Geometric similarity of aorta, venae cavae, andcertain of their branches in mammals

J. P. HOLT, E. A. RHODE, W. W. HOLT, AND H. KINESDepartment of Heart Research, University of Louisville School of Medicine Health SciencesCenter, Louisville, Kentucky 40202; and Department of Medicine, School of VeterinaryMedicine, 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 weGeometric similarity of aorta, venae cavae, and certain of presented evidence that allometric equations apply totheir branches in mammals. Am. J. Physiol. 241 (Regulatory the geometric and functional characteristics ofthe mam-Integrative Comp. Physiol. 10): R100-R104, 1981.—The diam- malian heart (8,10). The present studies were undertakeneters of the aorta and venae cavae at various points throughout to determine whether the geometry of these vessels cantheir lengths, the diameters of their major branches, and the be described by p0wer-law equations relating diameterlengths of various aortic and vena caval segments were meas- and { h bod ^. ^ (n)ured in plastic corrosion casts of the artenai and venous systems ° joof the normal adult mouse, rat, rabbit, dog, goat, horse, andcow, extending over a body weight range of 38,000-fold (arterial) METHODSand 1,100-fold (venous). It is shown that the diameters and t>i„„*:„ „«™„„;«», ™,^ ~e ™,-™oi ««4«1t ™,-o« r-nfc- *«u• . « • * » 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 t s o f n o r m a l a d u l t m i c e , r a t s , r a b -l e n g t h s o f t h e s e v e s s e l s a r e d e s c r i b e d b v p o w e r - l a w e q u a t i o n s . . , , , ,relating the particular diameter or length to body weight (BW) £lte, dogs goats, horses, and cows were prepared asraised to a particular power, i.e., diameter = a BW6. Equations follows: after attaining a deep anesthetic level, cannu-for the diameters and lengths of the vessels are given for slightly lation of the carotid and femoral arteries and exposure todistended vessels and for vessels distended in the physiological severe hemorrhage, the animals were killed by injectingr a n g e . e i t h e r a l a r g e d o s e o f p e n t o b a r b i t a l s o d i u m o r c o n c e n -

arteries; veins; mouse; rat; rabbit; dog; goat; cow; horse; similarity

following Thompson's discussion (21) of the effectsof scale in biology, Huxley (12) employed allometric orpower-law equations for somatic form analysis, and morerecently several investigators have described power-lawequations relating various physiological variables to bodyweight (5). Evidence has been presented that power-lawequations describe quantitative morphological and functional characteristics of the kidney, heart, respiratorysystem, and certain other organs over an approximately70 X 106-fold variation in body weight of mammals (1, 2,5, 8-10, 19). As this evidence has grown a number ofintriguing theories of biologic similarity have been formulated (6, 14).

A full understanding of hemodynamics is not possiblewithout a knowledge of the dimensions of the vascularsegments through which blood flows. For example theReynolds number, the Pouiselle-Hagen relation, andpressure gradients are related to the geometry of thetubular system. Although a number of investigators (16,17, 20) have reported quantitative measurements of diameters and lengths of certain vessel segments in onespecies, insofar as we are aware no data are availableconcerning the comparative quantitative geometric pat-

a u g ^ a E iof mammals, large and small, other than that of Clark(3) and Gunther (5) regarding the diameter and total

METHODS

Plastic corrosion casts of normal adult mice, rats, rabbits, dogs, goats, horses, and cows were prepared asfollows: after attaining a deep anesthetic level,1 cannu-lation of the carotid and femoral arteries and exposure tosevere hemorrhage, the animals were killed by injectingeither a large dose of pentobarbital sodium or concentrated KCL solution into the arch of the aorta. This wasfollowed by perfusion of the arterial system with physiological saline by way of the carotid artery for 2 min.then bleeding from the femoral artery for 2 min. BatsonVCompound (Polysciences, Paul Valley Industrial ParkWarrington, PA 18976) was then quickly mixed andinjected under 100 mmHg pressure into the carotid artery. Bleeding was continued from the femoral artenuntil plastic was seen to pass from the cannula at whichtime the plastic solution was injected under 100 mmHgpressure into the femoral artery as well as the carotid irall animals except mice. In these animals injection wa^by way of the left ventricle. The infusion was continuedmaintaining pressure at 100 mmHg, until the plastichardened. This took between 30 and 60 min in differen'.experiments. Following this, the animal was decapitatedskinned, and the carcass placed in concentrated potassium hydroxide solution (15-33%) for a period varyingfrom 18 h to 3 days. At the end of this time most of thtissue had been macerated and the remaining arteriacast was washed with water until it was free of tissue.

