Download Geometric similarity of aorta, venae cavae, and certain of their

Geometric similarity of aorta, venae cavae, and
certain of their branches in mammals
J. P. HOLT, E. A. RHODE, W. W. HOLT, AND H. KINES
Department of Heart Research, University of Louisville School of Medicine Health Sciences
Center, Louisville, Kentucky 40202; and Department of Medicine, School of Veterinary
Medicine, University of California, Davis, California 95616
Holt, J. P., E. A. Rhode, W. VV. Holt, and H. Kines. length of the mammalian aorta. In earlier reports we
Geometric similarity of aorta, venae cavae, and certain of presented evidence that allometric equations apply to
their branches in mammals. Am. J. Physiol. 241 (Regulatory the geometric and functional characteristics ofthe mamIntegrative Comp. Physiol. 10): R100-R104, 1981.—The diam- malian heart (8,10). The present studies were undertaken
eters of the aorta and venae cavae at various points throughout to determine whether the geometry of these vessels can
their lengths, the diameters of their major branches, and the be described by p0wer-law equations relating diameter
lengths of various aortic and vena caval segments were meas- and { h bod ^. ^ (n)
ured in plastic corrosion casts of the artenai and venous systems ° jo
of the normal adult mouse, rat, rabbit, dog, goat, horse, and
METHODS
cow, extending over a body weight range of 38,000-fold (arterial) METHODS
and 1,100-fold (venous). It is shown that the diameters and t>i„„*:„
™,^casts
««4«1t
*«u
ofunormal
mice,
rats,
• . « • * » l j u j u i P l a s t i c c o r r o s i o n c a s Plastic
t s o„«™„„;«»,
f corrosion
norm
a l ~ea d™,-™oi
lt m
i adult
c e , ™,-o«
r a t sr-nfc, r arab
blengths of these vessels are described bv power-law equations . . , , , ,
bits,
dogs,
goats,
horses,
and
cows
were
prepared
as
relating the particular diameter or length to body weight (BW)
£lte, dogs
goats,
horses,
and
cows were prepared
as
follows:
after
attaining
a deep
anesthetic
raised to a particular power, i.e., diameter = a BW6. Equations
follows:
after
attaining
a deep
anestheticlevel,1
level, cannucannulation
of of
thethe
carotid
for the diameters and lengths of the vessels are given for slightly
lation
carotidand
andfemoral
femoralarteries
arteriesand
and exposure
exposure to
to
severe
hemorrhage,
the
distended vessels and for vessels distended in the physiological
severe
hemorrhage,
theanimals
animalswere
werekilled
killedby
by injecting
injecting
a blarge
range.
either
a
large
dose
of
peither
ento
a r bdose
i t a l of pentobarbital
s o d i u m sodium
or
c or
o nconcen
centrated KCL solution into the arch of the aorta. This was
arteries; veins; mouse; rat; rabbit; dog; goat; cow; horse; similar followed by perfusion of the arterial system with physi
ity
ological saline by way of the carotid artery for 2 min.
then bleeding from the femoral artery for 2 min. BatsonV
Compound (Polysciences, Paul Valley Industrial Park
following Thompson's discussion (21) of the effects Warrington, PA 18976) was then quickly mixed and
of scale in biology, Huxley (12) employed allometric or injected under 100 mmHg pressure into the carotid ar
power-law equations for somatic form analysis, and more tery. Bleeding was continued from the femoral arten
recently several investigators have described power-law until plastic was seen to pass from the cannula at which
equations relating various physiological variables to body time the plastic solution was injected under 100 mmHg
weight (5). Evidence has been presented that power-law pressure into the femoral artery as well as the carotid ir
equations describe quantitative morphological and func all animals except mice. In these animals injection wa^
tional characteristics of the kidney, heart, respiratory by way of the left ventricle. The infusion was continued
system, and certain other organs over an approximately maintaining pressure at 100 mmHg, until the plastic
70 X 106-fold variation in body weight of mammals (1, 2, hardened. This took between 30 and 60 min in differen'.
5, 8-10, 19). As this evidence has grown a number of experiments. Following this, the animal was decapitated
intriguing theories of biologic similarity have been for skinned, and the carcass placed in concentrated potas
mulated (6, 14).
