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Similarity of renal glomerular hemodynamics in mammals
Similarity of renal glomerular hemodynamics in mammals J. P. Holt E. A. Rhode Louisville, Ky., and Davis, Calif.
Despite the approximately 70 million-fold variation in body weight of mammals (shrew to whale), the heart, lungs, kidneys, and other major organs show much similarity in morphology and function. Evidence has been presented t h a t over the widest range of mammals quantitative morphological and functional characteristics of these and other organs are described by power law equations relating the particular variable to body weight.1 5 13, 2s. 27StahFS. 2~has shown by application of engineering dimensional analysis to physiology, utilizing the cancellation of statistically fitted power law prediction formulas for various physiological parameters, that it is possible to obtain dimensionless constants and dimensionless "design criteria" which characterize integrated mammalian physiological functions. The glomerular circulation has been described as a system of capillaries in parallel circuits having cross-connections. 14. 15, 30 By applying Poiseuille's law to the flow of blood through these glomerular capillary circuits, and utilizing power law equations describing the relationship between body weight and glomerular number, glomerular volume, renal blood flow, and glomerular filtration rate, it has been possible to calculate the number and length of capillaries in the mammalian glomerulus. Evidence will be presented to show that the average length of the glomerular capillaries, average linear velocity of blood flow, average time that blood spends in the capillaries, From tbe Heart Research Laborato2T, Department of Medicine, University of Louisville School of Medicine, Louisville, Ky. and University of California, School of Veterinary Medicine, Davis, Calif. This investigation was supported, in part, by the Kentucky, Louisville, and Jefferson County Heart Associations, Public Health Service Research Grants 2075 and HE 5622 from the National Heart and Lung Institute. Received for publication June 2, 1975: Reprint requests: Dr. J. P. Holt, Department of Medicine, Health Sciences Center, Louisville, Ky. 40201.
October, t976, Vol. 92, No. 4, p p . 465-472
blood flow per unit time per unit glomerular capillary surface area, and the glomerular filtration rate per unit time per unit surface area are all constant regardless of the size of the mammal. Methods
Microscopic measurements of glomerular diameter and volume were made in plastic corrosion casts of three rats, two rabbits, four dogs, one goat, one horse, and one cow. Casts were prepared of the entire arterial system as follows: after anesthesia and cannulation of the carotid and femoral arteries, the animals were killed by injecting concentrated KC1 into the arch of the aorta and immediately bled. Batson's Compound* was then injected under 100 mm. Hg pressure into the carotid and femoral arteries and the pressure maintained until the plastic hardened. This took bet~veen 30 and 60 minutes in different experiments. Following this, the animal was macerated in concentrated potassium hydroxide solution (15 to 33 per cent) and the remaining arterial cast washed with water until it was free of all remaining tissue. The kidneys of these casts were removed and the diameters of 50 or more of the glomerular capillary tufts measured with a binocular microscope and the average diameter determined. Glomerular volume, V, was calculated by the equation: V = 4/3 ~ (D/2) 3 where D is diameter of the glomerulus. Data in different mammalian species were collected from the literature for glomerular volume,19, 5~, 53.30 number of glomeruli,rs. ~2-~4,30 effective renal plasma flow measured either with Diodrast or para-amino hippurate, 5, 6. . . . ,,. 15. ~s. 1~. 20. ~1.._,~.5s and for glomerular filtration *Polysciences, Inc., Paul Valley Industrial Park, Warrington, Pa. 18976.
American Heart Journal
465
Holt and Rhode
Table I. P o w e r l a w p a r a m e t e r s for g l o m e r u l a r v o l u m e , n u m b e r , f i l t r a t i o n r a t e , e f f e c t i v e r e n a l p l a s m a flow, a n d b o d y w e i g h t for a w i d e v a r i e t y o f m a m m a l s
(mouse to whale) Power law coefficients*
Variables (units)
Species
a
Glomerular capillary tuft volume (ml.) Rat, rabbit, goat, dog, horse, COW Glomerular volume (ml.) Mouse, kangaroo rat, rat, guinea pig, ground hog, opossum, rabbit, cat, monkey, dog, man, swine, cattle, elephant Glomerular number Rat, rabbit, dog, horse, cattle Glomerular number Mouse, kangaroo rat, rat, guinea pig, ground hog, opossum, rabbit, cat, monkey, dog, man, horse, cattle, swine, elephant, whale Total glomerular volume (ml.) Rat, rabbit, dog, horse, cattle Total glomerular volume (ml.) Mouse, kangaroo rat, rat, guinea pig, ground hog, opossum, rabbit, cat, monkey, dog, swine, man, cattle, elephant ERPF (mlosec. ~) Rat, rabbit, dog, man GFR (ml.sec. 1) Rat, rabbit, cat, dog, man, monkey, goat, sheep, swine, cattle
4.30 x 10~ 1.30 x 10~
Sb
0.26 0.94 0.29 0.94
6 14
22.6 24.7 0.06 23.4 36.4 0.06
1.42 x 10~ 0.57 0.99 9.53 • 10~ 0.62 0.98
5 16
86.9 37.1 0.18 29.3 48.3 0.06
5.91 x 10-~ 0.84 0.99 1.37 x 10~ 0.85 0.99
5 14
61.8 27.3 0.13 16.4 26.0 0.05
1.75 x 10-1 0.81 0.99 3.25 x 10-~ 0.79 0.99
12 15
20.1 25.1 0.07 17.1 22.2 0.05
*Statistical fit is to t h e equation, y = a B W b. B o d y w e i g h t is in kilograms; r, correlation coefficient; N, t o t a l n u m b e r of d a t a points; sa, 95 per cent confidence limits of a in p e r cent; SE, m e a n { _+ ) s t a n d a r d error of t h e e s t i m a t e in per cent; sb, 95 per cent confidence limits of b in slope units.
rate measured w i t h i n u l i n . 2 ~ ~....... 1~ 17.2...... . . . . . . ~-~T h e a v e r a g e v a l u e of t h e s e p a r a m e t e r s for a n a n i m a l o f a v e r a g e w e i g h t for e a c h s p e c i e s w a s c a l c u l a t e d a n d e m p l o y e d in t h e a n a l y s i s below. Log-log plots were prepared of the relationship between body weight and: glomerular number. glomerular volume, total glomerular volume. e f f e c t i v e r e n a l p l a s m a flow. a n d g l o m e r u l a r f i l t r a t i o n r a t e in o n e k i d n e y . T h e d a t a w e r e t r a n s f o r m e d t o b a s e 10 l o g a r i t h m s a n d t h e l i n e a r regression of the logarithmic values calculated by t h e m e t h o d o f l e a s t s q u a r e s t o give t h e p a r a m e t e r s in t h e p o w e r l a w f o r m u l a : y = aX b w h e r e y is a n y v a r i a b l e a n d X is m a s s o f b o d y w e i g h t in k i l o g r a m s . S t a t i s t i c a l a n a l y s i s o f t h e logarithmic equations included: the correlation coefficient (r), 95 p e r c e n t c o n f i d e n c e l i m i t s for r e p e a t e d line fits (sa a n d Sb), a n d t h e s t a n d a r d e r r o r o f t h e e s t i m a t e , S~, w h i c h h a s m u c h t h e s a m e s i g n i f i c a n c e for a l o g a r i t h m i c r e g r e s s i o n line as t h e s t a n d a r d d e v i a t i o n for a m e a n , i.e., t w o SE l i m i t s s h o u l d i n c l u d e 95 p e r c e n t o f t h e cases. W i t h l o g - l o g a n a l y s i s , + S E a n d --SE d i f f e r s l i g h t l y ; t h e v a l u e s s h o w n in t h e t a b l e a r e t h e mean of the two values.
466
Results T a b l e I p r e s e n t s t h e c o e f f i c i e n t s for t h e p o w e r law regression equations, as well as the statistical measures for the relationship of these variables to body weight.
