation for every single function of an organism that can be submitted to dimensional analysis.

The idea of mixed regimes had been formulated in 1838-1839 by Sarrus and Rameaux (64) in their communication to the Royal Academy of Medicine in Paris : Yorsque la nature peut atteindre un but par plusieurs moyen, elle n’emploie jamais exclusivement l’un d’eux jusqu’aux limites, elle les fait concourir de man&e que chacun de ces moyens tend a produire une part lgale de 19efet total. ”

The main differences between various similarity criteria are related to the coefficient for the time ratio exponent (y), whereas the parameters of the mass ratio exponent (a) and of the length ratio (p) are invariant, as shown below:

Similarity Criteria

Dynamic or mechanical Kinematic or biological Hydrodynamic or transport

Parameters of Exponents for

Mass a Length /3 Time y

1 l/3 l/6 1 113 l/3 1 l/3 z/3

McMahon (56) has proposed an alternative pathway to the problems of size, shape, and metabolic rate in mammals by applying elastic stability criteria in relation to buckling and bending loads in animals of different size. The quasi-cylindri- cal elements that make up the whole animal have a length @)-to-diameter (d) ratio of approximately 10, and each length (I) increases as the $4 power of the corresponding diameter (d). In consequence, the whole body surface should be proportional to the product of both factors (Id). Now, when length is expressed as a

October 1975 THEQRY OF BIOLOGICAL SIMILARITY 673

function of body weight (I a W4), and also diameters (da VV8), this yields for the surface area of the body IW8 or W”*63. Thus, the present model agrees with the experimental observations (see Table 12, item I).

A second assumption is necessary [McMahon (56)] based on the constancy of the tensile strength of a muscle (a = F/A) and of its shortening velocity (AZ/At), in order that the power output of a particular muscle depend only on its cross- sectional area (A). Since this area is a function of d2, then the maximal power output is proportional to ( W3/8)2 = ?V”*75, This fractional power of body weight is equivalent to Kleiber’s law of energy metabolism as a function of body weight.

Finally, from the proportionality of lung volumes (VL) to body weight (IV), and from the fact that alveolar ventilation (VA) goes as W”.75, the same author came to the conclusion that the respiratory frequency (pr) must scale as W-o*25, which is in close agreement with the empirical and theoretical allometric exponent for this function (see Table 11, item 9; Table 12, item IO).

In the approach of McMahon (56), three independent assumptions were necessary in order to deduce the numerical values of the allometric exponents (b) for the total surface area of the body, the metabolic rate, and the respiratory frequency. This is in contrast with the general theory of biological similarities, as mentioned above, since the latter calls for only one set of basic assumptions.

VI. ALLOMETRIC EQUATIONS : THEORETICAL FOUNDATION

Once the reduced exponents (b) have been calculated it is necessary to assess the validity of various similarity criteria; for this purpose the theoretically obtained values (b) should be compared to the empirical findings. The most simple, universal, and convenient mathematical expression is represented by Huxley’s (40) allometric equation y = aW, where y = any function definable in terms of the MLT system, a = empirical parameter, IV = body weight, and b = reduced exponent. The numerical values of both parameters can be easily obtained from the log-log plot of the experimental data, since the logarithmic expression is a straight line (log y = log a + b log IV). These equations are widely used because of the simplicity of their determination and the interpretability of their parameters (19).

Because of the biological importance of allometry, a brief discussion of the theoretical foundation of this equation might be justified. For this analysis we lean heavily on von Bertalanffy’s (6) enlightened treatment on the subject, which is particularly related to growth processes in general.

In a system, defined as a complex of interacting elements that in the present example are only of one kind, the “growth” of the system is directly proportional to the number of elements present:

dQ -Z dt WQ (26)

Depending on whether the constant al is positive or negative, the growth of the system is likewise positive or negative, and the system as a whole increases or decreases.


The solution of equation 26 is :

Q = Qo.8 W (27)

where Qo = number of elements at t = 0. This exponential law, found in many fields, is also known as the law of natural

growth. Furthermore, this type of equation can be used for studying the competi- tion between parts, assuming that the increase in each element depends only on this element itself. For two competing elements we have :

(28)

For one element, the integration leads to:

ln Q 1 = al-t + kl

where kl is the integration constant. The corresponding antilogarithm reads as follows :

Q = kl 1 e-e a1 t

and if ekl = Cl , we obtain finally for both elements:

Q ad 1 = Cl-e

Q a2t 2 = C2-e

(29)

In order to eliminate t, we obtain:

t = In Qr - In Cl = In & - In CZ (30)

a1 a2

The solution of this equ .ation is :

C

00

a = - .

c2: Q2

With C&2* = a, and if a = b, we finally have Huxley’s (40) allometric equation :

Q 1 = a*Qzb (31)

As has been stated by Gould (19), in his thorough review on the subject of allometry in ontogeny and phylogeny, the term “allometry” can be used to desig- nate proportion changes, correlated with variations in size either of body weight or of a specific part (organs), concerning the morphological, physiological, or chemical characteristics of organisms. In addition, any increase in size requires form changes or changes of its proportions, in order to remain functionally efficient. But the increase in size means a decrease of the area-volume ratio, which can be com- pensated either by: 1) a differential increase of surfaces by complication of the structure (branching); 2) changes in shape (flattening); and 3) incorporation of inactive, inert, or nonrespiring organic matter, within the volume occupied.


In advanced animals we may find an increase of the complication of the internal structures (circulatory, respiratory, digestive, excretory systems), which eventually may lead to the required increase of the corresponding exchange areas.

VII. EMPIRICAL ALLOMETRIC EQUATIONS

Though widely scattered in the literature, many reports demonstrating the effective use of allometric equations have been published. In this report we em- phasize the close agreement between the empirically obtained exponents (b) of the allometric equation (r = a* Wb) and the calculated values of the reduced exponent (b), mostly in accordance with the kinematic similarity criterion, although other criteria are sometimes applied.

A. Morphometric Studies

A quite extensive literature on the subject has been excellently reviewed by Adolph (1). Furthermore, an allometric study on five species of primates has been reported by Stahl and Gummerson (72), from which we have selected some representative allometric equations (Table 6), in order to illustrate that gross somatic measurements, some characteristics of the skeleton, and the weights of numerous organs can be defined adequately by means of allometric criteria. The same holds for the organ weights in mammals, which have been correlated with the corresponding body weights and plotted logarithmically to obtain the pertinent allometric equations (70).

Due to the statistical nature of the empirical allometric equations and to the multifarious relationships among organs, we should not expect a better correlation. This can be illustrated by the extensive study of Tenney and Bartlett (73) on the tracheal dimensions in a variety of mammals (Fig. 2). These authors found that the

_ tracheal length is almost proportional to PV”e2? and the tracheal diameter is proportional to VV o.3g Since in both instances we are dealing with a linear dimension, . the theoretically expected exponent should be the same, namely, b = 0.33. This comparative study (73) yielded two values that are nearly equal to the expected figure.