sium hydroxide solution (15-33%) for a period varying
A full understanding of hemodynamics is not possible from 18 h to 3 days. At the end of this time most of th
without a knowledge of the dimensions of the vascular tissue had been macerated and the remaining arteria
segments through which blood flows. For example the cast was washed with water until it was free of tissue.
Reynolds number, the Pouiselle-Hagen relation, and
In another group of animals venous casts were pre
pressure gradients are related to the geometry of the pared in a similar manner except that the plastic wa.c
tubular system. Although a number of investigators (16, injected by way of the femoral vein and a .catheter wa
17, 20) have reported quantitative measurements of di
1 Mouse (pentobarbital sodium, 120 mg/kg), rat (pentobarbital so
ameters and lengths of certain vessel segments in one
species, insofar as we are aware no data are available dium, 55 mg/kg), rabbit (pentobarbital sodium, 18" mg/kg; Dial-Ure
0.3 ml solution/kg), dog (morphine, 3 mg/kg; Dial-Urethant
concerning the comparative quantitative geometric pat- thane,
0.125 ml solution/kg; pentobarbital sodium, 7.5 mg/kg), goat (pento
aug^aEi
barbital sodium, 12-20 mg/kg; acepromazine, 0.15 mg/kg), horse
of mammals, large and small, other than that of Clark (chloral hydrate, 85-169 mg/kg), and cow (chloral hydrate, 76-122 mg/
(3) and Gunther (5) regarding the diameter and total kg).
0363-6119/81/0000-0000$01.25 Copyright© 1981 the American Physiological Societ
GEOMETRIC SIMILARITY OF AORTA, VENAE CAVAE, AND BRANCHES
placed in the vena cava near the right atrium by way of
the external jugular vein. The plastic was injected under
a pressure of 100 mmHg for 1 or 2 min until it was seen
to pass from the open end of the catheter in the vena
cava. At this time the pressure was decreased to 25
mmHg and the vena caval catheter occluded. Pressure
was maintained at 25 mmHg until the plastic hardened.
The remainder of the procedure was the same as that
employed in the arterial preparations.
The weight of each animal was recorded in kilograms
prior to an experiment. Twenty -one animals were utilized
in arterial injections; and an additional 14 animals were
used in venous injections. The individual body weights
are presented in Table 1.
■ Venous pressure varies considerably in different por
tions of the mammalian venous system and the state of
collapse of these vessels varies accordingly (4, 7). Early
experiments utilizing injection pressures of 5 mmHg con
sistently produced casts inadequate for the measurement
of vessel dimensions; many vessels were seen to be in
corrected for 1% shrinkage of the plastic that took place
after solidification. In some cases the vessels were slightly,
oval instead of circular. In these cases the average value
of the greatest and least diameters were recorded.
Log-log plots were prepared of the relationship be
tween body weight and diameter of the aorta at various
points throughout its length, diameter of each branch
from the aorta, and the length of aortic segments between
the points where each branch originated. Similar log-log
plots were prepared for the venae cavae and their
branches. The data were transformed to base 10 loga
rithms and the linear regression calculated by the method
of least squares to give the parameters in the power-law
equation
y\saXb
y is any variable
X is mass of body weight in kilograms
Statistical analysis of the logarithmic equations included:
the correlation coefficient (r), 95% confidence limits for
rpnnitn/J
measured.
The casts w
cavae with th
U.-.^
(Z
+
r.
I~
-.
_
J
„
\
1
^1
_j
_l
i
r.