Glomerular capillary tuft volume and body weight. T h e l o g a r i t h m i c r e l a t i o n s h i p b e t w e e n t h e experimentally determined glomerular capillary t u f t v o l u m e a n d b o d y w e i g h t in m a m m a l s e x t e n d i n g o v e r a 1,740-fold r a n g e ( r a t t o horse} in b o d y w e i g h t is s h o w n in Fig. 1, B a n d d e s c r i b e d b y the equation: G T V = 4.30 x 10 6 BW0.26 (1) w h e r e G T V is g l o m e r u l a r t u f t v o l u m e in m i l l i liters a n d B W is b o d y w e i g h t in k i l o g r a m s . A l s o , Fig. 1, B s h o w s t h e r e l a t i o n s h i p for g l o m e r u l a r v o l u m e a n d b o d y w e i g h t in m a m m a l s r a n g i n g 216 t h o u s a n d - f o l d in b o d y w e i g h t ( m o u s e t o elep h a n t ) , u t i l i z i n g v a l u e s r e p o r t e d in t h e l i t e r a t u r e . T h i s r e l a t i o n s h i p is d e s c r i b e d b y t h e e q u a t i o n : G V = 1.30 x 10 -6 B W ~ (2) Glomerular number and body weight. T h e n u m b e r o f g l o m e r u l i in o n e k i d n e y f o r t h e s a m e species o f a n i m a l s s h o w n in Fig. 1, B w a s d e t e r m i n e d f r o m s t u d i e s in t h e l i t e r a t u r e . T h e l o g a r i t h mic r e l a t i o n s h i p b e t w e e n t h e n u m b e r o f g l o m e r -
October, 1976, Vol. 92, No. 4
Similarity of renal glomerular hemodynamics in mammals i
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IO 0 10 2 I0 4 BODY WEIGHT (kg) Fig. 1. Logarithmic relationships between body weight and total glomerular volume in one kidney, number of glomeruli in one kidney, and average single glomerular volume in 18 species of mammals (mouse to whale). The closed symbols and solid lines represent data taken from the literature. The open symbols and broken lines represent data obtained by means of our plastic injection casts. Equations for the solid lines are given below the lines; equations for the broken lines are given above the lines. Total gl0merular volume and individual glomerular volume are in milliliters; body weight is in kilograms.
American Heart Journal
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CATTLE HORSE GOAT MAN SWINE DOG MONKEY CAT RABBIT OPOSSUM
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I00 I0 ! I02 BODY WEIGHT (kg)
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Fig. 2. Logarithmic relationships between body weight and effective renal plasma flow (ERPF,) and glomerular filtration rate (GFR) in 10 species of mammals (rat to bovine). The data were taken from the literature in which effective renal plasma flow was measured by means of either diodrast or para-amino hippurate and glomerular filtration rate was measured by means of inulin. Body weight is in kilograms,
uli (GN) and body weight for these mammals* is shown in Fig. 1, C. This relationship is described by the equation: G N -- 1.42 x 10 ~ B W ~,57 (3) where B W is i n k i l o g r a m s . Also, Fig. 1, C s h o w s t h e s a m e r e l a t i o n s h i p for m a m m a l s v a r y i n g 1.5 million-fold in body w e i g h t (mouse to whale), utilizing values rePorted in the literature. This r e l a t i o n s h i p is described b y t h e e q u a t i o n : G N -- 9.53 • 105 B W ~ (4) Total glomerular volume. T h e average glomer*Excluding
the goat.
467
Holt a n d R h o d e
ular volume and body weight were calculated for each species in which we prepared plastic glomerular t u f t casts. With the d a t a from the l i t e r a t u r e the n u m b e r of glomeruli was calculated for these animals. T h e p r o d u c t of glomerular v o l u m e and glomerular n u m b e r for each species was calculated, giving the t o t a l glomerular v o l u m e in one kidney for an animal having a body weight equal to the average body weight of the animals studied. T h e logarithmic relationship between total glomerular volume and body weight for these animals varying 1,740-fold in b o d y weight is shown in Fig. 1, A and is described b y the equation: T G V = 591 • 10 -3 B W ~ (5) where T G V is in milliliters and B W is in kilograms. In a similar manner, utilizing values from the literature, total glomerular v o l u m e was calculated for m a m m a l s ranging 216 thou,sand-fold in body weight (mouse to elephant). T h e logalithmic relationship between total glomerular volume and body weight for these animals is shown in Fig. 1, A and is described by the equation: T G V = 137 x 10 .3 BW TM. (6) Effective renal plasma flow and body weight. T h e logarithmic relationship between body weight and effective renal plasma flow t h r o u g h one kidney, measured by para-amino h i p p u r a t e or Diodrast clearance techniques in studies r e p o r t e d in the literature, for m a m m a l s varying 446-fold (rat to man) in body weight is shown in Fig. 2, A and described by the equation: E R P F = 1.75 x 10 -~ BW 0-~' (7) where E R P F is effective renal plasma flow in ml.. sec. ~ and BW is in kilograms. Assuming an h e m a t o c r i t of 0.45, this can be corrected to blood flow through one kidney to give: E R B F = 3.18 • 10 ~ S W ~ (8) Average glomerular capillary length and number. T h e volume of the glomerular capillary t u f t in our plastic injection casts consists of the injected capillaries, intraglomerular parts of the afferent and efferent arterioles, plus space between the capillaries. Although it is not k n o w n what fraction of the injected capillary tuft consists of space between capillaries, separation of the capillaries by breaking the cast into small pieces showed t h a t the capillaries were closely packed together and t h a t there was minimal space between them. Likewise, it is not k n o w n what fraction of the injected capillary t u f t
468
consists of afferent and efferent vessels, b u t presumably this volume is a small fraction of the total. Thus, in the analysis given below for length and n u m b e r of capillaries in the glomerulus we have assumed t h a t in the plastic glomerular t u f t the volume measured is 100 per cent capillaries in parallel circuits. Since the blood flow t h r o u g h all of the glomeruli in one kidney, from the d o w n s t r e a m end of the afferent arteriole to the u p s t r e a m end of the efferent arteriole, is through a system of capillaries in parallel, the total flow is described as a first approximation by Poiseuille's law as follows: Nc AP ~r (re) 4 FGB = (9) 8 Lc~ where F G B is the blood flow t h r o u g h all glomeruli in one kidney in ml. 9 sec. -1, Nc is the n u m b e r of capillaries in all of these glomeruli, rc is the average capillary radius, Lo is the average capillary length, AP is the average pressure drop from the beginning to the end of the capillary, and v is the coefficient of viscosity of blood. It is generally agreed t h a t the coefficient of viscosity of blood, the diameter of the capillaries, and arterial, capillary, and venous pressures do not differ significantly in m a m m a l s of different size. In contrast, the exact value of the pressure drop from beginning to end of the glomerular capillaries is not known. However, recent measurements of this pressure drop in the r a t a n d monkey have shown t h a t it is small, being no greater t h a n 2.5 mm. Hg. ~. 4 7 Substituting in Eq. (9) this value for h p as a first approximation, the mean value for blood viscosity of 0.0228 dynes 9 sec. o cm. -~, and capillary diameter of 8/~ gives:
FGB =
1.46 x 10-9 Nc L~
(i0)
where the flow is in ml. 9 -I Since this flow is the same as that given in Eq. (8), equating the two and solving for Nr gives:
Nc = 2.18 x 108 B W ~
Lo
(11)
where BW is in kilograms. Assuming t h a t the volume of the glomerular capillary tuft measured in our plastic injection specimens consists of 100 per cent capillaries, total glomerular t u f t volume in one kidney equals the number of capillaries in all of the glomeruli times the v o l u m e of the average capillary, i.e.. T G V = No (~rro2Lc).
