Geometric Similarity of Aorta, Venae Cavae, and Certain of Their Branches in Mammals

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 mam- Integrative 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 cow, extending over a body weight range of 38,000-fold (arterial) METHODSMETHODS •and .«•*» 1,100-fold l j(venous). u j u It i is Plastic shown that corrosion the diameters and casts t>i„„*:„Plastic of „«™„„;«», corrosion normal ™,^ casts ~e adult ™,-™oiof normal mice, ««4«1t adult ™,-o« mice, rats, r-nfc-rats, rab- rab *«u lengths of these vessels are described bv power-law equations . . , , , , relating the particular diameter or length to body weight (BW)bits, £lte, dogs, dogs goats, goats, horses, horses, and and cows cows werewere preparedprepared as raised to a particular power, i.e., diameter = a BW6. Equationsfollows: follows: after after attaining attaining a deepa deep anesthetic anesthetic level,1 level, cannu- for the diameters and lengths of the vessels are given for slightlylation lation of ofthe the carotid carotid and and femoral femoral arteries arteries and and exposureexposure to distended vessels and for vessels distended in the physiologicalsevere severe hemorrhage, hemorrhage, the the animals animals were were killed killed byby injectinginjecting range. either a large dose of pentobarbitaleither a large dose of pentobarbital sodium sodium or concen- or concen trated 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 ameters and lengths of certain vessel segments in one 1 Mouse (pentobarbital sodium, 120 mg/kg), rat (pentobarbital so species, insofar as we are aware no data are available dium, 55 mg/kg), rabbit (pentobarbital sodium, 18" mg/kg; Dial-Ure thane, 0.3 ml solution/kg), dog (morphine, 3 mg/kg; Dial-Urethant concerning the comparative quantitative geometric pat- 0.125 ml solution/kg; pentobarbital sodium, 7.5 mg/kg), goat (pento augâEi 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 corrected for 1% shrinkage of the plastic that took place the external jugular vein. The plastic was injected under after solidification. In some cases the vessels were slightly, a pressure of 100 mmHg for 1 or 2 min until it was seen oval instead of circular. In these cases the average value to pass from the open end of the catheter in the vena of the greatest and least diameters were recorded. cava. At this time the pressure was decreased to 25 Log-log plots were prepared of the relationship be mmHg and the vena caval catheter occluded. Pressure tween body weight and diameter of the aorta at various was maintained at 25 mmHg until the plastic hardened. points throughout its length, diameter of each branch The remainder of the procedure was the same as that from the aorta, and the length of aortic segments between employed in the arterial preparations. the points where each branch originated. Similar log-log The weight of each animal was recorded in kilograms plots were prepared for the venae cavae and their prior to an experiment. Twenty -one animals were utilized branches. The data were transformed to base 10 loga in arterial injections; and an additional 14 animals were rithms and the linear regression calculated by the method used in venous injections. The individual body weights of least squares to give the parameters in the power-law are presented in Table 1. ■ Venous pressure varies considerably in different por equation tions of the mammalian venous system and the state of y\saXb collapse of these vessels varies accordingly (4, 7). Early y is any variable experiments utilizing injection pressures of 5 mmHg con sistently produced casts inadequate for the measurement X is mass of body weight in kilograms of vessel dimensions; many vessels were seen to be in Statistical analysis of the logarithmic equations included: the correlation coefficient (r), 95% confidence limits for r p n n i t n / J U . - . ^ ( Z + r . I ~ - . _ J „ \ 1 ^ 1 _ j _ l i r . measured. The casts w ~..