The numerical data obtained for several organs are shown in Table 7; with the exception of the pituitary (b = 0.56), the adrenals (b = 0.80), and the brain (b = 0.70), for all the other organs the correlation with body weight is almost linear (b = l.O), as should be expected from a theoretical point of view.

However, as emphasized by Weibel (77) with regard to the factors determin- ing the diffusion capacity of the lung, other variables should be taken into account. These are the so-called “adaptation factors” (Ai), which eventually could be defined as power functions of body weight. In consequence, any biological characteristic, morphological or physiological, may be written as follows:


TABLE 6. Examples of allometric analysis of gross somatic, skeletal, and organ measurements in 5 species of adult primates

Item and Measurement Parameter (a) Exponent (b) Theoretical Exponent (b)

Gross somatic measurements, cm I Vertex-heel height 2 Trunk height 3 Chest circumference 4 Thoracic width

Skeletal lengths, cm 5 Humerus 6 Radius 7 Ulna 8 Femur 9 Fibula

Skeletal weights, g 10 Femur 11 Tibia I2 Fibula 13 Humerus I4 Ulna 15 Radius

Organ weights, g 16 Heart 17 Right lung 18 Empty stomach 19 Liver with gallbladder 20 Large intestine 21 Pancreas 22 Spleen

46.6 21.3 17.1

5.44

7.72 7.88 8.22 9.38 8.45

3.78 2.82 0.60 2.41 1.28 1 .oo

4.77 0.84' 3.30 1.02 5.80 0.96

30.9 0.80> 13.7 1 .oo

1.11 0.94 1.12 0.99,

0.30 0.28 0.37

0.33

0.32

0.35) 0.31 0.37 0.33 0.34 0.24

1.08) 1.02 1.04 1.07 1.15 1.17 1

1 .oo

1 .oo

Based on y = a* W6; W is body weight in kilograms; n = 35 animals. [From Stahl and Gummerson (72) .]

where Ai = adaptation functions (Al, A2 , As . . . A,) of endogenous or exogenous nature (metabolic rate; Po2; Pco2; temperature, etc).

Earlier work had suggested that organs, such as the kidneys, liver, and heart, may increase in size corresponding to the metabolic demands including a safety factor, viz., with an allometric exponent above 0.75, as has been pointed out by von Bertalanffy [quoted by Gould (19)].

For the endocrine glands, whose functions are mainly regulatory, the correlation with total body mass should not be of direct proportionality (b = 1.0).

The brain is another example of a marked discrepancy between the theoretically expected values (b = 1 .O) and the empirical findings (b = 0.70). As Rensch [quoted by Gould (19)] has pointed out, surface innervation is the major brain function, and consequently an interspecific value of b = 0.66 represents the surface dependency of brain growth. Furthermore, the cerebral cortex is a surface layer of gray matter that increases only by “surface” expansion while the internal white matter expands “volumetrically” [Le Gros Clark, quoted by Gould (19, p. 623)-J.

Otto ber 1975 THEORY OF BIOLOGICAL SIMILARITY

- Diameter (Slope - 0.39) Whale .

IO- o- - -0 Length (Slope : 0. 2 7 )

i;~ ~~~~~...~ er

0 O

0 .a a0 0 4

o 3

OYo 0

0 0

00 0

1 2

0.01’ 1 1 I 1 - - - - - - I

0 001 0.01 0. I I IO 100 1000 2

Body Weight. (kg )

FIG. 2. Logarithmic plot of tracheal diameter and tracheal length as a function of body weight in a representative collection of mammals. [From Tenney and Bartlett (73.1

Finally, certain structures must reach a certain size in order to function at all, but may gain little advantage from increasing beyond this threshold, as has been discussed by Gould (19). F or instance, the vertebrate eye shows a relative decrease in progressively larger animals of a related series; rods and cones do not vary with the size of the animal, since their dimensions are optically limited by the interfer- ence patterns of the waves of light, which set bounds to the production of clear retinal images [Thompson (75, p. 53)].

23. Phy siom etr ic Studies

1. Cardiovascular system

Thorough interspecies measurements have been performed in mammals, and thus the empirical allometric equations cover a wide field of functions. The correspondence between empirical findings and theoretical calculations is quite satisfactory, as shown in Tables 8, 9, and 10. In practically all cases, only one single similarity criterion was applied; occasionally transport similarity had to be used- for instance, when dealing with a molecular process such as blood viscosity (Table 9, item 2).

Another peculiar result was observed in some electrocardiogram (ECG) waves (Table 9, item 4), h w ose duration follows a mechanical similarity (b = 0.16) instead of kinematic or biological similarities (b = 0.27) commonly found in biological periodicities (Table 10, items 3 and 4).

The only striking discrepancy between empirical allometry and the theory of biological similarity was found in connection with the cardiac output in mammals (Table 10, item r>, as reported by Patterson et al. (60). Indeed, these authors ob-

678 B. GUNTHER

TABLE 7. Organ weights in mammals as functions of body weight

Vulwne 55

Item Organ Parameter (a) Exponent (b)

1 Adrenals

2 Brain

3 Heart

4 Kidneys

5 Liver

6 Lungs

7 Pancreas

8 Pituitary

9 Spleen

10 Thyroid

0.27 0.80

10 0.70

5.8 0.98

7.3 0.85

33.3 0.87

11.3 0.99

2.0 0.91

0.03 0.56

2.5 1.02

0.13 0.92

Based on y = a* Wb; y is organ weight in grams and W is body weight in kilograms. Theoretically calculated exponent is always b = 1 .OO. [From Stahl (70).]

TABLE 8. Allometric equations of cardiovascular variables in mammals

Item Function Parameter (a) Exponent (b) Theoretical -______ Reduced

Left Right Left ventricle

Right ventricle ventricle

Exponent ventricle lb)

-

Cardiac output, ml/min

End-systolic volume, ml

End-diastolic volume, ml

Heart weight, g

Heart rate, min-1

Stroke volume, ml

Stroke work, g-m

Total peripheral resistance, (dynes X s)/cm6

Ventricle weight, g

166 179 0.79 0.78 0.73

0.59 0.62 0.99 0.99 1 .oo

1,76 2.02 1.02 1.02 1 .oo 2.61 3.19 1.10 1.07 5. .oo

236 240 -0.25 -0.26 -0.27 0.66 0.75 1.05 1.04 1 .oo 0.59 0.12 1.15 1.14 1 .oo

487 X lo2 -0.76 -0.73

1.65 0.74 1.11 1.06 1 .oo

Based ony = a- W; W is body weight in kilograms. [From Holt et al. (39).]

tained exponent b = 0.9888, instead of the theoretically expected value of b = 0.73. The difference might be due to the fact that their data were obtained from a variety of sources, probably employing different techniques, with and without anesthesia, in animals under varying states of apprehension. In contrast, results agreeing with the theory were found by other investigators (see Table 8, item I; Table 9, item I; Table 10, item 8) : that is, when the same authors performed all the measurements, following strictly the same procedure in all interspecies deternrinations, the results are more reliable for the subsequent statistical treatment (Fig. 3).