~..^ v^vU..ui.v, i_-t, "iiii.ii nao mu^ii me sdiiie fcigiimcance
for a logarithmic regression line as the standard deviation
for a mean, i.e., two SE limits should include 95% of the
cases. With the log-log analysis, +SE and -SE differ
slightly; the values shown in Tables 1-3 are the mean of
the two absolute values.
Lengths of vessel segments were measured from the
midpoint of a branch to the midpoint of the next branch.
Larger vessels were measured with calipers and smaller
branches with a microscope. These measurements were RESULTS
table 1. Body weight of individual experimental
animals
Arterial Injections
Venous Injections
0.017
0.023
0.024
0.025
0.431
0.431
0.441
0.472
0.415
0.490
0.500
2.40
2.55
3.70
2.50
2.80
2.80
4.30
19.25
25.50
27.70
9.75
15.20
22.70
23.20
32.30
50.90
95.50
63.50
._ , Bovine
480.90
659.00
258.50
Equine
425.00
527.00
471.70
Rabbits
Canine
Table 2 presents the coefficients for the power-law
regression equations, as well as statistical measures for
the relationships to body weight of the' diameters and
lengths of the aorta, superior and inferior vena cava, and
their major branches. The results extend over a body
weight range of more than 38,000-fold for arteries and
1,100-fold for veins. In Table 2 and in the equations given
below the diameters and lengths are in centimeters and
body weight is in kilograms.
Aorta and its branches. The logarithmic relationship
between body weight and the diameter of the ascending
aorta, AID, the length of the ascending aorta to the point
where the brachiocephalic artery comes off, AIL, and the
total length of the aorta AL, are shown in Fig. 1, A-C.
Equations describing these relationships, as well as sim
ilar relationships for the left coronary, LCD, right renal
RRD, and right iliac, RID, arteries are given below
AID = 0.41 BW036 LCD = 0.097 BW036
AL = 16.12 BW032 RRD = 0.169 BW030
AIL = 1.00 BW028 RID = 0.177 BW031
It is of interest to note that whereas heart weight is a
function of BW1, kidney weight is a function of BW085
(5), while the diameters of the left coronary and right
renal arteries are functions of BW036 and BW030, respec
tively.
Venae cavae and their branches. The logarithmic
relationships between body weight and the diameters of
the superior, SVCD, and inferior vena cava, IVCD, where
HOLT, RHODE, HOLT, AND
table 2. Power-law parameters for diameters and lengths of the aorta, venae cavae and
their branches and body weight for a wide variety of mammals (mice to cattle)
Power-Law Coefficients
Power-Law Coefficients
Va r i a b l e s , c m —
Variables, cm
So
SR
s«
Diameter ascending aorta 0.41 0.36 0.99 20 5.8 10.9 0.02
Diameter
Diameter
SVC at
SVC
heartat heart 0. IFtfTlfTTTHTI
Diameter aorta at V* length 0.34 0.36 0.99 15 6.4 10.6 0.02
Diameter
IVC
at heart 0.48 0.41
Diameter
IVC at
heart
0.41 0.95
0.95 14
Diameter aorta at Vi length 0.32 0.33 0.99 15 5.8 9.6 0.02 Diameter
IVC
0.97 0.97
14 12.1
114.8
j 004
Diameter
IVCatat Vi
Vi length 0.83 0.26 0.26
14 12.1
I i4.8lo.64
Diameter aorta at % length 0.25 0.35 0.99 15 11.4 19.0 0.03 Diameter
DiameterIVC
IVC
at Vi
at length
Vi length 0.56 0.3
0.3
iiiii^ iiMim
Diameter aorta at bifurcation 0.25 0.34 0.98 15 12.4 20.6 0.04 Diameter IVC
IVC atat% %
length
length 0.40 0.36
0.95
22.6
0.36
0.9514 14
22.627.8
27.8 0.07