October, 1976, Vol. 92, No. 4
Similarity of renal glomerular hemodynamics in mammals
Taking rc to have a value of 4 • 10 -4 cm., t h e equation becomes: T G V = Nc Lc (5.02 • 10-~). (12) T h e t o t a l glomerular volume for the animals t h a t we studied, having distended glomeruli a n d described by Eq. (5), is the same as t h a t given in Eq. (12); equating the two and solving for Nc gives: 117.7 X 104 B W `)84 (13) Nc= Lo Since b o t h this equation and Eq. (11) give t h e total n u m b e r of glomerular capillaries in one kidney, t h e y must be equal to each other, equating the two and solving for Lc gives: Lc = 7.33 • 10 ~ B W ~176 (14) where Lc is in centimeters and B W is in kilograms. Since B W ~176is so near to B W ~ and is within t h e 95 per cent confidence limits of the power functions from which it was derived, it appears reasonable to assume t h a t its value is B W ~ which has a value of 1. T h u s the length of the average capillary is constant, regardless of the size of t h e mammal, and has a value of 733 microns. Substituting in Eq. (13) this value for Le gives the equation for the total n u m b e r of g l o m e r u l a r capillaries in one kidney, i.e., Nc -- 16.1 • 106 B W ~ (15) Velocity of flow in glomerular capillaries. T o t a l capillary cross-section in all glomeruli in one kidney is equal to the n u m b e r of capillaries in the parallel circuits of all glomeruli times t h e cross-section of an individual capillary, i.e., X S = Nc ~r (re) 2 where No is the n u m b e r of capillaries and re is t h e average capillary radius. Substituting the n u m b e r of distended capillaries given in Eq. (15) a n d taking the radius of the average capillary to be 4 microns gives: X S -- 809 • 10 -~ B W ~ (16) where X S is the t o t a l capillary cross-section in all glomeruli in one kidney in square centimeters, and B W is in kilograms. Linear velocity is equal to volumetric flow r a t e divided by cross-section. Thus, dividing Eq. (8) for the flow t h r o u g h all the glomeruli in one k i d n e y by Eq. (16) for the cross-section gives the average linear velocity in the average capillary: V = 0.039 B W ~ (17) where V is in c m . - sec.-' and B W is in kilograms. Since B W -~176in the equation is small and is within the variation of the 95 per cent confidence
American Heart Journal
limits of the power functions from which it was derived, it is assumed to be BW ~ and the e q u a t i o n becomes: V = 0.039 c m . . sec. 1 (18) Thus, the average velocity t h r o u g h the average capillary is constant regardless of the size of the mammal. Time spent in glomerular capillaries. T h e time spent in traversing the length of a capillary is equal to the length divided by the velocity. Since the average velocity t h r o u g h the glomerular capillaries as shown above is 0.039 cm. sec. -1, and the average capillary length is 7.33 x 10 2 cm., the average time blood spends in the capillaries is: T = 1.88 sec. (19) Thus, this time is c o n s t a n t regardless of the size of the mammal. Blood flow per unit capillary surface area in the mammalian glomerulus. Assuming that the glomerular capillary t u f t is m a d e up entirely of capillaries, the total capillary surface area in one kidney is equal to the product of the length and circumference of the average capillary multiplied by the n u m b e r of capillaries, i.e., SAo = Lc Nc 2~r ro. Substituting Eq. (14) for the length of the average capillary and Eq. (15) for the n u m b e r of capillaries and taking capillary radius to be 4 microns gives: ,, SA = 2966 B W ~ (20) where the surface area is in square centimeters and B W is in kilograms. T h e blood flow t h r o u g h all glomeruli in one kidney, regardless of the size of the mammal, is given by Eq. (8). Dividing this equation by Eq. (20) for the surface area of the capillaries in all glomeruli in one kidney gives: F G B / S A = 1.07 • 10 4 BW-003 (21) where F G B / S A is blood flow per unit capillary surface area in m l . . sec. -1- cm. 2 and B W is in kilograms. Since B W -~176is small and is within the variation of the 95 per cent confidence limits of the power functions from which it was derived, it is assumed to be B W ~ and the equation becomes: F G B / S A = 1.07 • 10 4 m l . . s e c . - ' , cm. -2 (22) Thus, the a m o u n t of blood flowing per unit surface area of glomerular capillary per unit time is constant regardless of the size of the mammal. Glomerular filtration rate and body weight.
469
Holt and Rhode
Table II. R e n a l g l o m e r u l a r
hemodynamic
constants
and variables in mammals*
Change of value shown in column A (in per cent) Glomerular capillary tuft Hemodynamic parameters
Distended A
AP I
IViscosityl Radius
I
Undistended B
1
(5.)
Hematocrit I
I
r
Capillary tuft I
I (75%)
Glomerular capillary: Length (cm.) 7.33 x 10 -~ Number 16.1 X l0 '~ BW"*:' Cross-section (sq. cm.) 809 x 10 .2 BW"*" Velocity (cm "sec. -~) 0.039 Surface area (sq. cm.) 2966 BW ~,~:~ Time (sec.) 1.88 FGB/SA 1.07 x 10-' (ml. 9cm. ~. sec. ') G F R / S A (ml.-cm. 2.sec.-')0.11 x 10 -~
2.50 x 10 -~ -29 -50 5.5 x 10" BW '~:' +41 +100 274 x 10 ~ BW ''~" +41 +100 0.116 -29 -50 342 BW '.*:~ 0 0 0.22 0 0 9.30 • 10 ~ 0 0 0.95 x 10 -~
0
0
+42 -30 -30 +42 0 0 0
+25 -49 -20 +25 -20 0 +25
+5 -5 -5 -3 0 +8 -8
-5 -6 -6 +6 -11 -10 +12
-13 -14 -14 +16 -25 -25 +33
0
+25
0
+12
+33
*The constant values, regardless of body size, of average glomerular capillary length, velocity of blood flow, time that blood spends in the glomerular capillaries, blood flow per unit capillary surface area (FGB/SA), and glomerular filtration rate per unit capillary surface area (GFR/SA); and the power law equations describing the number of capillaries, total cross-section of capillaries, and total surface area of capillaries in all glomeruli in one kidney are given. The values of the distended capillaries were calculated from our plastic injection casts; the values for undistended capillaries were calculated with the use of data from the literature in which the glomerular capillaries were not distended. These values and equations were calculated taking blood viscosity to be 0.0228 dynes 9sec 9cm. ~-,hematocrit 0.45, capillary radius 4~, the pressure drop from beginning to end of the capillary to be 2.5 mm. Hg, and glomerular tuft volume to be 100 per cent of the measured volume. The values in the columns under "Per cent changes from values shown in Column A" were calculated with the same values employed in the calculations for Column A except that one of the values was changed for the calculations in each column as shown, i.e., hp of 1.25 and 0.625 instead of 2.5 mm. Hg, blood viscosity of 0.0114 dynes 9sec 9cm. -~ instead of 0.0228, capillary radius of 5it instead of 4, hematocrit of 0.40 instead of 0.45, and capillary tuft volumes of 90 and 75 per cent of the measured volume instead of 100 per cent. The logarithmic
relationship
between
glomerular
filtration rate in one kidney and body weight in mammals ranging 438-fold (rat to man) in body w e i g h t is s h o w n i n F i g . 2, B . T h i s r e l a t i o n s h i p is described by the equation: G F R = 3.25 x 10 -~ B W ~ (23) w h e r e G F R i s i n m l . 9 s e c . - ' a n d B W is i n k i l o grams. D i v i d i n g E q . (23) f o r g l o m e r u l a r f i l t r a t i o n r a t e b y E q . (20) f o r t o t a l g l o m e r u l a r c a p i l l a r y s u r f a c e area gives the glomerular filtration rate per unit surface area of glomerular capillaries, GFR/SA, i.e.; GFR/SA = 0.11 • 10 -4 B W ...... (24) w h e r e G F R / S A is i n m l . o c m . - 2 . sec. -1 o f g l o m e r ular capillary surface. Since the power function of B W -~176 is s m a l l a n d w i t h i n t h e 9 5 p e r c e n t confidence limits of the power functions from w h i c h i t w a s d e r i v e d , i t is a s s u m e d t o b e z e r o . Thus, the glomerular filtration rate per unit c a p i l l a r y s u r f a c e a r e a is c o n s t a n t r e g a r d l e s s o f t h e s i z e o f t h e m a m m a l a n d h a s a v a l u e o f 1.1 x 10 -5 m l . 9 c m . -~ 9 s e a . - '
Discussion I t is t o b e e m p h a s i z e d t h a t t h e d a t a g i v e n h e r e are for normal adult mammals varying greatly in
470