2. Respiratory functions

Multiple aspects of the respiratory functions in mammals have been investi- gated, in order to establish the corresponding allometric equations. These experi-


TABLE 9. Empirical allometric equations of blood and circulatory system in mammals

Item Function Parameter (a) Exponent (b) Calculated

Exwgy; @A .

5

6 7

8

9

10

11

12

13

Blood flow, g X s-1

Blood viscosity rel. Hz0

Cross-sectional area of aorta, cm2

Duration, ms P wave PR interval QRS interval QT interval

Elasticity of aorta, dynes/cm6

Heart mass, g

Hemoglobin mass, g

Length of aorta, cm

Mean blood velocity (aorta), cm/s

Pulse time, s

Systemic arterial pressure, dynes/ cm2

Total blood volume, cm3

Total peripheral resistance, (dynes X s)/cm6

3.3 x 10-Z 0.74 0.73

2.3 X loo 0.067 o.o*

1.8 X lo-3 0.67 0.66

13.97 19.15 11.12 17.59

8.4 X lo8

6.6 x 10-S

1.3 x 10-Z

1.8 X loo

1.84 X 101

4.3 x 10-Z

1.2 x 106

0.168 0.187 0.165 0.30 1 1.19

0.98

0.99

0.32

0.07

0.27

0.032

5.5 x 10-Z 0.99 1 .o

2.8 X lo6 -0.68 -0.73

0.16-0.27

-0.95

1 .o

1 .o

0.33

0.07

0.27

0.00

Based on y = a* Wb; IV is body weight in grams. [Gunther and Leon de la Barra (29).] * According to transport similarities.

mental studies were performed without any underlying theoretical framework that might inadvertently have influenced the final results. For this reason, the agreement between empirical results and theoretical predictions is particularly striking. On the other hand, in the present review the redundancy of experimental results has not been avoided in order to illustrate the similarity of results reported by independent investigators. The correspondence between empirical allometric exponents (b) and the theoretically calculated reduced exponent (b) is shown in Tables 11, 12, and 13.

In general, the kinematic (biological) similarity criterion was applied with satisfactory results, with the exception of diffusion processes in the lung, where the “transport” similarity should be applied, if we consider that in this particular case we are dealing with a process located at the molecular level.

Tenney and Remmers (74) examined 26 representative mammals-from the shrew to the whale-and established an allometric correlation between lung volume and body weight with an empirical exponent b = 1.02, which agrees with the theoretically predicted value (b = 1.00) and with values established by other investigators (Table 12, item 16; Table 13, item 5). Furthermore, they found a linear correlation (double logarithmic plot) between alveolar surface area (Sa) and basal oxygen consumption (V02 a W”.74), as well as a linear relationship between alveolar diameters (microns) and the metabolic rate per unit body weight, which yields an allometric exponent of b = -0.7 1. It is interesting that the alveolar

680 B. GUNTHER Volume 5.5

TABLE 10. Allometric equations of cardiovascular functions in mammals and birds

Item Function Parameter (a) Exponent lb)

Theoret ical Exponent Ref. GO, Eq. 17

Cardiac output, liters/ min

Force of heart, dynes X 103

Heart frequency in mammals, mine1

Heart frequency in birds (body wt in g) , min-l

Heart work, erg X 103

Heart power, erg X s-1

Internal surface, left ventricle, cm2

Minute volume, ml/min

Plasma, colloid osmotic pressure cmH 20

Stroke volume, cm3

Tension in aortic wall (wt in g), (dynes X 103)/cm

Tension-time index per systole, dynes X s X lo3

Tension-time index per min, dynes X lo3

0.1017 0.9988 0.73 60 8.32 X lo5 0.69 0.66 43

241 -0.25 -0.27 71

I

2

3

4 763 -0.23 -0.27 10

5

6

7

8.91 X 10” 1.06 1 .oo 43 3.8 X lo5 0.77 0.73 43

6.17 0.68 0.66 43

8

9 204 0.75 0.73 43

23.2 0.075 0.00 81

10

II

0.74 1.03 1 .oo 43

2,436 0.35 0.33 79

I2 9.12 X 104 0.94 0.94 43

13 2.63 X lo7 0.66 0.66 43

Based ony = a* Wb; W is body weight in kilograms.

; ; RAT, , GUINEA I IPIG I I

i RA86lT I I I

I MAN, /HEART POWER (~0 iv

,I( /

Lrnl\UlHL

OUTPUT (0) I

- b = -w-.

i I I I I * ’ I I 1 1

2 s 1 0 1 2 3 log BODY WEIGHT (Kg) -

FIG. 3. Double logarithmic plot of various cardiovascular functions in mammals of different size. [Numerical data from JuiniE and Klensch (43, Table 1) .]

October 197.5 THEORY OF BIOLOGICAL SIMILARITY 681

TABLE 1 1. Empirical allometrie equations of respiratory system in mammals -- ________~_____I__-

Calculated Item Function Parameter (a) Exponent (b) Ewge;; @),

. _____

1 Elastic work respiration, per erg 2.17.x 102 0.965 1 .oo 2 Elastic respiration, erg/s power per 1.03 x lo2 0.70 0.73 3 Flow resistance, respiration, 1.30 x lo* -0.87 -0.73

(dynes X s&m5

4 Functional residual capacity, cm3 3.0 x 1o-3 1.19 1 .oo

5 Intrapleural dynes/cm2 pressure, 3.04 x 102 0.004 0.00 6 Length of trachea (man), cm 2.8 X 10-I 0.30 0.33 7 Lung mass, g 1.24 x 10-Z 0.99 1 .oo

8 Metabolic rate, erg/s 2.14 X 106 0.734 0.73 9 Respiratory frequency, s-1 5.5 x 100 -0.28 -0.27

10 Respiratory minute volume, cm3/s 2.8 X 1O-2 0.76 0.73 11 Thoracic compliance, cmb/dynes 1.65 X 10-S 0.82 1 .oo

12 Thoracic and lung, compliance, 1.2 x 1o-G 1 .oo 1 l oo cmb/dynes

13 Tidal volume, cm3 6.3 X 1O-3 1 .OO 1 .oo

I4 Vital capacity, cm3 6.3 X 10-i 1 .oo 1 .oo

Based on y = a* Wb; W is body weight in grams. [From Giinther and Le6n de la Barra

Gwl

diameters are almost of the same magnitude, irrespective of body size; however, if an animal has a higher metabolic rate, its alveoli will be of smaller size.