Diameter aorta at R renal 0.26 0.34 0.99 14 9.8 15.3 0.03 Diameter
DiameterIVC
IVC atat bifurcation
bifurcation 0.43 0.330.33
0.96
14 14
18.1
0.96
18.122.2
22.2 0^06
0.06
artery
Diameter
IVC
at
hepatic
v e i n Diameter
0 . 6 0 IVC
0 . at
3 0hepatic
0 . 9vein
5
14
0.30
18.0
0.95 2 14
2 . 18.0
2
22.2
0 0.06
06
Diameter aorta at L renal 0.250.25
0.33
0.33
0.99
0.99141411.4
11.4 18.5
18.5 0.03
0.03 Diameter
DiameterIVC
IVC atat R renal
renal vein
vein 0.61 0.31
0.31 0.99
0.99 14
14 6.8
6.8 8.4
8.4 0.02
14
artery
Diameter
IVC
at
L
renal
v e i n Diameter
0 . 5 6 IVC0at. 3L 0renal0vein
.97
1 4 0.30
1 30.97
.9
1 713.9
. 1 17.1
6 .6.05
05
Diameter L coronary artery 0.10
0.36 0.97
0.97 181820.1
20.1
26.3
Diameter
hepatic
0.92
14 14
20.9
0.10 0.36
26.3
0.030.03Diameter
hepatic
vein vein 0.60 0.260.26
0.92
20.925.7
25.7 o!o?
0.07
Diameter brachiocephalic 0.24
0.24
0.37
0.370.99
0.99 21
121 9.1
9.1 18.3
18.3 0.02
0.02 Diameter
renal
0.920.92
14 23.7
29.2
Diameter RRrenal
veinvein 0.34 0.30 0.30
14 23.7
29.2 o!os
0.08
artery
Diameter
L
renal
vein
0 . 4 6Diameter
0 . 2L 5renal vein
0.92
12
10.25
9 . 40.92 212
2 .19.4
8 22.8
0 .0.08
08
Diameter R renal artery 0.17
0.02 Diameter
iliac
0.950.95
14 21.5
26.4
0.170.30
0.300.99
0.99 15
15 5.1
5.1 8.5 0.02
Diameter RRiliac
veinvein 0.29 0.33 0.33
14 21.5
26.4 o!o7
0.07
Diameter L renal artery 0.15
0.15
0.31
0.310.99
0.99 14
14 9.4
9.4 15.2
15.2 0.03
0.03 Diameter
iliac
0.94 0.94
14 25.6
31.7
Diameter L Liliac
veinvein 0.30 0.37 0.37
14 25.6
31.7o'fJ8
0.08
Diameter R iliac artery 0.18 0.18
0.310.31
0.96
0.96151519.1
19.1 32.0
32.0 0.06
0.06 Length
IVC, heart
heart
0.98 0.98
14 13.2
16.2
14 13.2
16.2o!o-4
0.04
Length IVC,
to to 13.26 0.33 0.33
Diameter L iliac artery 0.160.16
0.33
0.330.98
0.98 15
15 11.7
11.7 19.5
19.50.04
0.04 bifurcation
bifurcation
Diameter intercostal artery 0.050.05
0.36
0.360.98
0.981515 14.3
14.3 23.9
23.9 0.04 Length
heart totohepatic
hepatic 1.70 0.46
0.46 0.99
0.99 14
14 7.6
7.6 9.4
9.4 0.03
Length IVC,
IVC, heart
16.12 0.32
0.32 0.99
0.99 15
Length aorta, valves to 16.12
15 6.6
6.610.9
10.90.02
0.02 vein
vein
bifurcation
Length,
IVC,
heart
to
R
r eLength,
n a l IVC,
6 . 7 heart
5
0to. 3R9renal0 . 9 9
1 0.39
4
90.99
. 4 141 1
. 6 11.60 .0.03
03
9.4
0.28 0.96
0.96 I 21
21 14.6
I 0.04
Length aorta, valves to 1.001.00
0.28
14.629.6
29.6
0.04 vein
vein
b r a c h i o c e p h a l i c a r t e r y L e n g t h I V C , h e a r t t Length
o L rIVC,
e n aheart
l 7 to
. 4L8renal
0 . 3 7 0 . 9 0.37
9 1 30.997 .13
9 7.9
9 . 49.40 .0.03
03
11.68
0.330.33
0.990.99
! 14 !I 5.6
0.020.02 vein
Length aorta, valves to L renal
11.68
14 9.0
5.6 I 9.0
artery
Length aorta, valves to R renal11.18
11.180.34
0.340.99
0.9914145.6
5.69.2
9.2j 0.02
artery
12.821.3
21.3 0.04
0.04
Length aorta, between 0.61 0.61
0.380.38
0.990.99
15 1512.8
intercostal arteries
Arterial measurements were made on mice, rats, rabbits, dogs, goats, horses, and cattle whereas venous measurements included all of these
animals except mice. The value given for the intercostal arteries is the average of 5 pairs. Statistical fit is to the equation, y = a BW*. Body weight
is in kilograms: r, correlation coefficient; n, total number of data points; s„, 95% confidence limits of a in percent; S« mean ± SE of the estimate