^ v^vU..ui.v, i_-t, "iiii.ii nao muîi me sdiiie fcigiimcance for a logarithmic regression line as the standard deviation cavae with th 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 Lengths of vessel segments were measured from the the two absolute values. 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 Table 2 presents the coefficients for the power-law animals 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 Arterial Injections Venous Injections their major branches. The results extend over a body 0.017 weight range of more than 38,000-fold for arteries and 0.023 0.024 1,100-fold for veins. In Table 2 and in the equations given 0.025 below the diameters and lengths are in centimeters and body weight is in kilograms. 0.431 0.415 Aorta and its branches. The logarithmic relationship 0.431 0.490 between body weight and the diameter of the ascending 0.441 0.500 0.472 aorta, AID, the length of the ascending aorta to the point where the brachiocephalic artery comes off, AIL, and the Rabbits 2.40 2.50 total length of the aorta AL, are shown in Fig. 1, A-C. 2.55 2.80 Equations describing these relationships, as well as sim 3.70 2.80 ilar relationships for the left coronary, LCD, right renal 4.30 RRD, and right iliac, RID, arteries are given below Canine 19.25 9.75 25.50 15.20 AID = 0.41 BW036 LCD = 0.097 BW036 27.70 22.70 AL = 16.12 BW032 RRD = 0.169 BW030 23.20 AIL = 1.00 BW028 RID = 0.177 BW031 32.30 63.50 50.90 It is of interest to note that whereas heart weight is a 95.50 function of BW1, kidney weight is a function of BW085 ._ , Bovine 480.90 258.50 (5), while the diameters of the left coronary and right 659.00 renal arteries are functions of BW036 and BW030, respec Equine tively. 425.00 471.70 Venae cavae and their branches. The logarithmic 527.00 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 V a 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 0.02Diameter Diameter SVC atSVC heart at heart 0. IFtfTlfTTTHTI Diameter aorta at V* length 0.34 0.36 0.99 15 6.4 10.6 0.02Diameter Diameter IVC at IVC heart at heart 0.48 0.410.41 0.95 14 Diameter aorta at Vi length 0.32 0.33 0.99 15 5.8 9.6 0.02 DiameterDiameter IVC IVC at at ViVi length 0.83 0.26 0.26 0.97 0.97 14 12.114 12.1 114.8 I i4.8lo.64 j 004 Diameter aorta at % length 0.25 0.35 0.99 15 11.4 19.0 0.03 Diameter Diameter IVC IVC at Vi at length Vi length 0.56 0.30.3 iiiii^ iiMim Diameter aorta at bifurcation 0.25 0.34 0.98 15 12.4 20.6 0.04 Diameter IVC IVC at at % %length length 0.40 0.360.36 0.95 0.95 14 14 22.6 22.6 27.8 27.8 0.07 Diameter aorta at R renal 0.26 0.34 0.99 14 9.8 15.3 0.03 DiameterDiameter IVC IVC at at bifurcation bifurcation 0.43 0.330.33 0.96 0.96 14 14 18.1 18.1 22.2 22.2 0^060.06 artery Diameter IVC at hepatic vein Diameter 0.60 IVC 0.30 at hepatic 0.95 vein 14 0.30 18.0 0.95 22.214 18.0 22.2 0 0.06 06 Diameter aorta at L renal 0.250.25 0.33 0.33 0.99 0.99 14 14 11.4 11.4 18.5 18.5 0.03 0.03 DiameterDiameter IVC IVC atat R renal renal vein vein 0.61 0.310.31 0.99 1414 6.8 6.8 8.4 8.4 0.02 artery Diameter IVC at L renal vein Diameter 0.56 IVC 0.30 at L renal 0.97 vein 14 0.30 13.9 0.97 14 17.1 13.9 17.1 6.05 6.05 Diameter L coronary artery 0.100.10 0.36 0.97 0.97 18 18 20.1 20.1 26.3 26.3 0.03 0.03Diameter Diameter hepatic hepatic vein vein 0.60 0.26 0.26 0.92 0.92 14 