The gas exchange “surface” not only depends on body size (PV”eG6), but the average level of oxygen consumption (%,) also should be taken into account, as has been pointed out by Weibel (77). Th us, the allometric exponent of the alveolar surface area (Sa), expressed as a function of body weight (IV), is b = 0.95 for free- living species (shrews, monkeys, and dogs) and b = 0.88‘for captive species (mice, rats, and rabbits). This hypothesis is strongly supported by the study of normal mice compared with Japanese waltzing mice [Geelhaar and Weibel 18)], since the permanent motor activity characteristic of the waltzing mice is associated with a significant increase of oxygen consumption, alveolar surface area, and the hypo- thetical value for the diffusion capacity of the lungs.

The morphometrically estimated diffusion capacity [Weibel (76, 77)], as shown in Table 12 (item 3, 4) and Table 13 (items 2, 3), agrees with the theoretical reduced exponent (b = 1.0) in accordance with the transport similarity, since the diffusion capacity (ml l min-l l mmHg+) has the dimension [DL] = [M-l l L4' T].

In basal conditions, oxygen and carbon dioxide diffusion should be correlated with oxygen transport by the circulatory system and the oxygen consumption by all cells of the body; viz., the expected reduced exponent should be determined in agreement with the biological similarity criterion (b = 0.73). The estimated higher value (b 2 1.00) may be due to the application of another similarity criterion, within a mixed regime (transport similarity), or else the adaptative functions (Ai) suggested by Weibel (77) should be introduced :

DL = Al* (vo2)*A2(P02)'Aowb (33)


TABLE 12. Allometric equations of respiratory variables in mammals

Item Functions Parameter (a> Calculated Exponent

@I Reduced Exponent

W, 3. 17

1

2 3

4

5

6 7

8

9

10

11

a2

13

14

15

16

17

18

19

Body area, m2

Dead space, ml

Diffusion capacity for Oz9 ml/min per mmHg

Diffusion capacity for CO, ml/min per mmHg

Functional residual capacity, ml

Lung compliance, ml/cmH 20

Minute volume, ml* min-1

Oxygen uptake, ml* min-r

Power of breathing, g-cm/min

Respiratory rate, min-’

Thoracic compliance, ml/cmH 20

Tidal volume, ml

Total airway resistance, cmH *O/liter per s

Total blood volume, ml

Total compliance, ml/cmH 20

Total lung capacity, ml

Total lung weight, g

Vital capacity, ml

Work per breath, g-cm

0.11 0.65 0.66

2.76 0.96 1 .OO

0.16 1.18 1 .oo*

0.22

24.1 1.13 1 .OO

2.10 1.08 1 .oo

379 0.80 0.73

11.6 0.76 0.73

962 0.78 0.73

53.5 -0.26 -0.27

4.52 0.86 1 .oo

7.69 1.04 1.00

24.4 -0.70 -0.73

65.6 1.02 1 .oo 1.56 1.04 1 .oo

53.5 1.06 1 .oo 11.3 0.99 1 .oo

56.7 1.03 1 .oo

17.1 1.08 1 .oo

1.14 1 .oo*

Based on y = a* Wb; W is body weight in kilograms. [From Stahl (71).] to transport similarity criteria.

* According

Finally, it is worth mentioning that as early as 19 12 Dreyer, Ray, and Walker (11) had established a close correlation among the cross-sectional area of the trachea, the aortic area, the blood volume, and the surface area. They found that for all these variables the best allometric exponents (6) are within a narrow range (0.70-0.72).

3. Energy metabolism and body size

It would be beyond the scope of this paper to go into a detailed discussion of the problem, since the quite extensive literature has been excellently reviewed in several monographs [Benedict (2), Brody (9), Kleiber (45), Kayser and Heusner (44), Hemmingsen (36), and Zeuthen (80)].

After Lavoisier and Laplace introduced the calorimetric methods into biological research, in order to determine quantitatively heat generation (H) per unit time, it was found that the metabolic rate increases with body weight (Ha r/v> and conversely decreases when expressed as heat production per unit body mass

THEORY OF IXOLOGICAL SIMILARITY 683 October 1975

TABLE 13. Comparison of empirical allometric equations for respiratory functions in mammals and birds

-____ _____-I__ -I____--- -----___- ------.__--- -l______l_ - Calculated

Item Function Parameter (a) Exponent (b) Exp;;e;; (b), Ref. .

I

2

3

4

9

IO

II

I2

Alveolar surface area, m2

Diffusion capacity (morphometric estimation in mammals), ml O,/ min per mmHg

Diffusion capacity (birds, excluding

passerines (body wt in kg), ml CO/min per mmHg

Diffusion capacity of mammals (mor-

phometric estimation), m2 Free-living species Captive species

Lung volume, ml

Oxygen consumption, ml OJmin

Respiratory rate in mammals, min-l

Respiratory frequency of 45 species

of birds, min-’

Respiratory frequency, passerine birds, min-l

Respiratory frequency, nonpasserine birds, min-i

Thickness of air-blood barrier, pm

Tidal volume (mammals), ml

0 q 0038 0.98 0.66 76

0.0083 0.96 1 .oo* 76

0.55?

0.0122 0.96 0.66 0.0119 0.84 0.66

0.032 1.05 1 .oo

0.122 0.72 0.73

354 -0.28 -0.27

182 -0.33 -0.27

158 -0.28 -0.27 10

146 -0.31 -0.27 10

0.300 0.05 o.ot 76 0.0062 1 .Ol 1 .oo 1

1.14? 1 .oo* 51

76

76

76 1

10

Based on y = a* Wb; W is body weight in grams. * Calculated according to transport similarity . t Calculated according to cellular similarity.

@Y/W). A certain uniformity was attained when Sarrus and Rameaux in 1838- 1839, Rubner in 1883, and Richet in 1885 recommended application of the so- called “surface law” : that is, expression of the metabolic rate (kcal/h) per square meter and hour (H/m2 per h).

Although the surface rule has been generally accepted and is still widely used as a reference value (basal metabolism, cardiac index), this alleged biological law has been criticized severely by numerous investigators, since the experimental findings do not follow the postulated relationship of N o= IV21 a. As shown in Table 14 most investigators have found that the allometric exponent (0) varies around 0.75 not only in homeotherms (mammals and birds) but also in poikilotherms (amphibians and reptiles) and even in unicellular organisms. The total weight range studied is enormous ( lOIs), from the lo-l2 g unicellular organisms to the several-ton elephant (> lo6 g). The allametric exponent is practically the same (b = 0.751) and only a change of parameter (a) could be reported by Hemmingsen (36), as shown in Fig. 4.