In another group of animals venous casts were prepared in a similar manner except that the plastic wa.cinjected by way of the femoral vein and a .catheter wa

1 Mouse (pentobarbital sodium, 120 mg/kg), rat (pentobarbital sodium, 55 mg/kg), rabbit (pentobarbital sodium, 18" mg/kg; Dial-Urethane, 0.3 ml solution/kg), dog (morphine, 3 mg/kg; Dial-Urethant0.125 ml solution/kg; pentobarbital sodium, 7.5 mg/kg), goat (pentobarbital sodium, 12-20 mg/kg; acepromazine, 0.15 mg/kg), horse(chloral hydrate, 85-169 mg/kg), and cow (chloral hydrate, 76-122 mg/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 ofthe external jugular vein. The plastic was injected undera pressure of 100 mmHg for 1 or 2 min until it was seento pass from the open end of the catheter in the venacava. At this time the pressure was decreased to 25mmHg and the vena caval catheter occluded. Pressurewas maintained at 25 mmHg until the plastic hardened.The remainder of the procedure was the same as thatemployed in the arterial preparations.

The weight of each animal was recorded in kilogramsprior to an experiment. Twenty -one animals were utilizedin arterial injections; and an additional 14 animals wereused in venous injections. The individual body weightsare presented in Table 1.■ Venous pressure varies considerably in different portions of the mammalian venous system and the state ofcollapse of these vessels varies accordingly (4, 7). Earlyexperiments utilizing injection pressures of 5 mmHg consistently produced casts inadequate for the measurementof vessel dimensions; many vessels were seen to be in

measured.The casts w

cavae with th

Lengths of vessel segments were measured from themidpoint of a branch to the midpoint of the next branch.Larger vessels were measured with calipers and smallerbranches with a microscope. These measurements were

table 1. Body weight of individual experimentalanimals

Rabbits

Canine

._ , Bovine

Equine

Arterial Injections Venous Injections0.0170.0230.0240.025

0.4310.4310.4410.472

0.4150.4900.500

2.402.553.70

2.502.802.804.30

19.2525.5027.70

9.7515.2022.7023.20

32.3050.9095.50

63.50

480.90659.00

258.50

425.00527.00

471.70

corrected for 1% shrinkage of the plastic that took placeafter solidification. In some cases the vessels were slightly,oval instead of circular. In these cases the average valueof the greatest and least diameters were recorded.

Log-log plots were prepared of the relationship between body weight and diameter of the aorta at variouspoints throughout its length, diameter of each branchfrom the aorta, and the length of aortic segments betweenthe points where each branch originated. Similar log-logplots were prepared for the venae cavae and theirbranches. The data were transformed to base 10 logarithms and the linear regression calculated by the methodof least squares to give the parameters in the power-lawequation

y\saXby is any variableX is mass of body weight in kilograms

Statistical analysis of the logarithmic equations included:the correlation coefficient (r), 95% confidence limits forr p n n i t n / J U . - . ^ ( Z + r . I ~ - . _ J „ \ 1 ^ 1 _ j _ l i r .

~..^ v^vU..ui.v, i_-t, "iiii.ii nao mu^ii me sdiiie fcigiimcancefor a logarithmic regression line as the standard deviationfor a mean, i.e., two SE limits should include 95% of thecases. With the log-log analysis, +SE and -SE differslightly; the values shown in Tables 1-3 are the mean ofthe two absolute values.

RESULTS

Table 2 presents the coefficients for the power-lawregression equations, as well as statistical measures forthe relationships to body weight of the' diameters andlengths of the aorta, superior and inferior vena cava, andtheir major branches. The results extend over a bodyweight range of more than 38,000-fold for arteries and1,100-fold for veins. In Table 2 and in the equations givenbelow the diameters and lengths are in centimeters andbody weight is in kilograms.