size and extending over many species. We believe that the relationships described formulate the similarity criteria which define the normal adult mammalian design of the cardiovascular functions studied. These relationships differ to some extent from species to species and from animal to animal, depending on the degree of adaptation to different environmental situations during the course of evolutionary development. These differences from the normal pattern are limited and define to some degree the deviation from the normal that an animal or species might have and yet still survive. If, i n a p a r t i c u l a r s p e c i e s o r i n d i v i d u a l a n i m a l , i t is k n o w n t h a t o n e o r m o r e o f t h e h e m o d y n a m i c variables differ from the values used in the calculations employed here, then more accurate calculations may be made employing the variab l e s w h i c h d i f f e r f r o m t h e o n e s w e u s e d . T h i s is o f particular importance in the case of the pressure d r o p b e c a u s e a l l t h a t is k n o w n is t h a t i t i s n o g r e a t e r t h a n 2.5 m m . H g . I t m a y b e t h a t i t h a s a v a l u e o f s o m e f r a c t i o n o f 2.5 m m . H g . I n T a b l e I I e x a m p l e s a r e s h o w n w h e r e t h e p r e s s u r e d r o p is t a k e n t o b e 1.25 a n d 0.625 m m . H g , b l o o d v i s c o s i t y 0.0114 p o i s e , c a p i l l a r y r a d i u s 5 m i c r o n s , h e m a t o c r i t 0.40, a n d g l o m e r u l a r t u f t v o l u m e s o f 9 0 a n d 75
October, 1976, Vol. 92, No. 4
Similarity of renal glomerular hemodynamics in mammals
per cent of our m e a s u r e d volume. I t will be n o t e d in the table t h a t the deviation, f r o m t h e values e m p l o y e d in our calculations, was less t h a n 50 per cent in all cases except t w o (capillary n u m b e r a n d cross-section) in which t h e deviation was 100 per cent when the pressure d r o p was t a k e n to be 0.625 m m . Hg. T h i s difference is small w h e n considered in relationship to the over-all p a t t e r n of f u n c t i o n which extends m o r e t h a n 446-fold in b o d y weight. I t is interesting to note in T a b l e I I t h a t t h e t i m e t h a t blood spends in g l o m e r u l a r capillaries is i n d e p e n d e n t of pressure drop, blood viscosity, a n d radius. T h i s suggests t h a t if, in a p a r t i c u l a r species or animal, one or m o r e of these f a c t o r s deviate f r o m t h e values t h a t we h a v e e m p l o y e d in t h e calculations, t h e n this a n i m a l or species h a s an a p p r o p r i a t e change in capillary l e n g t h a n d n u m b e r such t h a t the blood t r a n s i t t i m e t h r o u g h the g l o m e r u l a r capillaries is c o n s t a n t . U n t i l such t i m e as m o r e i n f o r m a t i o n is available on the pressure drop in large a n d small m a m m a l s we c a n n o t be certain t h a t t h e pressure drop is the s a m e regardless of m a m m a l size. I f the pressure drop differs f r o m a n i m a l to a n i m a l or species to species, however, it is to be e m p h a s i z e d t h a t regardless of this the t i m e t h a t blood spends in the capillaries r e m a i n s c o n s t a n t (Table II). In view of the fact t h a t , regardless of m a m m a l size, the factors involved in the f o r m a t i o n of the protein-free glomerular filtrate are a p p r o x i m a t e ly the same, i.e., (1} h y d r o s t a t i c pressure in the capillary forcing filtrate t h r o u g h t h e c a p i l l a r y wall, (2) colloidal osmotic pressure of t h e blood proteins, (3} h y d r o s t a t i c pressure in the collecting tubules of the kidney, and, as shown above, (4) blood flow per unit g l o m e r u l a r capillary surface area per unit time, it would a p p e a r t h a t the n u m b e r a n d average length of capillaries in the glomerulus are related in such a m a n n e r t h a t blood spends a c o n s t a n t t i m e in the capillaries in order t h a t the g l o m e r u l a r filtration r a t e p e r unit surface area of capillaries is c o n s t a n t regardless of m a m m a l size. T h e large a m o u n t of g l o m e r u l a r filtrate formed in the large m a m m a l comes a b o u t as a result of the greater t o t a l g l o m e r u l a r surface area. T h e volumes of the distended g l o m e r u l a r capillary t u f t s t h a t we m e a s u r e d were c o n s i d e r a b l y larger t h a n the glomeruli of the s a m e species reported by o t h e r investigators, 1 9 9 22.23. 30 in which the glomeruli were n o t distended a n d an u n k n o w n a m o u n t of shrinkage h a d t a k e n place
American Heart Journal
during histological p r e p a r a t i o n . I t m i g h t be argued t h a t our distended g l o m e r u l a r capillaries were larger t h a n during life, a n d t h a t as a result the calculated values for t h e h e m o d y n a m i c variables were in error. C o m p a r i s o n s of our values for the above variables with the values o b t a i n e d when calculated f r o m d a t a on u n d i s t e n d e d glomeruli described in the literature, a s s u m i n g as did V i m t r u p 3~ t h a t the glomeruhis consists of 50 per cent capillaries, are given in T a b l e II. As s h o w n in the table, the v~lues of the various p a r a m e t e r s for undistended glomeruli differ considerably f r o m our values. T h e h y d r o s t a t i c pressure in the glom e r u l a r capillaries has been r e p o r t e d to be 45 m m . Hg, and it would a p p e a r t h a t the v o l u m e s of the glomerular t u f t s in o u r plastic casts, in which a high injection pressure was m a i n t a i n e d until t h e plastic hardened, were closer to the v o l u m e s existing during life t h a n would be the v o l u m e with no distending h y d r o s t a t i c pressure in the capillaries. W h e n i n f o r m a t i o n becomes available on a wider range of m a m m a l s , p a r t i c u l a r values given for the p a t t e r n of function t h a t we h a v e described will no d o u b t be changed to s o m e extent. W e believe, however, t h a t these differences will be small, and will n o t alter the description of t h e general similarity p a t t e r n of f u n c t i o n as presented. The authors express their thanks to W. Powell, T. Peterson, M. May, and M. R. Bledsoe for technical assistance. REFERENCES
1. Adolph,E. A.: Quantitative relations in the physiological constitutions of mammals, Science 109:579, 1949. 2. Bolomey, A. A., Michie, A. J., Michie, C., Breed, E. S., oSchreiner, G. E., and Lauson, H. D.: Simultaneous measurement of effective renal blood flow and cardiac output in resting normal subjects and patients with essential hypertension, J. Clin. Invest. 28:10, 1949. 3. Brenner, B. M., Troy, J. L., and Daugharty, T. M.: The dynamics of glomerular ultrafiltration in the rat, J. Clin. Invest. 50:1776, 1971. 4. Brenner, B. M., Troy, J. L., Daugharty, T. M., Deen, W. M., and Robertson, C. R.: Dynamics of glomerular ultrafiltration in the rat. II. Plasma-flow dependence of GFR, Am. J. Physiol. 223"1191, 1972. 5. Brody, S.: Bioenergetics and growth, New York, 1945, Reinhold Publishing Company. 6. Chasis, H., Redish, J., Goldring, W., Ranges, H. A., and Smith, H. W.: The use of sodium p-aminohippurate for the functional evaluation of the human kidney, J. Clin. Invest. 24:583, 1945. 7. Deen, W. M., Robertson, C. R., and Brenner, G. M.: Glomerular ultrafiltration, Fed. Proc. 33:14, 1974. 8. DeJongh, D. K., and Kok, K.: Investigation into the estimation of inulin and PAH clearances in rats, Acta Physiol. Pharmacol. Neerl. 4:222, 1955.
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9. 10. 11. 12.
13.
14.
15.
16. 17. 18.
19.
472
Dicker, S. E., and Heller, H.: The mechanism of water diuresis in normal rats and rabbits as analyzed by inulin and diodone clearances, J. Physiol. 103:449, 1945. Forster, R. P.: An examination of some factors which alter glomerular activity in the rabbit kidney, Am. J. Physiol. 150"523, 1947. Friedman, S. M., Polley, J.R., and Friedman, C. L.: The clearance of inulin and sodium p-aminohippurate in the rat, Am. J. Physiol. 150:340, 1947. Goldring, W., Chasis, H., Ranges, H. A., and Smith, H. W.: Relations of effective renal blood flow and glomerular filtration to tubular excretory mass in normal man, J. Clin. Invest. 19:739, 1940. Gtinther, B.: On theories of biological similarity, in Fortschritte der experimentellen und theoretischen Biophysik, Leipzig, 1975, Georg Thieme Verlag, Band 19. Hall, B. V.: Studies of normal glomerular structure by electron microscopy, in Proceedings of the Fifth Annual Conference on the Nephrotic Syndrome, New York, 1953, National Nephrosis Foundation, pp. 1-39. Hall, B. V.: Further studies of the normal structure of the renal glomerulus, Proceedings of the Sixth Annual Conference on the Nephrotic Syndrome, New York, 1954, National Nephrosis Foundation, pp. 1-39. Houck, C. R.: Statistical analysis of filtration rate and effective renal plasma flow related to weight and surface area in dogs, Am. J. Physiol. 153:169, 1948. Kaplan, B. I., and Smith, H. W.: Excretion of inulin, creatinine, xylose and urea in the normal rabbit, Am. J. Physiol. 113:354, 1935. Korner, P. I.: Renal blood flow, glomerular filtration rate, renal PAH extraction ratio, and the role of the renal vasomotor nerves in the unanesthetized rabbit, Circ. Res. 12:353, 1963. Kunkel, P. A.: The number and size of the glomeruli in the kidney of several mammals, Bull. Johns Hopkins Hosp. 47:285, 1930.