in percent; sb 95% confidence limits of b in slope units. SVC, superior vena cava; IVC, inferior vena cava; R, right; L, left.
they enter the heart, and the length of the inferior vena
cava from the heart to the bifurcation, IVCl, are shown
in Fig. 2, A-C. Equations describing similar relationships
for the diameters of the right renal, RRVD, right iliac,
RIVD, and hepatic Hd, veins are given below.
SVCD= 0.46 BW°U RRVD
RRVd= =0.34
0.34 BW030
IVCd = 0.48 BW041
RIVD =0.29BW0-33
IVCl = 13.26 BW033
Hd = 0.60 BW026
As shown in Table 2, the scatter of the data for the
venous system was somewhat greater than that for the
aorta, the correlation coefficient being greater than 0.92
and the standard estimate of the error less than 38%.
Similar relationships for the diameters of the inferior
vena cava at various points throughout its length, the left
iliac and left renal veins, as well as the lengths of various
segments of the venae cavae are shown in Table 2.
DISCUSSION
■■■
■;
The scatter of the data, as shown in Table 2, was
smaller in the arterial than in the venous system. Al
though the reason for this difference is not known it may
be related to the fact that the small injection pressure of
25 mmHg in the venous system, as compared to 100
rnrnHg in the arterial system, led to greater variation in
the diameters in the venous segments. This view is sup
ported by the fact that in preliminary experiments in
which the venous system was injected with a pressure of
only 10 mmHg there was more variation in the diameter
of the venous segments. Whereas, during life the arterial
system is always distended with a relatively high pres
sure, the pressure distending the veins varies consider
ably from place to place and is affected to a greater
degree by changes in body position. As, for example, in
the vertical position the pressure in the iliac veins is
higher than that in the superior vena cava, which may
be in a partially collapsed state (7). Thus, venous meas
urements reported here do not represent the condition in
any particular physiological state, instead they represent
the maximum capacity of distension of these vessels at a
distending pressure approaching 25 mmHg.
Although the arterial injection pressure was 100 mmHg
the pressure distending the arterial tree at the time of
hardening of the plastic was much less. Evidence for this
was obtained in several experiments in which pressure
was measured in the aorta throughout the plastic injec
tion period. At the beginning ofthe injection, the pressure
in the aorta was approximately 100 mmHg but after a
few minutes it fell to between 35 and 75 mmHg. Thus,
the diameters and lengths of the arteries reported are for
slightly distended vessels and not for vessels in the phys
iological state distended with 100 mmHg pressure. This
is confirmed by the fact that the diameter of the ascend
ing aorta calculated by the equation
D = 0.41 BW°36
GEOMETRIC SIMILARITY OF AORTA, VENAE CAVAE, AND BRANCHES
in vessel diameters in the living dog, as compared to the
values calculated from the equations in Table 2, are
shown in Table 3. It will be noted that the ascending and
descending thoracic aorta and their branches when dis
tended increase their diameters to a considerably greater
degree than the abdominal aorta and its branches.