Aorta and its branches. The logarithmic relationshipbetween body weight and the diameter of the ascendingaorta, AID, the length of the ascending aorta to the pointwhere the brachiocephalic artery comes off, AIL, and thetotal length of the aorta AL, are shown in Fig. 1, A-C.Equations describing these relationships, as well as similar relationships for the left coronary, LCD, right renalRRD, and right iliac, RID, arteries are given below

AID = 0.41 BW036 LCD = 0.097 BW036AL = 16.12 BW032 RRD = 0.169 BW030AIL = 1.00 BW028 RID = 0.177 BW031

It is of interest to note that whereas heart weight is afunction of BW1, kidney weight is a function of BW085(5), while the diameters of the left coronary and rightrenal arteries are functions of BW036 and BW030, respectively.

Venae cavae and their branches. The logarithmicrelationships between body weight and the diameters ofthe 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 andtheir branches and body weight for a wide variety of mammals (mice to cattle)

Power-Law Coefficients Power-Law Coefficients

S o S R s «

0.25 0.33 0.99 14 11.4 18.5 0.03

0.10 0.36 0.97 18 20.1 26.3 0.030.24 0.37 0.99 121 9.1 18.3 0.02

0.17 0.30 0.99 15 5.1 8.5 0.020.15 0.31 0.99 14 9.4 15.2 0.030.18 0.31 0.96 15 19.1 32.0 0.060.16 0.33 0.98 15 11.7 19.5 0.040.05 0.36 0.98 15 14.3 23.9 0.04

16.12 0.32 0.99 15 6.6 10.9 0.02

1.00 0.28 0.96 I 21 14.6 29.6 I 0.04

11.68 0.33 0.99 ! 14 I 5.6 9.0 I 0.02

11.18 0.34 0.99 14 5.6 9.2 j 0.02

0.61 0.38 0.99 15 12.8 21.3 0.04

IFtfTlfTTTHTI0.41 0.950.26 0.970.30.36 0.950.33 0.960.30 0.950.31 0.990.30 0.970.26 0.920.30 0.920.25 0.920.33 0.950.37 0.940.33 0.98

V a r i a b l e s , c m —

Diameter ascending aorta 0.41 0.36 0.99 20 5.8 10.9 0.02 Diameter SVC at heart 0.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 0.95 14Diameter aorta at Vi length 0.32 0.33 0.99 15 5.8 9.6 0.02 Diameter IVC at Vi length 0.83 0.26 0.97 14 12.1 114.8 j 004Diameter aorta at % length 0.25 0.35 0.99 15 11.4 19.0 0.03 Diameter IVC at Vi length 0.56 0.3Diameter aorta at bifurcation 0.25 0.34 0.98 15 12.4 20.6 0.04 Diameter IVC at % length 0.40 0.36 0.95 14 22.6 27.8 0.07Diameter aorta at R renal 0.26 0.34 0.99 14 9.8 15.3 0.03 Diameter IVC at bifurcation 0.43 0.33 0.96 14 18.1 22.2 0^06

a r t e r y D i a m e t e r I V C a t h e p a t i c v e i n 0 . 6 0 0 . 3 0 0 . 9 5 1 4 1 8 . 0 2 2 . 2 0 0 6Diameter aorta at L renal 0.25 0.33 0.99 14 11.4 18.5 0.03 Diameter IVC at R renal vein 0.61 0.31 0.99 14 6.8 8.4 0.02

a r t e r y D i a m e t e r I V C a t L r e n a l v e i n 0 . 5 6 0 . 3 0 0 . 9 7 1 4 1 3 . 9 1 7 . 1 6 . 0 5Diameter L coronary artery 0.10 0.36 0.97 18 20.1 26.3 0.03 Diameter hepatic vein 0.60 0.26 0.92 14 20.9 25.7 o!o?Diameter brachiocephalic 0.24 0.37 0.99 21 9.1 18.3 0.02 Diameter R renal vein 0.34 0.30 0.92 14 23.7 29.2 o!os

a r t e r y D i a m e t e r L r e n a l v e i n 0 . 4 6 0 . 2 5 0 . 9 2 1 2 1 9 . 4 2 2 . 8 0 . 0 8Diameter R renal artery 0.17 0.30 0.99 15 5.1 8.5 0.02 Diameter R iliac vein 0.29 0.33 0.95 14 21.5 26.4 o!o7Diameter L renal artery 0.15 0.31 0.99 14 9.4 15.2 0.03 Diameter L iliac vein 0.30 0.37 0.94 14 25.6 31.7 o'fJ8Diameter R iliac artery 0.18 0.31 0.96 15 19.1 32.0 0.06 Length IVC, heart to 13.26 0.33 0.98 14 13.2 16.2 o!o-4Diameter L il iac artery 0.16 0.33 0.98 15 11.7 19.5 0.04 bifurcationDiameter intercostal artery 0.05 0.36 0.98 15 14.3 23.9 0.04 Length IVC, heart to hepatic 1.70 0.46 0.99 14 7.6 9.4 0.03Length aorta, valves to 16.12 0.32 0.99 15 6.6 10.9 0.02 vein