20. 21. 22. 23.
24. 25. 26. 27. 28. 29.
30. 31. 32.
Lippman, R. W.: Clearances as a measure of renal function in the rat, Am. J. Physiol. ] 52:27, 1948. Moustgaard, J.: Inulin-hippodin clearance in healthy dogs, Skand. Veterinar-tidskr. 35:508, 1945. Oliver, J.: Nephrons and kidneys, New York, 1968, Harper & Row, Publishers, p. 48. Rytand, D. A.: The number and size of mammalian glomeruli as related to kidney and to body weight with methods for their enumeration and measurement, Am. J. Anat. 62:507, 1937. Smith, H. W.: The kidney: Structure and function in health and disease, Oxford, 1951, Oxford University Press, p. 570. Smith, W. W.: Renal blood flow and mannitol diuresis in the rabbit, Bull. Mt. Desert Island Biol. Lab. 43:25, 1941. Stahl, W. R.: Scaling of respiratory variables in mammals, J. Appl. Physiol. 22:453, 1967. Stahl, W. R.: Similarity and dimensional methods in biology, Science 137-138:205, 1962. Stamler, J., Katz, L. N., and Rodbard, S.: Serial renal clearances in dogs with nephrogenic and spontaneous hypertension, J. Exp. Med. 90:511, 1949. Sweet, A. Y., Levitt, M. F., and Hodes, H. L.: Kidney function, body fluid compartments and water and electrolyte metabolism in the monkey, Am. J. Physiol. 201:975, 1961. Vimtrup, B. J.: On the number, shape, structure, and surface area of the glomeruli in the kidneys of man and mammals, Am. J. Anat. 41:123, 1928. Vogel, G.: Zur Bestimmung Von glomerul~rer Filtrationsrate und filtrierender Flt/che in der Niere einiger Hausstiugetiere, Pflfigers Arch. 269:264, 1959. Wirz, H.: Untersuchungen tiber die Nierenfunktion bei adrenalektomierten katzen., Helv. Physiol. Acta 3:589, 1945.
October, 1976, Vol. 92, No. 4
Despite the approximately 70 million-fold variation in body weight of mammals (shrew to whale), the heart, lungs, kidneys, and other major organs show much similarity in morphology and function. Evidence has been presented t h a t over the widest range of mammals quantitative morphological and functional characteristics of these and other organs are described by power law equations relating the particular variable to body weight.1 5 13, 2s. 27StahFS. 2~has shown by application of engineering dimensional analysis to physiology, utilizing the cancellation of statistically fitted power law prediction formulas for various physiological parameters, that it is possible to obtain dimensionless constants and dimensionless "design criteria" which characterize integrated mammalian physiological functions. The glomerular circulation has been described as a system of capillaries in parallel circuits having cross-connections. 14. 15, 30 By applying Poiseuille's law to the flow of blood through these glomerular capillary circuits, and utilizing power law equations describing the relationship between body weight and glomerular number, glomerular volume, renal blood flow, and glomerular filtration rate, it has been possible to calculate the number and length of capillaries in the mammalian glomerulus. Evidence will be presented to show that the average length of the glomerular capillaries, average linear velocity of blood flow, average time that blood spends in the capillaries, From tbe Heart Research Laborato2T, Department of Medicine, University of Louisville School of Medicine, Louisville, Ky. and University of California, School of Veterinary Medicine, Davis, Calif. This investigation was supported, in part, by the Kentucky, Louisville, and Jefferson County Heart Associations, Public Health Service Research Grants 2075 and HE 5622 from the National Heart and Lung Institute. Received for publication June 2, 1975: Reprint requests: Dr. J. P. Holt, Department of Medicine, Health Sciences Center, Louisville, Ky. 40201.
October, t976, Vol. 92, No. 4, p p . 465-472
blood flow per unit time per unit glomerular capillary surface area, and the glomerular filtration rate per unit time per unit surface area are all constant regardless of the size of the mammal. Methods
Microscopic measurements of glomerular diameter and volume were made in plastic corrosion casts of three rats, two rabbits, four dogs, one goat, one horse, and one cow. Casts were prepared of the entire arterial system as follows: after anesthesia and cannulation of the carotid and femoral arteries, the animals were killed by injecting concentrated KC1 into the arch of the aorta and immediately bled. Batson's Compound* was then injected under 100 mm. Hg pressure into the carotid and femoral arteries and the pressure maintained until the plastic hardened. This took bet~veen 30 and 60 minutes in different experiments. Following this, the animal was macerated in concentrated potassium hydroxide solution (15 to 33 per cent) and the remaining arterial cast washed with water until it was free of all remaining tissue. The kidneys of these casts were removed and the diameters of 50 or more of the glomerular capillary tufts measured with a binocular microscope and the average diameter determined. Glomerular volume, V, was calculated by the equation: V = 4/3 ~ (D/2) 3 where D is diameter of the glomerulus. Data in different mammalian species were collected from the literature for glomerular volume,19, 5~, 53.30 number of glomeruli,rs. ~2-~4,30 effective renal plasma flow measured either with Diodrast or para-amino hippurate, 5, 6. . . . ,,. 15. ~s. 1~. 20. ~1.._,~.5s and for glomerular filtration *Polysciences, Inc., Paul Valley Industrial Park, Warrington, Pa. 18976.
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Holt and Rhode
Table I. P o w e r l a w p a r a m e t e r s for g l o m e r u l a r v o l u m e , n u m b e r , f i l t r a t i o n r a t e , e f f e c t i v e r e n a l p l a s m a flow, a n d b o d y w e i g h t for a w i d e v a r i e t y o f m a m m a l s
(mouse to whale) Power law coefficients*
Variables (units)
Species
a
Glomerular capillary tuft volume (ml.) Rat, rabbit, goat, dog, horse, COW Glomerular volume (ml.) Mouse, kangaroo rat, rat, guinea pig, ground hog, opossum, rabbit, cat, monkey, dog, man, swine, cattle, elephant Glomerular number Rat, rabbit, dog, horse, cattle Glomerular number Mouse, kangaroo rat, rat, guinea pig, ground hog, opossum, rabbit, cat, monkey, dog, man, horse, cattle, swine, elephant, whale Total glomerular volume (ml.) Rat, rabbit, dog, horse, cattle Total glomerular volume (ml.) Mouse, kangaroo rat, rat, guinea pig, ground hog, opossum, rabbit, cat, monkey, dog, swine, man, cattle, elephant ERPF (mlosec. ~) Rat, rabbit, dog, man GFR (ml.sec. 1) Rat, rabbit, cat, dog, man, monkey, goat, sheep, swine, cattle
4.30 x 10~ 1.30 x 10~
Sb
0.26 0.94 0.29 0.94
6 14
22.6 24.7 0.06 23.4 36.4 0.06
1.42 x 10~ 0.57 0.99 9.53 • 10~ 0.62 0.98
5 16
86.9 37.1 0.18 29.3 48.3 0.06
5.91 x 10-~ 0.84 0.99 1.37 x 10~ 0.85 0.99
5 14
61.8 27.3 0.13 16.4 26.0 0.05
1.75 x 10-1 0.81 0.99 3.25 x 10-~ 0.79 0.99
12 15
20.1 25.1 0.07 17.1 22.2 0.05
*Statistical fit is to t h e equation, y = a B W b. B o d y w e i g h t is in kilograms; r, correlation coefficient; N, t o t a l n u m b e r of d a t a points; sa, 95 per cent confidence limits of a in p e r cent; SE, m e a n { _+ ) s t a n d a r d error of t h e e s t i m a t e in per cent; sb, 95 per cent confidence limits of b in slope units.
