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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. 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