The allometric exponent for the weight-specific metabolic rate (&‘W’) is pq70.73/J4/71.00 = f/j/7--0.2?) which has been confirmed by IMacMillen and Nelson (54)

684 B. GijNTHER

TABLE 14. Allometrie equations of basal metabolism in d$jTerent species

Votume 55

-

Item Unit of Measurement

jmphibians and rep tiles

kcallday

3irds Nonpasserine Passerine

ml OJmin ml OJmin

3asyurid marsupials

cm3 Od(g* h)

Sbernating animals

Summer Winter

kcaljday kcal/day

Homeotherms kcal* h-l

Mammals ml Oz/min

7 Placental mammals

cm3 02/(g- h)

8 Poikilotherms kcal* h-r

9 Unicellular orga- kcal* h-r nisms

Bdt’

kg

g g

g

Body T:zP, Parameter (a> Exponent

w Ref.

- 18 1.78 0.728 44

38 0.078 0.723 51 38 0.126 0.724 51

37.2 2.45 -0.261 54

37 63.6 0.62 10 2.09 0.69

44

39 2 x IO-2 ’ 0.751 36

38 0.057 0.75 0.061 0.76 0.064 0.734

37

20

20

3.8 -0.265

45 71

9

54

6i.9 X 1O-4 0.751

0.751

36

8.43 X lo-+ 36

Based on y = a* W? Theoretical reduced exponent is b = 0.73 in all instances or b = -0.27 for weight-specific metabolism.

both for placental mammals and for dasyurid marsupials, despite the fact that both have a parallel evolutionary history of 70 million years (see Table 14, items 3 and 7).

Only in hibernating animals, during summer and winter, Kayser and Heus- ner (44) found that the allometric exponent varied around 0.66 (see Table 14, item a), which might be interpreted as confirming the surface law.

However, if we accept that organisms should be considered as mixed regimes, the possible range for the reduced exponent (b) of metabolism can be defined from the mechanical (dynamical) similarity criterion (b = 1.16) to the kinematic (biological) similarity (b = 0.66). Any experimental result within the range 0.66 < - b < 1.16 can be accepted as valid. -

It is interesting to note that van Bertalanffy (4, 5) had described three metabolic types closely correlated to three growth types: I) in the first type, the metabolic rate is proportional to the surface area (b = 0.66); 2) in the second type, metabolism is proportional to weight itself (b = 1 .OO) ; and 3) in the third type, the

October 1975 THEORY QF BIOLOGICAL SIMILARITY 685

-6

-8

[og 6ody weight (g)

FIG. 4. Relationship between standard energy metabolism and body weight (double logarithmic scale) in unicellular, poikilothermic, and homeotherrnic organisms. All values are corrected to 39°C. [Modified from Hemmingsen (36).]

metabolic rate holds a halfway position between weight and surface proportionality (0.66 < b < l.O)- viz., the so-called “intermediate” form. Each metabolic type is correlated with a particular kind of growth curve. Nevertheless, the upper limit is b = 1.0 and not b = 1.16 as postulated by the dynamic similarity criterion. But, surprisingly enough, in tropical snakes GalvSo et al. (17) have found that oxygen consumption expressed as a function of body weight for 16 different-size snakes (Boidae) yielded an allometric exponent of b = 1.09. A similar result (b = 1.12) was reported by GalvZo et al. (17) based on Benedict’s data, published many years ago, obtained from 12 different snakes also belonging to Boidae. These results sub- stantiate the theoretical upper limit (b = 1.16), which is valid for any dynamic similarity. Indeed, tropical snakes represent a very peculiar model because of their filiform shape, their particular kind of locomotion, and their warm habitat.

It should be mentioned that the surface law does not hold in the tropics, as has been demonstrated (14) for dogs, since heat production (H), expressed in kilo- calories per hour as a function of body weight ( W) in kilograms, can be described as follows :

H= 2.000w”~go (34)

686 B. GtiNTHER Volume 55

For lean men living in the same tropical zone (Sao Paulo, Brazil) Galvao (15) found that

H= 2.083 Woe8 3 (35)

and for well-proportioned men he obtained the following allometric equation (16) :

H= 2.334W0’78 (36)

On the other hand, for well-proportioned men living in cold climates, Galvgo (16) calculated that

H= 4.088 W”-67 (37)

and in the latter case the allometric exponent is strictly surface proportional (b = 0.66). Galvgo concluded that the surface law was not a mere coincidence, but a thermal causal law, since in men and animals living in cold climates the metabolic rate is surface proportional, whereas in tropical zones much higher values were found (0.75 < b < 1 .OO); this may suggest that heat production becomes proportional to the metabolically active mass.

An alternative explanation for the three different metabolic types (0.66 < b < 1.00) could be visualized if the surface-volume relationship is studied at the cellular level. Davison [quoted by Gould (19)] has pointed out that an increase in body mass can be accomplished by either: I> cell enlargement, and consequently the cell surface could vary as the Fs power of weight; or 2) formation of additional cells of fixed size, and then cell surface could increase proportionately with weight.

Unicellular organisms may follow the first alternative (M a W”*66), multicellular organisms may approach the second alternative (A4 a Wl-O), or an intermediate form (0.66 < b < 1.00) could be the result of a compromise solution (mixed regime).

Recently Maldonado et al. (55) have studied cell diameters at the nuclear level in tissues of some homeotherms (mouse, dog, bovine) and poikilotherms (frogs, toads), keeping strictly invariant the fixation and staining techniques. From the statistical analysis of these measurements the correlation between cell diameter (D) and body weight (W) could be established.

For homeotherms, the allometric equation was:

D= 13.26 W”-028 (38)

For poikilotherms they found :

D= 16.79 W”-04g (39)

where the cell diameter (D) is expressed in microns and the body weight (W) in grams.

In conclusion, the cell diameter increases only slightly while the body size varies markedly, in accordance to the postulated assumption of the cellular similarity criterion. 3

3 Cellular similarity should be applied to processes at the cell level, approximately X0 or M/O*O.

where L& is not A, but

Otto ber 1975 THEORY OF BIOLOGICAL SIMILARITY 687

0.8

W

- 0.8

N= 80

Y = 1.02 x - 0.006

I = 0.99

I r I 1 I 1 I 1 I I -0.8 - cl.4 Q.0 0.4 0.8 ----z-- .

THEORETICAL EXPONENT (b)

FIG. 5. Correlation between experimental and theoretical reduced exponents (6) fix 8Q empirical allometric equations. [From Giinther (22) .]