b i f u r c a t i o n L e n g t h , I V C , h e a r t t o R r e n a l 6 . 7 5 0 . 3 9 0 . 9 9 1 4 9 . 4 1 1 . 6 0 . 0 3Length aorta, valves to 1.00 0.28 0.96 21 14.6 29.6 0.04 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 o L r e n a l 7 . 4 8 0 . 3 7 0 . 9 9 1 3 7 . 9 9 . 4 0 . 0 3Length aorta, valves to L renal 11.68 0.33 0.99 ! 14 5.6 9.0 0.02 vein

arteryLength aorta, valves to R renal 11.18 0.34 0.99 14 5.6 9.2 0.02

arteryLength aorta, between 0.61 0.38 0.99 15 12.8 21.3 0.04

intercostal arteriesArterial 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 weightis in kilograms: r, correlation coefficient; n, total number of data points; s„, 95% confidence limits of a in percent; S« mean ± SE of the estimatein percent; sb 95% confidence limits of b in slope units. SVC, superior vena cava; IVC, inferior vena cava; R, right; L, left.

Variables, cm

Diameter SVC at heartDiameter IVC at heartDiameter IVC at Vi lengthDiameter IVC at Vi lengthDiameter IVC at % lengthDiameter IVC at bifurcationDiameter IVC at hepatic veinDiameter IVC at R renal veinDiameter IVC at L renal veinDiameter hepatic veinDiameter R renal veinDiameter L renal veinDiameter R iliac veinDiameter L iliac veinLength IVC, heart to

bifurcationLength IVC, heart to hepatic

veinLength, IVC, heart to R renal

veinLength IVC, heart to L renal

14 12.1 I i4.8lo.64i i i i i ^ i i M i m

14 22.6 27.8 0.0714 18.1 22.2 0.0614 18.0 22.2 0.0614 6.8 8.4 0.0214 13.9 17.1 6.0514 20.9 25.7 0.0714 23.7 29.2 0.0812 19.4 22.8 0.0814 21.5 26.4 0.0714 25.6 31.7 0.0814 13.2 16.2 0.04

0.46 0.99 14 7.6 9.4 0.03

0.39 0.99 14 9.4 11.6 0.03

0.37 0.99 13 7.9 9.4 0.03

■■■;■

they enter the heart, and the length of the inferior venacava from the heart to the bifurcation, IVCl, are shownin Fig. 2, A-C. Equations describing similar relationshipsfor the diameters of the right renal, RRVD, right iliac,RIVD, and hepatic Hd, veins are given below.

SVCD= 0.46 BW°U RRVD = 0.34 BW030IVCd = 0.48 BW041IVCl = 13.26 BW033

RRVd = 0.34 BW030RIVD =0.29BW0-33Hd = 0.60 BW026

As shown in Table 2, the scatter of the data for thevenous system was somewhat greater than that for theaorta, the correlation coefficient being greater than 0.92and the standard estimate of the error less than 38%.Similar relationships for the diameters of the inferiorvena cava at various points throughout its length, the leftiliac and left renal veins, as well as the lengths of varioussegments of the venae cavae are shown in Table 2.

DISCUSSION

The scatter of the data, as shown in Table 2, wassmaller in the arterial than in the venous system. Although the reason for this difference is not known it maybe related to the fact that the small injection pressure of25 mmHg in the venous system, as compared to 100rnrnHg in the arterial system, led to greater variation inthe diameters in the venous segments. This view is supported by the fact that in preliminary experiments in

which the venous system was injected with a pressure ofonly 10 mmHg there was more variation in the diameterof the venous segments. Whereas, during life the arterialsystem is always distended with a relatively high pressure, the pressure distending the veins varies considerably from place to place and is affected to a greaterdegree by changes in body position. As, for example, inthe vertical position the pressure in the iliac veins ishigher than that in the superior vena cava, which maybe in a partially collapsed state (7). Thus, venous measurements reported here do not represent the condition inany particular physiological state, instead they representthe maximum capacity of distension of these vessels at adistending pressure approaching 25 mmHg.