rate measured w i t h i n u l i n . 2 ~ ~....... 1~ 17.2...... . . . . . . ~-~T h e a v e r a g e v a l u e of t h e s e p a r a m e t e r s for a n a n i m a l o f a v e r a g e w e i g h t for e a c h s p e c i e s w a s c a l c u l a t e d a n d e m p l o y e d in t h e a n a l y s i s below. Log-log plots were prepared of the relationship between body weight and: glomerular number. glomerular volume, total glomerular volume. e f f e c t i v e r e n a l p l a s m a flow. a n d g l o m e r u l a r f i l t r a t i o n r a t e in o n e k i d n e y . T h e d a t a w e r e t r a n s f o r m e d t o b a s e 10 l o g a r i t h m s a n d t h e l i n e a r regression of the logarithmic values calculated by t h e m e t h o d o f l e a s t s q u a r e s t o give t h e p a r a m e t e r s in t h e p o w e r l a w f o r m u l a : y = aX b w h e r e y is a n y v a r i a b l e a n d X is m a s s o f b o d y w e i g h t in k i l o g r a m s . S t a t i s t i c a l a n a l y s i s o f t h e logarithmic equations included: the correlation coefficient (r), 95 p e r c e n t c o n f i d e n c e l i m i t s for r e p e a t e d line fits (sa a n d Sb), a n d t h e s t a n d a r d e r r o r o f t h e e s t i m a t e , S~, w h i c h h a s m u c h t h e s a m e s i g n i f i c a n c e for a l o g a r i t h m i c r e g r e s s i o n line as t h e s t a n d a r d d e v i a t i o n for a m e a n , i.e., t w o SE l i m i t s s h o u l d i n c l u d e 95 p e r c e n t o f t h e cases. W i t h l o g - l o g a n a l y s i s , + S E a n d --SE d i f f e r s l i g h t l y ; t h e v a l u e s s h o w n in t h e t a b l e a r e t h e mean of the two values.
466
Results T a b l e I p r e s e n t s t h e c o e f f i c i e n t s for t h e p o w e r law regression equations, as well as the statistical measures for the relationship of these variables to body weight.
Glomerular capillary tuft volume and body weight. T h e l o g a r i t h m i c r e l a t i o n s h i p b e t w e e n t h e experimentally determined glomerular capillary t u f t v o l u m e a n d b o d y w e i g h t in m a m m a l s e x t e n d i n g o v e r a 1,740-fold r a n g e ( r a t t o horse} in b o d y w e i g h t is s h o w n in Fig. 1, B a n d d e s c r i b e d b y the equation: G T V = 4.30 x 10 6 BW0.26 (1) w h e r e G T V is g l o m e r u l a r t u f t v o l u m e in m i l l i liters a n d B W is b o d y w e i g h t in k i l o g r a m s . A l s o , Fig. 1, B s h o w s t h e r e l a t i o n s h i p for g l o m e r u l a r v o l u m e a n d b o d y w e i g h t in m a m m a l s r a n g i n g 216 t h o u s a n d - f o l d in b o d y w e i g h t ( m o u s e t o elep h a n t ) , u t i l i z i n g v a l u e s r e p o r t e d in t h e l i t e r a t u r e . T h i s r e l a t i o n s h i p is d e s c r i b e d b y t h e e q u a t i o n : G V = 1.30 x 10 -6 B W ~ (2) Glomerular number and body weight. T h e n u m b e r o f g l o m e r u l i in o n e k i d n e y f o r t h e s a m e species o f a n i m a l s s h o w n in Fig. 1, B w a s d e t e r m i n e d f r o m s t u d i e s in t h e l i t e r a t u r e . T h e l o g a r i t h mic r e l a t i o n s h i p b e t w e e n t h e n u m b e r o f g l o m e r -
October, 1976, Vol. 92, No. 4
Similarity of renal glomerular hemodynamics in mammals i
t
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IO 0 10 2 I0 4 BODY WEIGHT (kg) Fig. 1. Logarithmic relationships between body weight and total glomerular volume in one kidney, number of glomeruli in one kidney, and average single glomerular volume in 18 species of mammals (mouse to whale). The closed symbols and solid lines represent data taken from the literature. The open symbols and broken lines represent data obtained by means of our plastic injection casts. Equations for the solid lines are given below the lines; equations for the broken lines are given above the lines. Total gl0merular volume and individual glomerular volume are in milliliters; body weight is in kilograms.
American Heart Journal
0
10-2
I0 -I
/ /,./,~Y A i
#
i.,Y~
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A ~.L" ///~
10 6
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E -t~
GN = 1.42• 105 BW 0"57 / / ~ o -
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r 0
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CATTLE HORSE GOAT MAN SWINE DOG MONKEY CAT RABBIT OPOSSUM
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to 7
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ERPF = 1.75x I0-' BW o.81
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I00 I0 ! I02 BODY WEIGHT (kg)
i03
Fig. 2. Logarithmic relationships between body weight and effective renal plasma flow (ERPF,) and glomerular filtration rate (GFR) in 10 species of mammals (rat to bovine). The data were taken from the literature in which effective renal plasma flow was measured by means of either diodrast or para-amino hippurate and glomerular filtration rate was measured by means of inulin. Body weight is in kilograms,
uli (GN) and body weight for these mammals* is shown in Fig. 1, C. This relationship is described by the equation: G N -- 1.42 x 10 ~ B W ~,57 (3) where B W is i n k i l o g r a m s . Also, Fig. 1, C s h o w s t h e s a m e r e l a t i o n s h i p for m a m m a l s v a r y i n g 1.5 million-fold in body w e i g h t (mouse to whale), utilizing values rePorted in the literature. This r e l a t i o n s h i p is described b y t h e e q u a t i o n : G N -- 9.53 • 105 B W ~ (4) Total glomerular volume. T h e average glomer*Excluding
the goat.
467
Holt a n d R h o d e
ular volume and body weight were calculated for each species in which we prepared plastic glomerular t u f t casts. With the d a t a from the l i t e r a t u r e the n u m b e r of glomeruli was calculated for these animals. T h e p r o d u c t of glomerular v o l u m e and glomerular n u m b e r for each species was calculated, giving the t o t a l glomerular v o l u m e in one kidney for an animal having a body weight equal to the average body weight of the animals studied. T h e logarithmic relationship between total glomerular volume and body weight for these animals varying 1,740-fold in b o d y weight is shown in Fig. 1, A and is described b y the equation: T G V = 591 • 10 -3 B W ~ (5) where T G V is in milliliters and B W is in kilograms. In a similar manner, utilizing values from the literature, total glomerular v o l u m e was calculated for m a m m a l s ranging 216 thou,sand-fold in body weight (mouse to elephant). T h e logalithmic relationship between total glomerular volume and body weight for these animals is shown in Fig. 1, A and is described by the equation: T G V = 137 x 10 .3 BW TM. (6) Effective renal plasma flow and body weight. T h e logarithmic relationship between body weight and effective renal plasma flow t h r o u g h one kidney, measured by para-amino h i p p u r a t e or Diodrast clearance techniques in studies r e p o r t e d in the literature, for m a m m a l s varying 446-fold (rat to man) in body weight is shown in Fig. 2, A and described by the equation: E R P F = 1.75 x 10 -~ BW 0-~' (7) where E R P F is effective renal plasma flow in ml.. sec. ~ and BW is in kilograms. Assuming an h e m a t o c r i t of 0.45, this can be corrected to blood flow through one kidney to give: E R B F = 3.18 • 10 ~ S W ~ (8) Average glomerular capillary length and number. T h e volume of the glomerular capillary t u f t in our plastic injection casts consists of the injected capillaries, intraglomerular parts of the afferent and efferent arterioles, plus space between the capillaries. Although it is not k n o w n what fraction of the injected capillary tuft consists of space between capillaries, separation of the capillaries by breaking the cast into small pieces showed t h a t the capillaries were closely packed together and t h a t there was minimal space between them. Likewise, it is not k n o w n what fraction of the injected capillary t u f t
468
consists of afferent and efferent vessels, b u t presumably this volume is a small fraction of the total. Thus, in the analysis given below for length and n u m b e r of capillaries in the glomerulus we have assumed t h a t in the plastic glomerular t u f t the volume measured is 100 per cent capillaries in parallel circuits. Since the blood flow t h r o u g h all of the glomeruli in one kidney, from the d o w n s t r e a m end of the afferent arteriole to the u p s t r e a m end of the efferent arteriole, is through a system of capillaries in parallel, the total flow is described as a first approximation by Poiseuille's law as follows: Nc AP ~r (re) 4 FGB = (9) 8 Lc~ where F G B is the blood flow t h r o u g h all glomeruli in one kidney in ml. 9 sec. -1, Nc is the n u m b e r of capillaries in all of these glomeruli, rc is the average capillary radius, Lo is the average capillary length, AP is the average pressure drop from the beginning to the end of the capillary, and v is the coefficient of viscosity of blood. It is generally agreed t h a t the coefficient of viscosity of blood, the diameter of the capillaries, and arterial, capillary, and venous pressures do not differ significantly in m a m m a l s of different size. In contrast, the exact value of the pressure drop from beginning to end of the glomerular capillaries is not known. However, recent measurements of this pressure drop in the r a t a n d monkey have shown t h a t it is small, being no greater t h a n 2.5 mm. Hg. ~. 4 7 Substituting in Eq. (9) this value for h p as a first approximation, the mean value for blood viscosity of 0.0228 dynes 9 sec. o cm. -~, and capillary diameter of 8/~ gives:
FGB =
1.46 x 10-9 Nc L~
(i0)
where the flow is in ml. 9 -I Since this flow is the same as that given in Eq. (8), equating the two and solving for Nr gives:
Nc = 2.18 x 108 B W ~
Lo
(11)
where BW is in kilograms. Assuming t h a t the volume of the glomerular capillary tuft measured in our plastic injection specimens consists of 100 per cent capillaries, total glomerular t u f t volume in one kidney equals the number of capillaries in all of the glomeruli times the v o l u m e of the average capillary, i.e.. T G V = No (~rro2Lc).