VIII. EMPIRICAL ALLOMETRIC EQUATIONS, THEQRETICAL REDUCED EXPONENTS,

AND OPTIMALITY PRINCIPLES

In the light of the varied experimental evidence and of the almost universal applicability of the allometric equation in biology, it seemed worthwhile to study the possible correlation between the empirical allometric exponents and the calculated reduced exponents [Gunther (ZZ)]. F or any function the reduced exponent (b) was calculated according to equation 17 and compared with 80 empirical allometric exponents (b) found in the literature, which are reported in part in the present paper (see Tables 6-14). The corresponding regression equation reads as follows :

Y = 1.02x - 0.006 (40)

Since the correlation coefficient is r = 0.99, we may conclude that the correspondence between the empirical values and the theoretical predictions is satisfactory (Fig. 5).

According to Rosen (63), the allometric laws are closely related to ‘“optimality” considerations- viz., the minimization processes defined on the set of all possible relations. Hence, the allometric form functionals will appear as immediate consequences of optimality considerations. Assuming that this interpretation is

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valid for each allometric equation, then the assemblage within a multivariable system should lead to a perfect morphological and functional adaptation among all these variables. Therefore, the just-described correlation between the theoretically calculated allometric exponents and the power functions derived from the statistical analysis of the empirical data should not be a surprise.

In addition, each organ is constructed in such a fashion that it will function in an optimal manner for efficiency, energy expenditure, and ease of maintenance (62). However, it is convenient to distinguish between the “absolute” optimum characterizing the adult form, and a number of “relative” optima found in a developing organism (growth), which is considered as an open system passing through a succession of relati.vely stable steady states until it ultimately attains a stable state (adult form). These systems are necessarily related by an allometric law, whereby the allometric parameters (a and b) are defined by Rosen (63) as “structure constants.” For instance, parameter a is essentially a ratio of initial values and exponent b is the ratio of a pair of rate constants.

The allometric formalism is independent of time: viz., the time “cancels out.)) Nevertheless, the absence of absolute time is more apparent than real, since one form functional (y) is no longer measured with regard to absolute time (t), but rather with respect to another form functional (x), a relationship that now plays the role of a “physiological clock.”

As a particular optimiration problem consider Wilson’s (78) study on the “minimum entropy production” as a design criterion for the respiratory system in man. He shows that calculated and observed values of pulmonary ventilation agree within the accuracy of the experimental data for oxygen consumptions below 2 liters/min.

IX. POSSIBLE RELATIONSHIPS BETWEEN TWO ALLOMETRIC EQUATIONS

As has been shown, single allometric equations are very useful for the quantitative description of interspecies morphometric and physiometric relationships. Nevertheless, it is possible to extend these applications to the algebraic treatment of the relations among organ weights, any particular function, and body weight (20). The need for this kind of treatment is illustrated by the following statement of Adolph (1) : “It seems likely that an organism is an integrated system by virtue of the fact that none of its properties is entirely uncorrelated, but that most are demon- strably interlinked.”

From a theoretical point of view four kinds of correlation can be established between pairs of allometric equations (22) :

A) The weight of two organs (~1 and ~2) can both be expressed as functions of body weight (VV).

B) Two independent functions (fl andfz) can be described quantitatively with regard to body weight (IV).

C) The weight of an organ (w) and any corresponding function (f) of this particular organ are both correlated with body weight (Mr).

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SEVERAL INITIAL ALLOMETRIC FINAL ALLOMETRIC

ITEM RELATIONSHIPS EQUATIONS EQUATIONS

w4 = a, Wbl

I

A al

W= b, 4

’ b w2

=a,W 2 I ca,) b,/b2cwzJ

B f, =a, Wbl

al

f2 =a2 Wb2

f, = [a >b /‘b (f2’b1’b2

2 ’ 2

f =a, Wbl

f= al b,‘b2

b W =a2W 2

(a2)bl/bJw)

D f =a, w b,

W l W b,/b2

f b

=a,W 2

FIG. 6. Allometric type of relationships among organ weights (w), functions v), and body weight (W).

D) One function (f) h as b een determined both with regard to body weight (W) and with the corresponding organ weight (w).

Since in the first three instances all variables are expressed as functions of body weight (IV), the same algebraic treatment can be employed; in the last case the situation is slightly different and deserves another approach (Fig. 6). However, in all four instances the final equations are typical allometric equations, once the parameters are reduced to the simplest form.

X. DIMENSIONLESS AND INVARIANT NUMBERS

Dimensional analysis and similarity criteria lead finally to the disclosure of dimensionless and invariant numbers, namely to quantitative multivariable relationships, that are valid for all animals belonging to a certain model, irrespective of body mass.

Two procedures have been recommended to obtain dimensionless products: I) the “allometric canceling technique” advocated by Stahl (67-69, 71), and 2) the “dimensional and solution matrix method” that we have applied to a wide variety of physiological problems (24, 29-32, 52).

690 B. G&VTHER Volume 55

A. Allometric Canceling Technique

The adequate combination of allometric equations leads to dimensionless numbers (MO *Lo 0 TO), whose residual mass exponent (RIME) should be as close to zero as possible. Any value greater than zero means that the resulting number is subjected to a scale effect-viz., that the dimensionless number varies in accordance with the magnitude of the final exponent of body mass (IV).

In order to illustrate the technique proposed by Stahl (67) we have summarized a few of these invariant numbers in Table 15. Some are simple ratios of two allometric equations; in other instances these allometric ratios are formed by three or more empirical allometric equations. The final parameters (a) and the exponents (b) or RME are also shown in Table 15. In general, RME is very close to zero; only occasionally the result is exactly b = O,OO, as one would expect from the reduced exponents calculated in accordance with the theory of biological similarity.

Nevertheless, RME near the ideal value of zero is useful, both from a theoretical and a practical point of view.

TABLE 15. Some allometric ratios obtained by canceling technique -_____I__

Item

I

6

Composition

Mass of blood

Mass of heart

Urine output flow

Mass of kidneys

Life time

Breath time

Air flow X breath time

Lung volume

Oxygen flow X pulse time

Blood volume

Oxygen flow X breath time

Body mass

Basal power X lifetime

Mass

Allometric Ratio Form

0.055 MO* gg

0.0064 Mo.*2

0.0212 M”JJ6

8.85 X lo3 1MO.29

4.7 x 1O-6 M”a2*

4.7 X lo+’ M”o2* X 120 Mo.74

1.24 X 10-Z MO-99

3.8 Mo.73 x 1 .2 X lo+ MO.2’

5.5 x 1O-2 Moo99

3.8 MO”‘3 X 4.7 X lo+ M”e2*

1 .O Ml.O”

70.5 MO.‘3 X 365 X 7.52 MO.29

1 .O M1.O

Resulting Parameter

(4

8.3

0.3 cm3/g-h

8.3 X 10-4

1.8 X 1O-4

1.9 x lo6 kg-Cal/kg

M = mass of organism. Gram-cubic centimeter-hour system of Adolph (1) is used: in item 3, where mass of animal is in kilograms and time is in hours. [From Stahl (67).]