The interrelationship of hemodynamic phenomena
and vascular segment geometry is fundamental. As an
example, cardiac output which is proportional to BW0,79
(10) and cross-sectional area ofthe ascending aorta (pro
portional to BW0-72) determine that the mean velocity of
blood flow in the ascending aorta is proportional to
BW00'. Thepower 0.07 is almost equal to zero thus the
term, BW00', closely approximates unity. The mean ve-
D-0.41 BW
L- 16.1 BW
,0.51 D-0.46BW
m MOUSE
♦ R AT
O RABBIT
-
/
*
A 0OG
O G O AT
-
□ HORSE
■ COW
<r
l-i.o bw0-28 A£ >er q"
0.48 BW
- IO'0'
•
1
1
1
i
BODY WEIGHT (Kg)
FIG. 1. Logarithmic relationships between body weight and diame
ter of ascending aorta, length of ascending aorta to point where bra
chiocephalic artery comes off, and total length of aorta in 7 species of
normal adult mammals extending over a 38,000-fold range of body
weight (mice to cattle). Asc, ascending; D, diameter, L, length.
gives values in general agreement with autopsy values
reported by Clark (3) for a wide variety of mammals
(mouse to whale).
If it is assumed, as a first approximation, that the
aegree of vessel distension of the living dog is represent
ative of that for mammals in general, the diameter values
§£en by the equations reported here for the aorta and
BOOY WEIGHT (Kg)
l«e diameters in the anesthetized dog with distending £*"£ •«' Y' , ° T cava-lvtJ> Md ien&* oi
Pressures
«h„c;„l«„;„Qi
™„™
tu percent
" g ,nfenor
vena
cava
m 6 weight
sPecies
a 1,100-fold
^ures in in
theth«
physiological
range.
The
increase
range
in body
(ratoftomammals
cattle). D, extending
diameter, L,over
length
HOLT, RHODE, HOLT, AND KINES
': :
table 3. Percent increase in vessel diameter with
pressures in the physiological range
Mean Pres
sure, mmHg
Diameter In
crease 100 D„/
(D = a BW4),
%
Ascending aorta
Descending thoracic aorta upper Mi
Descending thoracic aorta middle Vh
Descending thoracic aorta lower V3
Abdominal aorta upper Vb
Abdominal aorta lower Vfi
External iliac artery
Renal artery
Brachiocephalic artery
Intercostal arterv
118
162
108
148
108
140
108
140
97
110
97
119
93
102
97
97
118
141
109
127
Do, diameter of vessels in a 22.1-kg living dog when distended with
pressures shown, as reported by Patel et al. (17). (D = a BW*) is
diameter of vessel calculated by equations from data in Table 2. See
text for discussion.
locity is nearly the same in the control state of large and
small mammals. It has been proposed by others (13, 18)
that cardiac output is proportional to body surface area
(BW067), to BW1-0, or to an intermediate value, the
present value of BW079 is based on measurement of
cardiac output in mammals varying 1,790-fold in body
weight, from rat to horse.
Quantitative relations of vascular similarity have been
demonstrated based on data for normal adult mammals
varying as much as 38,000-fold in body weight. The
diameters and lengths of vessel segments are described
by power-law equations relating their diameters and
lengths to body weight.
The authors thank W. Powell, M. R. Bledsoe, P. Bewley, J. P. Holt,
Jr., T, Peterson, and M. Max for technical assistance, and K. Shotts
and J. Hart for assistance in preparing the programs for the computer.
This work was conducted at the Heart Research Laboratory, Uni
versity of Louisville School of Medicine, Louisville, KY 40202, and
School of Veterinary Medicine, University of California Davis CA
95616.