Although the arterial injection pressure was 100 mmHgthe pressure distending the arterial tree at the time ofhardening of the plastic was much less. Evidence for thiswas obtained in several experiments in which pressurewas measured in the aorta throughout the plastic injection period. At the beginning ofthe injection, the pressurein the aorta was approximately 100 mmHg but after afew minutes it fell to between 35 and 75 mmHg. Thus,the diameters and lengths of the arteries reported are forslightly distended vessels and not for vessels in the physiological state distended with 100 mmHg pressure. Thisis confirmed by the fact that the diameter of the ascending aorta calculated by the equation

D = 0.41 BW°36

GEOMETRIC SIMILARITY OF AORTA, VENAE CAVAE, AND BRANCHES

D-0.41 BW

in vessel diameters in the living dog, as compared to thevalues calculated from the equations in Table 2, areshown in Table 3. It will be noted that the ascending anddescending thoracic aorta and their branches when distended increase their diameters to a considerably greaterdegree than the abdominal aorta and its branches.

The interrelationship of hemodynamic phenomenaand vascular segment geometry is fundamental. As anexample, cardiac output which is proportional to BW0,79(10) and cross-sectional area ofthe ascending aorta (proportional to BW0-72) determine that the mean velocity ofblood flow in the ascending aorta is proportional toBW00'. Thepower 0.07 is almost equal to zero thus theterm, BW00', closely approximates unity. The mean ve-

L- 16.1 BW

, 0 . 5 1 D - 0 . 4 6 B W

- / *

m MOUSE♦ RATO RABBITA 0 O GO G O AT□ H O R S E■ C O W

-

<rl-i.o bw0-28 A£ >er q"

•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 brachiocephalic artery comes off, and total length of aorta in 7 species ofnormal adult mammals extending over a 38,000-fold range of bodyweight (mice to cattle). Asc, ascending; D, diameter, L, length.

gives values in general agreement with autopsy valuesreported by Clark (3) for a wide variety of mammals(mouse to whale).

If it is assumed, as a first approximation, that theaegree of vessel distension of the living dog is representative of that for mammals in general, the diameter values§£en by the equations reported here for the aorta and

- I O ' 0 '

0.48 BW

BOOY WEIGHT (Kg)

l«e d iameters in the anesthet ized dog wi th d is tending £*"£ •« ' Y ' , ° T cava- lv tJ> Md ien&* o iPressures in th« «h„c;„l«„;„Qi ™„™ tu " g ,nfenor vena cava m 6 sPecies of mammals extending over a 1,100-fold^ures in the physiological range. The percent increase range in body weight (rat to cattle). D, diameter, L, length

': :table 3. Percent increase in vessel diameter withpressures in the physiological range

Diameter InMean Pres crease 100 D„/sure, mmHg (D = a BW4),

%Ascending aorta 118 162Descending thoracic aorta upper Mi 108 148Descending thoracic aorta middle Vh 108 140Descending thoracic aorta lower V3 108 140Abdominal aorta upper Vb 97 110Abdominal aorta lower Vfi 97 119External iliac artery 93 102Renal artery 97 97Brachiocephalic artery 118 141Intercostal arterv 109 127

Do, diameter of vessels in a 22.1-kg living dog when distended withpressures shown, as reported by Patel et al. (17). (D = a BW*) isdiameter of vessel calculated by equations from data in Table 2. Seetext for discussion.

locity is nearly the same in the control state of large andsmall mammals. It has been proposed by others (13, 18)that cardiac output is proportional to body surface area

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HOLT, RHODE, HOLT, AND KINES

(BW067), to BW1-0, or to an intermediate value, thepresent value of BW079 is based on measurement ofcardiac output in mammals varying 1,790-fold in bodyweight, from rat to horse.

Quantitative relations of vascular similarity have beendemonstrated based on data for normal adult mammalsvarying as much as 38,000-fold in body weight. Thediameters and lengths of vessel segments are describedby power-law equations relating their diameters andlengths 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. Shottsand J. Hart for assistance in preparing the programs for the computer.

This work was conducted at the Heart Research Laboratory, University of Louisville School of Medicine, Louisville, KY 40202, andSchool of Veterinary Medicine, University of California Davis CA95616.

This investigation was supported in part by Grants HE-5622 and2075 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.

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