RZid- Mass Expo- nent

-0.03

-~ Except


B. Matrix Method

From a set of empirical allometric equations it is possible to establish exact numerical relationships that are adimensional and invariant for animals belonging to the same model. This assumption is based on the ?r theorem4 set up by Bucking- ham in 19 14, which is a formula for deciding how many dimensionless numbers to expect. According to Ipsen’s (41) definition: “The theorem states that the number of independent dimensionless groups is equal to the difference between the number of variables that go to make them up and the number of independent dimensions involved.” It does not depend on the number of dimensions actually used, but rather on the minimal number that might have been used. Buckingham’s 7r theorem only tells us that there must be at least a certain amount of dimensionless numbers involved.

The first step in this procedure is to select the empirical allometric equations that are of particular interest for the problem under study. Second, the variables- the allometric equations-are arranged in a “dimensional matrix” and then the corresponding “equations of conditions” are established (n = MO l Lo* TO). Finally, from the ‘%olution matrix” we obtain a complete set of 7r numbers whose adequate combination yields the “invariant numbers.” They are valuable for interspecies comparisons, because these dimensionless numbers are numerically constant; i.e., the product of the corresponding allometric equations does not change from one species to another. The invariant numbers are the same, irrespective of body size, provided that the residual mass exponent is zero; otherwise a scale effect may be observed.

C . Invariant Number of Mammalian Circulatory System

The combination of two invariant numbers (29) yielded a circulatory invariant number (IC) that included six empirical allometric equations of interspecies validity :

Ic = p.~“. T3

- 308W”*01 iioZ*TPR*Qb -

where : P = systemic arterial pressure (dynes. cmB2) = 1 l 17 X lo5 W”*023 v = mean velocity of blood in the aorta

(cm l s-l) = 1.84 X lo1 W”.07 T = duration of one cardiac cycle (s) =z 4.3 X low2 W”*27 x702 = basal oxygen consumption (cm3 l s-l> = 1.06 X lo- 3 W” s7 3 TPR = total peripheral resistance

(dynesscmV5) = 3.35 X lo6 VV”s6*

4 Use of r to denote a dimen soever with the number 3.1 4159.

sionless product is conventional and has no relationship

(41)

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Qb = total blood volume (cm3) = 5.5 X 10W2 W”*gg W = body weight (g) This dimensionless and invariant number (I,) is practically constant for all

mammals, since the scale effect is relatively small and the RME is only b = 0.01. For a 20-g mouse Ic = 3 17; for a 4-ton elephant Ic = 358, despite the enormous body weight ratio of 1 :ZOO,OOO.

The invariance of Ic indicates that this dimensionless number is not dependent on body size and that under basal conditions six allometric equations can be represented by a single figure (I& because exact numerical relations exist among all these variables of the circulatory system of mammals.

D. Invariant Number of Respiratory System of Mammals

One of the invariant numbers valid for the respiratory system of mammals (IR) was obtained (32) through dimensional and solution matrices applied to the corresponding allometric equations :

IR = VE 0.03

(Cpv~~r~GH)1/2 = o*0233w (42)

where : VE = volume flow of expired air (cm3@) = 2.8 X 10S2 W”mT6 Cpv = thoracic and pulmonary compliance

(cm5 l dynes-‘) = 1.23 X lo-+ M/l-O0 I+ = respiratory frequency (s-l) = 5.5 X loo Wwo*28 GH = basal metabolic rate (ergs-‘) = 2.14 X lo5 W”+734 W = body weight (g) The small scale effect (RME = 0.03) results from substituting the four em-

pirical allometric equations in equation 42. As has been said, only when the four theoretically deduced exponents (b) are used may an RME of exactly zero be expected.

E. Invariant Number of Cardiores-iratory System of Mammals

The product of the invariant numbers (Ic and IR) yields the following multivariable relationship [Gunther (32)] :

I p.t3. y?&

CR = %2*TPR*Qb(Cpv$roGH)“2

= 7.176 W”*04 (43)

The resulting invariant number (I,,) represents a constant relationship among 10 variables of the circulatory and respiratory systems valid for all mammals. The residual mass exponent is slightly higher (b = 0.04) than in the original invariant numbers (Ic and &), but it is still small enough to be used as an interspecies index.

The disclosure of adimensional and invariant numbers seems to indicate that

October 1975 THEORY OP BIOLOGICAL SIMILARITY 693

among organs and functions exact numerical relationships should exist, irrespective of the size and nature of the model concerned.

XI. ICONIC TREATMENT OF PHYSIOLOGICAL PROBLEMS

A. Block Diagram of Systemic Ci?dation of A4ammals

The cardiovascular system of mammals can be analyzed as a causal chain of allometric equations, characterized in each instance by the corresponding reduced exponent (6).

The algorithmic treatment of a functional sequence of allometric equations has been derived from an iconic representation of algebraically interrelated functions, as reported previously by Milhorn (57), Guyton and Coleman (35), Gunther and Hodgson (34), and Gunther (22).

The sequence of this analysis may be characterized numerically in the following manner (Fig. 7) :

1) During the cardiac cycle, the telediastolic volume (VTD) is reduced (minus sign) in the amount of the stroke volume (Vs), while the telesystolic volume (VTS)

remains in the heart. All these volumes (see Table 3) theoretically should have the same reduced exponent (b = 1.00).

2) The product of the stroke volume (Vs) times the frequency of the heartbeat (I?h) yields the minute volume (MV). In the present case the following allometric operation should be made:

M/'l.O x w-o.27 = M/O"73

3) The minute volume (MV) times the total peripheral resistance (TPR) gives the mean systemic arterial pressure (Pas), as a result of:

M/o.73 x w-o.73 = pj,ro.oo

4) The mean systemic pressure (Pas) times the conductance of the arteriolar outflow (FK) yields again the minute volume (MV).

5) The minute volume (MV) times the d,uration of the diastolic period of

FIG. 7. Block diagram and allometric algorithm of cardiovascular system of mammals. For description, see text. [From Giin- ther (24) .]

694 EL G‘ijNTHER Volume 55

the heart (TD) determines the amount of blood (AV) that is to fill the heart cham- bers during diasto le. This allometric product reads as follows:

wO.73 x wO.27 = j,$71.00

6) The telesystolic volume (VTS) plus the diastolic volume increment (AV) gives the telediastolic volume (VTD), and thus we return to the starting point of our analysis.