This investigation was supported in part by Grants HE-5622 and
2075 from the National Heart and Lung Institute and the Kentucky,
Louisville, and Jefferson County Health Associations.
Received 22 August 1980; accepted in final form 17 January 1981.
REFERENCES
1. Aoolph, E. F. Quantitative relations in the physiological constitu
tion of mammals. Science 109: 579-585, 1949.
2. Brody, S. Bioenergetics and Growth. New York: Hafner 1974 p
352-398, 575-642.
3. Clark, A. J. Comparative Physiology ofthe Heart. London: Cam
bridge, 1927, p. 115, 149-153.
4. Duomarco, J. L., R. Rimini, C. E. Giambruno, R. Seoane, and
M. Haendel. Systemic venous collapse in man. Am. J. Cardiol.
11:357-361, 1963.
5. Gunther, B. On theories of biological similarity. Fortschr. Exp.
Theor. Biophys. 19: 33-85, 1975.
6. Gunther, B., and B. Leon De La Barra. A unified theory of
biological similarities. J. Theor. Biol. 13: 48-59, 1966.
7. Holt, J. P. Flow through collapsible tubes and through in situ
veins. IEEE Trans. Bio-Med. Eng. 16: 274-283, 1969.
8. Holt, J. P., H. Kines, and E. A. Rhode. Pattern of function of
left ventricle of mammals. Am. J. Physiol. 209: 22-32, 1965.
9. Holt, J. P., and E. A. Rhode. Similarity of renal glomerular
hemodynamics in mammals. Am. Heart J. 92: 465-472, 1976.
10. Holt, J. P., E. A. Rhode, and H. Kines. Ventricular volumes and
body weight in mammals. Am. J. Physiol. 215: 704-715, 1968.
11. Holt, W. W., E. A. Rhode, and J. P. Holt. Geometric similarity
in the vascular system (Abstract). Federation Proc. 37: 823, 1978.
12. Huxley, J S. Problems of Relative Growth. New York: Dover
1972, p.267-302.
13. Iberall, A. S. Blood flow and oxygen uptake in mammals. Ann.
Biomed. Eng. 1: 1-8, 1972.
14. Iberall, A. S. Growth, form, and function in mammals. Ann. NY
Acad. Sci. 231: 77-84, 1974.
15. Johnstone, R. E., and M. W. Thring. Pilot Plants, Models and
Scale-Up Methods of Chemical Engineering. New York: McGraw,
1957, p. 12-42, 74-97.
16. Mall, F. P. Die Blut und Lymphwege im Dunndarm des Hundes.
Koenig. Saechs. Ges. Wiss., Abh. Math. Phys. Kl. 14: 153-189,
1887.
17. Patel, D., F. De Freitas, J. Greenfield, and D. Fry. Relation
ship of radius to pressure along the aorta in living dogs. J. Appl.
Physiol. 18: 1111-1117, 1963.
18. Patterson, J. L., Jr., R. H. Goetz, J. T. Doyle, J. V. Warren,
O. H. Gauer, D. K. Detweiler, S. I. Said. H. Hoernicke, M.
McGregor, E. N. Keen, M. H. Smith, Jr., E. L. Hardie, M.
Reynolds, W. P. Flatt, and D. R. Waldo. Cardiorespiratory
dynamics in the ox and giraffe, with comparative observations on
man and other mammals. Ann. NY Acad. Sci. 127: 393-418, 1965.
19. Stahl, W. R. Scaling of respiratorv variables in mammals. J. Appl.
Physiol. 22: 453-460, 1967.
20. Suwa, N., T. Niwa, H. Fukasawa, and Y. Sasaki. Estimation of
intravascular blood pressure gradient by mathematical analysis of
arterial casts. Tohoku J. Exp. Med. 79: 168-198, 1963.
21. Thompson, D. W. On Growth and Form. London: Cambridge,
1952, vol. 1, p. 22-77.