14 20.9 20.9 25.7 25.7 o!o?0.07 Diameter brachiocephalic 0.240.24 0.37 0.37 0.99 0.99 21 121 9.1 9.1 18.3 0.02 0.02 DiameterDiameter R R renal renal vein vein 0.34 0.30 0.30 0.92 0.92 14 23.714 23.7 29.2 29.2 o!os0.08 artery Diameter L renal vein 0.46Diameter 0.25 L renal vein 0.92 12 19.40.25 0.92 22.812 19.4 22.8 0.08 0.08 Diameter R renal artery 0.170.17 0.30 0.30 0.99 0.99 15 15 5.1 5.1 8.5 0.02 0.02 DiameterDiameter R R iliac iliac vein vein 0.29 0.33 0.33 0.95 0.95 14 21.514 21.5 26.4 26.4 o!o70.07 Diameter L renal artery 0.150.15 0.31 0.31 0.99 0.99 14 14 9.4 9.4 15.2 0.03 0.03 DiameterDiameter L L iliac iliac vein vein 0.30 0.37 0.37 0.94 0.94 14 25.614 25.6 31.7 31.7 o'fJ8 0.08 Diameter R iliac artery 0.18 0.18 0.31 0.31 0.96 0.96 15 15 19.1 19.1 32.0 32.0 0.06 0.06 LengthLength IVC, IVC, heart heart to to 13.26 0.33 0.33 0.98 0.98 14 13.214 13.2 16.2 16.2 o!o-4 0.04 Diameter L iliac artery 0.160.16 0.33 0.33 0.98 0.98 15 15 11.7 19.5 19.5 0.04 0.04 bifurcationbifurcation Diameter intercostal artery 0.050.05 0.36 0.36 0.98 0.98 15 15 14.3 14.3 23.9 23.9 0.04 LengthLength IVC, IVC, heart to to hepatic hepatic 1.70 0.460.46 0.99 0.99 14 14 7.6 7.6 9.4 0.03 Length aorta, valves to 16.1216.12 0.32 0.32 0.99 0.99 15 15 6.6 6.6 10.9 10.9 0.02 0.02 veinvein bifurcation Length, IVC, heart to R renalLength, IVC, 6.75 heart 0.39to R renal 0.99 140.39 9.4 0.99 14 11.6 9.4 11.6 0.03 0.03 Length aorta, valves to 1.001.00 0.28 0.28 0.960.96 I 2121 14.6 14.6 29.6 29.6 I 0.04 0.04 veinvein brachiocephalic artery Length IVC, heart toLength L renalIVC, heart 7.48 to L renal 0.37 0.990.37 13 0.99 7.9 13 7.9 9.4 9.4 0.03 0.03 Length aorta, valves to L renal11.68 11.68 0.33 0.33 0.99 0.99 ! 14 !I 5.6 14 9.0 5.6 I 9.00.02 0.02 vein artery Length aorta, valves to R renal11.18 11.18 0.34 0.34 0.99 0.99 14 14 5.6 5.6 9.2 9.2 j 0.02 artery Length aorta, between 0.61 0.61 0.38 0.38 0.99 0.99 15 15 12.8 12.8 21.3 21.3 0.04 0.04 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 which the venous system was injected with a pressure of cava from the heart to the bifurcation, IVCl, are shown only 10 mmHg there was more variation in the diameter in Fig. 2, A-C. Equations describing similar relationships of the venous segments. Whereas, during life the arterial for the diameters of the right renal, RRVD, right iliac, system is always distended with a relatively high pres RIVD, and hepatic Hd, veins are given below. sure, the pressure distending the veins varies consider SVCD= 0.46 BW°U RRVDRRVd = = 0.34 0.34 BW030 ably from place to place and is affected to a greater degree by changes in body position. As, for example, in IVCd = 0.48 BW041 RIVD =0.29BW0-33 the vertical position the pressure in the iliac veins is higher than that in the superior vena cava, which may IVCl = 13.26 BW033 Hd = 0.60 BW026 be in a partially collapsed state (7). Thus, venous meas As shown in Table 2, the scatter of the data for the urements reported here do not represent the condition in venous system was somewhat greater than that for the any particular physiological state, instead they represent aorta, the correlation coefﬁcient being greater than 0.92 the maximum capacity of distension of these vessels at a and the standard estimate of the error less than 38%. distending pressure approaching 25 mmHg. Similar relationships for the diameters of the inferior Although the arterial injection pressure was 100 mmHg vena cava at various points throughout its length, the left the pressure distending the arterial tree at the time of iliac and left renal veins, as well as the lengths of various hardening of the plastic was much less. Evidence