This sequence is valid for the cardiovascular system of the mammalian model, irrespective of the size of the animal, since all basic allometric equations were derived from interspecies experimental findings.

In each block the corresponding theoretical allometric reduced exponent (b) has been indicated, to illustrate the algorithmic sequence and to permit the algebraic control.

B. Block Diagram of Gas Transport System: Blood, Circulatory, and Respiratory Systems

Figure 8 shows the algorithmic sequence of the gas transport systems until the cellular level is reached:

I) The respiratory minute volume both for inspired (VI) or expired (VE) air is the result of the tidal volume (VT) times the respiratory frequency (%‘r) per minute.

2) The oxygen uptake of the organism during 1 min (vo,) results from the difference between the quantity of oxygen present in the inspired air (VI X FI&

minus the quantity of oxygen in the expired air (VE X FE&, in agreement with the well-known respiratory equation :

x702 = 91 X F1oP - VE X FEDS

3) The aerobic metabolism of all cells belonging to an organism is associated with oxygen consumption (Vo2) and to carbon dioxide production (VCO~). As

CO2 is leaving the cells, it should have a negative sign. 4) The oxygen delivered to the cells (002) can be calculated in accordance

with Fick’s principle [VOW = MV (Ca,, - C&J.

FIG. 8. Block diagram and algorithm of allometric equations

corresponding to circulatory, respiratory, and metabolic systems of mammals. For description, see

text. [From Gianther (24).]

October 1975 THEORY OF BIOLOGICAL SXMILAWITY 695

5) As mentioned previously (Fig. 7) the stroke volume (Vs) times the heart rate (l?h) yields the minute volume (MV) of blood ejected by the heart.

6) Carbon dioxide transport can also be described by Fick’s principle. 7) Finally, the total amount of carbon dioxide produced per minute (ho,)

divided by the CO2 fraction of the expired air (FEATS) yields the minute volume of expired air, in agreement with the well-known respiratory equation :

Within each block of the diagram (Fig. 8) we can find the corresponding allometric reduced exponents (b), in accordance with equation 17 and Table 3.

C. Signal-Flow Diagram of Cardiovascular System of Mammals

Instead of the block diagram, it is possible to use for the same purpose the signal-flow algebra developed by S. J. Mason in 1953 in order to describe the algorithm of allometric equations belonging, for instance, to the cardiovascular system of mammals.

According to Ogata (59), the signal-flow graph consists of a network in which “nodes” are connected by “directed branches,” where each node represents a system variable and each branch, connected between two nodes, acts as a signal multiplier. As we might expect, a signal-flow graph contains essentially the same information as the more conventional block diagram, but has great advantages with regard to the mathematical solutions of complicated networks with multiple- feedback circuits.

For example, let us consider the cardiovascular system of mammals as a “path” from the “source” represented in this case by the stroke volume (Vs) until reaching the “sink,” which is represented by the mean systemic arterial pressure (Pas). Figure 9A shows that the stroke volume (Vs) multiplied by the gain or transmittance, namely the heart rate (l?h), yields the minute volume (MV), and multiplied by the total peripheral resistance (TPR) it gives the systemic arterial pressure (Pas).

Figure 9B illustrates the algorithmic sequence of the algebraic operations, starting from the stroke volume (b = 1.00) to the systemic arterial pressure (b = 0.00); Figure 9C shows the total transmittance of the original cascaded branches; and Figure 9D confirms that the product of the three variables yields the correct dimensional answer (Fig. 9E).

XII. SUMMARY

From this review we may conclude the following : 1) The body weight of an organism is an adequate reference index for the

correlation of morphological and physiological characteristics. In comparative physiology, body weight can be recommended as a unifying

frame of reference, particularly if the ponderal scale includes several decades, in order to apply logarithmic scales for the variables involved.


0 A

0 B

0 c

VS ih h/W TPR Pi% 0 \ / v / I.0 -027 0.73 -0.73 a0

ihx,TPR 3 /

VS PCS

FIG. 9. Signal-flow diagrams of circulatory system, corresponding allometric exponents (b), and dimensional analysis. [From Giinther (24) .]

0 B Vsx ihxTPR= PC

2) The statistical analysis of the experimental data can be represented con- veniently by means of the logarithmic equivalent of Huxley’s allometric equation

(Y = a- W), which is the most simple and at the same time the most versatile mathematical expression for intra- or interspecies comparisons.

The exponents (b) for the allometric equations can be predicted for all biological variables definable in terms of the 1MLT system of physics (M = mass, L = length, T = time) or of a four-dimensional system MLTt, where t = temperature.

3) By means of dimensional analysis and the theory of biological similarity a range of similarity criteria can be established: a> mechanical or dynamic similarity; b) kinematic or biological similarity; and c) hydrodynamic or transport similarity.

Most functions obey the so-called biological (kinematic) similarity, particularly when the concept of operational time is introduced into Lambert-Teissier’s original theory.

4) A satisfactory correlation (r = 0.99) for 80 empirical allometric exponents (b) describing morphological and physiological characteristics of living beings was found. These results are discussed in relation to Rosen’s optimality principles in biology.

5) Organisms should be considered as mixed regimes. This means that no single similarity criterion can predict the allometric exponent (b) of all functions that dimensionally belong to ML 5P or MLTt systems, despite the fact that in the great majority of cases kinematic similarity will satisfactorily predict the reduced exponent (b). Nevertheless, in some instances mechanical (dynamic) similarity must be applied, and in other circumstances hydrodynamic (transport) similarity.

6) Cellular or molecular levels are not in the domain of the present theory, since neither cell dimensions nor molecular processes (viz., blood viscosity, diffusion capacity) can be predicted by biological similarity criteria.

7) From the treatment of pairs of allometric equations new algebraic relation-

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ship were deduced with the aim of using not only body weight as a reference system but also any organ weight or any function. In all instances new allometric equations were obtained.

By means of the empirical canceling method of Stahl, or by dimensional and solution matrices, dimensionless and invariant numbers were obtained for the cardiovascular and respiratory systems of mammals. These invariant numbers may be used for characterizing multivariable systems, irrespective of the size of the organisms concerned.

8) ,411ometric equations can be associated into causal chains through an algorithmic treatment of a complete set of functional relationship, by means of block diagram or signal-flow diagram representations.

9) Dimensional analysis and similarity criteria constitute a valuable procedure for the prediction of multiple morphological and physiological parameters in animals.

I am indebted to Dr. Federico Leighton for helpful criticism and to Miss Alicia Vicente for typing and preparation of the manuscript. I thank the authors and publishers who kindly granted permission for reproduction of Figures 2, 5, and 7-9.

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Dimensional Analysis and Theory of Biological Similarity Present address : Instituto de Ciencias Medico-Biologicas, Departamento de Fisiopato- logia, Universidad de Conception, Chile