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Similarity of renal glomerular hemodynamics in mammals
Taking rc to have a value of 4 • 10 -4 cm., t h e equation becomes: T G V = Nc Lc (5.02 • 10-~). (12) T h e t o t a l glomerular volume for the animals t h a t we studied, having distended glomeruli a n d described by Eq. (5), is the same as t h a t given in Eq. (12); equating the two and solving for Nc gives: 117.7 X 104 B W `)84 (13) Nc= Lo Since b o t h this equation and Eq. (11) give t h e total n u m b e r of glomerular capillaries in one kidney, t h e y must be equal to each other, equating the two and solving for Lc gives: Lc = 7.33 • 10 ~ B W ~176 (14) where Lc is in centimeters and B W is in kilograms. Since B W ~176is so near to B W ~ and is within t h e 95 per cent confidence limits of the power functions from which it was derived, it appears reasonable to assume t h a t its value is B W ~ which has a value of 1. T h u s the length of the average capillary is constant, regardless of the size of t h e mammal, and has a value of 733 microns. Substituting in Eq. (13) this value for Le gives the equation for the total n u m b e r of g l o m e r u l a r capillaries in one kidney, i.e., Nc -- 16.1 • 106 B W ~ (15) Velocity of flow in glomerular capillaries. T o t a l capillary cross-section in all glomeruli in one kidney is equal to the n u m b e r of capillaries in the parallel circuits of all glomeruli times t h e cross-section of an individual capillary, i.e., X S = Nc ~r (re) 2 where No is the n u m b e r of capillaries and re is t h e average capillary radius. Substituting the n u m b e r of distended capillaries given in Eq. (15) a n d taking the radius of the average capillary to be 4 microns gives: X S -- 809 • 10 -~ B W ~ (16) where X S is the t o t a l capillary cross-section in all glomeruli in one kidney in square centimeters, and B W is in kilograms. Linear velocity is equal to volumetric flow r a t e divided by cross-section. Thus, dividing Eq. (8) for the flow t h r o u g h all the glomeruli in one k i d n e y by Eq. (16) for the cross-section gives the average linear velocity in the average capillary: V = 0.039 B W ~ (17) where V is in c m . - sec.-' and B W is in kilograms. Since B W -~176in the equation is small and is within the variation of the 95 per cent confidence
American Heart Journal
limits of the power functions from which it was derived, it is assumed to be BW ~ and the e q u a t i o n becomes: V = 0.039 c m . . sec. 1 (18) Thus, the average velocity t h r o u g h the average capillary is constant regardless of the size of the mammal. Time spent in glomerular capillaries. T h e time spent in traversing the length of a capillary is equal to the length divided by the velocity. Since the average velocity t h r o u g h the glomerular capillaries as shown above is 0.039 cm. sec. -1, and the average capillary length is 7.33 x 10 2 cm., the average time blood spends in the capillaries is: T = 1.88 sec. (19) Thus, this time is c o n s t a n t regardless of the size of the mammal. Blood flow per unit capillary surface area in the mammalian glomerulus. Assuming that the glomerular capillary t u f t is m a d e up entirely of capillaries, the total capillary surface area in one kidney is equal to the product of the length and circumference of the average capillary multiplied by the n u m b e r of capillaries, i.e., SAo = Lc Nc 2~r ro. Substituting Eq. (14) for the length of the average capillary and Eq. (15) for the n u m b e r of capillaries and taking capillary radius to be 4 microns gives: ,, SA = 2966 B W ~ (20) where the surface area is in square centimeters and B W is in kilograms. T h e blood flow t h r o u g h all glomeruli in one kidney, regardless of the size of the mammal, is given by Eq. (8). Dividing this equation by Eq. (20) for the surface area of the capillaries in all glomeruli in one kidney gives: F G B / S A = 1.07 • 10 4 BW-003 (21) where F G B / S A is blood flow per unit capillary surface area in m l . . sec. -1- cm. 2 and B W is in kilograms. Since B W -~176is small and is within the variation of the 95 per cent confidence limits of the power functions from which it was derived, it is assumed to be B W ~ and the equation becomes: F G B / S A = 1.07 • 10 4 m l . . s e c . - ' , cm. -2 (22) Thus, the a m o u n t of blood flowing per unit surface area of glomerular capillary per unit time is constant regardless of the size of the mammal. Glomerular filtration rate and body weight.
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Holt and Rhode
Table II. R e n a l g l o m e r u l a r
hemodynamic
constants
and variables in mammals*
Change of value shown in column A (in per cent) Glomerular capillary tuft Hemodynamic parameters
Distended A
AP I
IViscosityl Radius
I
Undistended B
1
(5.)