for this segments of the venae cavae are shown in Table 2. was obtained in several experiments in which pressure was measured in the aorta throughout the plastic injec DISCUSSION tion period. At the beginning ofthe injection, the pressure in the aorta was approximately 100 mmHg but after a The scatter of the data, as shown in Table 2, was few minutes it fell to between 35 and 75 mmHg. Thus, smaller in the arterial than in the venous system. Al the diameters and lengths of the arteries reported are for though the reason for this difference is not known it may be related to the fact that the small injection pressure of slightly distended vessels and not for vessels in the phys iological state distended with 100 mmHg pressure. This ■■■■; 25 mmHg in the venous system, as compared to 100 is conﬁrmed by the fact that the diameter of the ascend rnrnHg in the arterial system, led to greater variation in the diameters in the venous segments. This view is sup ing aorta calculated by the equation ported by the fact that in preliminary experiments in 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 D-0.41 BW 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-

L- 16.1 BW

,0.51 D-0.46BW

m MOUSE ♦ RAT O RABBIT -/* A 0 O G - O G O AT □ HORSE ■ COW er q" 0.48 BW

• - I O ' 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 Pressuresûres in inthe th« physiological «h„c;„l«„;„Qi range. ™„™ The tu percent " g ,nfenor increase vena range cava in body m 6 weight sPecies (rat of to mammalscattle). D, extendingdiameter, L, over length a 1,100-fold HOLT, RHODE, HOLT, AND KINES table 3. Percent increase in vessel diameter with (BW067), to BW1-0, or to an intermediate value, the ': : pressures in the physiological range present value of BW079 is based on measurement of cardiac output in mammals varying 1,790-fold in body Diameter In Mean Pres crease 100 D„/ weight, from rat to horse. sure, mmHg (D = a BW4), Quantitative relations of vascular similarity have been % demonstrated based on data for normal adult mammals Ascending aorta 118 162 varying as much as 38,000-fold in body weight. The Descending thoracic aorta upper Mi 108 148 diameters and lengths of vessel segments are described Descending thoracic aorta middle Vh 108 140 by power-law equations relating their diameters and Descending thoracic aorta lower V3 108 140 Abdominal aorta upper Vb 97 110 lengths to body weight. Abdominal aorta lower Vfi 97 119 External iliac artery 93 102 The authors thank W. Powell, M. R. Bledsoe, P. Bewley, J. P. Holt, Renal artery 97 97 Jr., T, Peterson, and M. Max for technical assistance, and K. Shotts Brachiocephalic artery 118 141 and J. Hart for assistance in preparing the programs for the computer. Intercostal arterv 109 127 This work was conducted at the Heart Research Laboratory, Uni Do, diameter of vessels in a 22.1-kg living dog when distended with versity of Louisville School of Medicine, Louisville, KY 40202, and pressures shown, as reported by Patel et al. (17). (D = a BW*) is School of Veterinary Medicine, University of California Davis CA diameter of vessel calculated by equations from data in Table 2. See 95616. text for discussion. This investigation was supported in part by Grants HE-5622 and 2075 from the National Heart and Lung Institute and the Kentucky, locity is nearly the same in the control state of large and Louisville, and Jefferson County Health Associations. small mammals. It has been proposed by others (13, 18) that cardiac output is proportional to body surface area Received 22 August 1980; accepted in final form 17 January 1981.

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