Hematocrit I
I
r
Capillary tuft I
I (75%)
Glomerular capillary: Length (cm.) 7.33 x 10 -~ Number 16.1 X l0 '~ BW"*:' Cross-section (sq. cm.) 809 x 10 .2 BW"*" Velocity (cm "sec. -~) 0.039 Surface area (sq. cm.) 2966 BW ~,~:~ Time (sec.) 1.88 FGB/SA 1.07 x 10-' (ml. 9cm. ~. sec. ') G F R / S A (ml.-cm. 2.sec.-')0.11 x 10 -~
2.50 x 10 -~ -29 -50 5.5 x 10" BW '~:' +41 +100 274 x 10 ~ BW ''~" +41 +100 0.116 -29 -50 342 BW '.*:~ 0 0 0.22 0 0 9.30 • 10 ~ 0 0 0.95 x 10 -~
0
0
+42 -30 -30 +42 0 0 0
+25 -49 -20 +25 -20 0 +25
+5 -5 -5 -3 0 +8 -8
-5 -6 -6 +6 -11 -10 +12
-13 -14 -14 +16 -25 -25 +33
0
+25
0
+12
+33
*The constant values, regardless of body size, of average glomerular capillary length, velocity of blood flow, time that blood spends in the glomerular capillaries, blood flow per unit capillary surface area (FGB/SA), and glomerular filtration rate per unit capillary surface area (GFR/SA); and the power law equations describing the number of capillaries, total cross-section of capillaries, and total surface area of capillaries in all glomeruli in one kidney are given. The values of the distended capillaries were calculated from our plastic injection casts; the values for undistended capillaries were calculated with the use of data from the literature in which the glomerular capillaries were not distended. These values and equations were calculated taking blood viscosity to be 0.0228 dynes 9sec 9cm. ~-,hematocrit 0.45, capillary radius 4~, the pressure drop from beginning to end of the capillary to be 2.5 mm. Hg, and glomerular tuft volume to be 100 per cent of the measured volume. The values in the columns under "Per cent changes from values shown in Column A" were calculated with the same values employed in the calculations for Column A except that one of the values was changed for the calculations in each column as shown, i.e., hp of 1.25 and 0.625 instead of 2.5 mm. Hg, blood viscosity of 0.0114 dynes 9sec 9cm. -~ instead of 0.0228, capillary radius of 5it instead of 4, hematocrit of 0.40 instead of 0.45, and capillary tuft volumes of 90 and 75 per cent of the measured volume instead of 100 per cent. The logarithmic
relationship
between
glomerular
filtration rate in one kidney and body weight in mammals ranging 438-fold (rat to man) in body w e i g h t is s h o w n i n F i g . 2, B . T h i s r e l a t i o n s h i p is described by the equation: G F R = 3.25 x 10 -~ B W ~ (23) w h e r e G F R i s i n m l . 9 s e c . - ' a n d B W is i n k i l o grams. D i v i d i n g E q . (23) f o r g l o m e r u l a r f i l t r a t i o n r a t e b y E q . (20) f o r t o t a l g l o m e r u l a r c a p i l l a r y s u r f a c e area gives the glomerular filtration rate per unit surface area of glomerular capillaries, GFR/SA, i.e.; GFR/SA = 0.11 • 10 -4 B W ...... (24) w h e r e G F R / S A is i n m l . o c m . - 2 . sec. -1 o f g l o m e r ular capillary surface. Since the power function of B W -~176 is s m a l l a n d w i t h i n t h e 9 5 p e r c e n t confidence limits of the power functions from w h i c h i t w a s d e r i v e d , i t is a s s u m e d t o b e z e r o . Thus, the glomerular filtration rate per unit c a p i l l a r y s u r f a c e a r e a is c o n s t a n t r e g a r d l e s s o f t h e s i z e o f t h e m a m m a l a n d h a s a v a l u e o f 1.1 x 10 -5 m l . 9 c m . -~ 9 s e a . - '
Discussion I t is t o b e e m p h a s i z e d t h a t t h e d a t a g i v e n h e r e are for normal adult mammals varying greatly in
470
size and extending over many species. We believe that the relationships described formulate the similarity criteria which define the normal adult mammalian design of the cardiovascular functions studied. These relationships differ to some extent from species to species and from animal to animal, depending on the degree of adaptation to different environmental situations during the course of evolutionary development. These differences from the normal pattern are limited and define to some degree the deviation from the normal that an animal or species might have and yet still survive. If, i n a p a r t i c u l a r s p e c i e s o r i n d i v i d u a l a n i m a l , i t is k n o w n t h a t o n e o r m o r e o f t h e h e m o d y n a m i c variables differ from the values used in the calculations employed here, then more accurate calculations may be made employing the variab l e s w h i c h d i f f e r f r o m t h e o n e s w e u s e d . T h i s is o f particular importance in the case of the pressure d r o p b e c a u s e a l l t h a t is k n o w n is t h a t i t i s n o g r e a t e r t h a n 2.5 m m . H g . I t m a y b e t h a t i t h a s a v a l u e o f s o m e f r a c t i o n o f 2.5 m m . H g . I n T a b l e I I e x a m p l e s a r e s h o w n w h e r e t h e p r e s s u r e d r o p is t a k e n t o b e 1.25 a n d 0.625 m m . H g , b l o o d v i s c o s i t y 0.0114 p o i s e , c a p i l l a r y r a d i u s 5 m i c r o n s , h e m a t o c r i t 0.40, a n d g l o m e r u l a r t u f t v o l u m e s o f 9 0 a n d 75
October, 1976, Vol. 92, No. 4
Similarity of renal glomerular hemodynamics in mammals
per cent of our m e a s u r e d volume. I t will be n o t e d in the table t h a t the deviation, f r o m t h e values e m p l o y e d in our calculations, was less t h a n 50 per cent in all cases except t w o (capillary n u m b e r a n d cross-section) in which t h e deviation was 100 per cent when the pressure d r o p was t a k e n to be 0.625 m m . Hg. T h i s difference is small w h e n considered in relationship to the over-all p a t t e r n of f u n c t i o n which extends m o r e t h a n 446-fold in b o d y weight. I t is interesting to note in T a b l e I I t h a t t h e t i m e t h a t blood spends in g l o m e r u l a r capillaries is i n d e p e n d e n t of pressure drop, blood viscosity, a n d radius. T h i s suggests t h a t if, in a p a r t i c u l a r species or animal, one or m o r e of these f a c t o r s deviate f r o m t h e values t h a t we h a v e e m p l o y e d in t h e calculations, t h e n this a n i m a l or species h a s an a p p r o p r i a t e change in capillary l e n g t h a n d n u m b e r such t h a t the blood t r a n s i t t i m e t h r o u g h the g l o m e r u l a r capillaries is c o n s t a n t . U n t i l such t i m e as m o r e i n f o r m a t i o n is available on the pressure drop in large a n d small m a m m a l s we c a n n o t be certain t h a t t h e pressure drop is the s a m e regardless of m a m m a l size. I f the pressure drop differs f r o m a n i m a l to a n i m a l or species to species, however, it is to be e m p h a s i z e d t h a t regardless of this the t i m e t h a t blood spends in the capillaries r e m a i n s c o n s t a n t (Table II). In view of the fact t h a t , regardless of m a m m a l size, the factors involved in the f o r m a t i o n of the protein-free glomerular filtrate are a p p r o x i m a t e ly the same, i.e., (1} h y d r o s t a t i c pressure in the capillary forcing filtrate t h r o u g h t h e c a p i l l a r y wall, (2) colloidal osmotic pressure of t h e blood proteins, (3} h y d r o s t a t i c pressure in the collecting tubules of the kidney, and, as shown above, (4) blood flow per unit g l o m e r u l a r capillary surface area per unit time, it would a p p e a r t h a t the n u m b e r a n d average length of capillaries in the glomerulus are related in such a m a n n e r t h a t blood spends a c o n s t a n t t i m e in the capillaries in order t h a t the g l o m e r u l a r filtration r a t e p e r unit surface area of capillaries is c o n s t a n t regardless of m a m m a l size. T h e large a m o u n t of g l o m e r u l a r filtrate formed in the large m a m m a l comes a b o u t as a result of the greater t o t a l g l o m e r u l a r surface area. T h e volumes of the distended g l o m e r u l a r capillary t u f t s t h a t we m e a s u r e d were c o n s i d e r a b l y larger t h a n the glomeruli of the s a m e species reported by o t h e r investigators, 1 9 9 22.23. 30 in which the glomeruli were n o t distended a n d an u n k n o w n a m o u n t of shrinkage h a d t a k e n place
American Heart Journal
during histological p r e p a r a t i o n . I t m i g h t be argued t h a t our distended g l o m e r u l a r capillaries were larger t h a n during life, a n d t h a t as a result the calculated values for t h e h e m o d y n a m i c variables were in error. C o m p a r i s o n s of our values for the above variables with the values o b t a i n e d when calculated f r o m d a t a on u n d i s t e n d e d glomeruli described in the literature, a s s u m i n g as did V i m t r u p 3~ t h a t the glomeruhis consists of 50 per cent capillaries, are given in T a b l e II. As s h o w n in the table, the v~lues of the various p a r a m e t e r s for undistended glomeruli differ considerably f r o m our values. T h e h y d r o s t a t i c pressure in the glom e r u l a r capillaries has been r e p o r t e d to be 45 m m . Hg, and it would a p p e a r t h a t the v o l u m e s of the glomerular t u f t s in o u r plastic casts, in which a high injection pressure was m a i n t a i n e d until t h e plastic hardened, were closer to the v o l u m e s existing during life t h a n would be the v o l u m e with no distending h y d r o s t a t i c pressure in the capillaries. W h e n i n f o r m a t i o n becomes available on a wider range of m a m m a l s , p a r t i c u l a r values given for the p a t t e r n of function t h a t we h a v e described will no d o u b t be changed to s o m e extent. W e believe, however, t h a t these differences will be small, and will n o t alter the description of t h e general similarity p a t t e r n of f u n c t i o n as presented. The authors express their thanks to W. Powell, T. Peterson, M. May, and M. R. Bledsoe for technical assistance. REFERENCES
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