Morphometry of pig coronary venous system W-

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GHASSAN S. KASSAB, DANIEL H. LIN, AND YUAN-CHENG B. FUNGCenter for Biomedical Engineering, University of California, San Diego, La 'Jolla, California 92093

Kassab, Ghassan S., Daniel H. Lin, and Yuan-ChengB. Fung. Morphometry of pig coronary venous system. Am. J.Physiol. 267 (Heart Circ. Physiol. 36): H2100-H2113,1994.—This is a third part of tripartite morphometric data of the pigcoronary blood vessels, giving a complete quantitative description of the arterial tree [Kassab et al., Am. J. Physiol. 265(Heart Circ. Physiol. 34): H350-H365, 1993], capillary network [Kassab and Fung, Am. J. Physiol. 267 (Heart Circ.Physiol. 36): H319-H325, 1994], and venous tree (this article). Together they provide the quantative anatomic foundation for coronary hemodnamics. The coronary venules have aunique morphology. Unlike coronary arterioles, which havecylindrical cross sections and a fairly constant diameter ineach segment, the venules have approximately elliptical crosssections, are usually wavy in the longitudinal direction,, andoften converge like fingers to a hand. Measurements weremade "with the silicone elastomer casting method on five pighearts. Data on smaller vessels were obtained from histologicalspecimens by optical sectioning. Data on larger vessels wereobtained from vascular casts. Arcading veins and anastomoseson the epicardial surface have a unique topology. Data on thenumber of vessels in each order, the major and minor axes,length, connectivity matrix, and the fractions ofthe vessels ofa given order connected in series in all orders of vessels of thesinusal and thebesian veins are presented. It is shown that ofthe blood in the coronary blood vessels of a pig heart 27.4% isin the arteries (>200 pm), 37.1% is in veins (>200 pm), and35.5% is in microcirculation ( < 200 u.m), of which 89.4% is inthe capillaries.heart; sinusal veins; thebesian veins; venules; arcades; anastomoses; connectivity matrix; diameter-defined Strahler system

there is no doubt that coronary vascular disease is oneofthe most important problems ofthe American peopleand that the current research on coronary blood flow isstagnant. One of the reasons for the lack of progress isthat some basic data are missing: for example, a complete quantitative description of the architecture of thecoronary blood vessels, the three-dimensional structureof the capillary network in the whole heart, detailedknowledge of the structure, materials, and mechanicalproperties of the coronary blood vessel wall, and theirbiological regulation by stress, strain, and time. Athero-genesis is related to stress and strain. Without the basicdata named above, however, stress and strain cannot becalculated (neither is there an instrument for in vivomeasurement, nor can the nuclear magnetic resonancemethod meet the challenge). A difficulty in obtaining thedesired basic data is the lack of new ideas in theoreticalorganization of the morphological data on one hand[Horton's idea (12) was published 49 years ago, itsimprovement by Strahler (29) was published in 1952,and its use by Horsfield in lung was began in 1968] and,on the other hand, a lack of new theories of elasticity andviscoelasticity of the blood vessel, which is composed ofnonhomogeneous anisotropic finite strain and nonlinearH2100

materials regulated biologically by stress and strain.Any advance would have to break these barriers.

Our objective is to elucidate quantitative morphometric characteristics of the coronary venous system. Theimportance of this objective is substantiated by epicardial pressure measurements, which have shown that thecoronary venous system may account for up to 33% oftotal coronary resistance during vasodilation (5). Acomplete set of morphometric data of the coronary ,venous blood vessels ofthe pig, which is often consideredas a good model for humans (24), is presented on thebasis of three new ideas described in the text: thediameter-defined ordering system, the connectivity matrix, and the series-parallel distinction. These dataprovide a basis to investigate coronary circulation, topredict the pressure-flow relationship, to determine thelongitudinal pressure and blood volume distribution, tounderstand the distribution of coronary blood flowduring venous retroperfusion to the ischemic myocardium, and to quantitate vascular remodeling in heartdisease.

There are two routes by which coronary venous flowreturns to the heart. In one route, blood flows from thegreat cardiac vein, the posterior vein ofthe left ventricle,the posterior interventricular vein, the oblique vein ofMarshall, and the small cardiac vein into the coronarysinus and anterior cardiac veins on the epicardial surface and empties directly into the right atrium. Inanother route, blood flows through the smallest cardiacveins of Thebesius to the endocardial surface and drainsdirectly into the heart chambers. Detailed anatomicstudies of these coronary veins began in the 19thcentury and received increased attention in the 20thcentury, particularly from the clinical point of viewregarding the benefits of retroperfusion to the ischemicmyocardium (2, 23). For a detailed review of the coronary venous anatomy and circulation, we refer thereader to The Coronary Sinus (21). Our data are separated into the categories of sinusal and thebesian veins.

METHODSStudies were carried out on five farm pigs (Yorkshire and

Duroc crosses) weighing 29-31 kg (30.1 ± 0.7 kg) and 3-4 moof age. The animal and isolated-heart preparations are identical to those described previously (19). Briefly, a KCl-arrestedadenosine-dilated heart was perfused with freshly catalyzedsilicone elastomer through its major coronary arteries [rightcoronary artery (RCA), left anterior descending coronaryartery (LAD), and left circumflex coronary artery (LCX)]. Thearterial perfusion pressure was maintained at 80 mmHg, andthe venous outlet pressure was zero (atmospheric) until theelastomer was hardened. The heart was then refrigerated insaline for several days to increase the strength ofthe siliconeelastomer before preparation for histological and cast studies.

For histological studies ofthe first 4 orders of small venules,a total of 12 plugs of myocardial tissue were removed fromeach ofthe right and left ventricles of four pigs. Each plug was

0363-6135/94 $3.00 Copyright o 1994 the American Physiological Society

MORPHOMETRY OF CORONARY VEINS H2101~4 x 4 mm in cross section and extended from epicardium toendocardium. Each plug was mounted on a freezing microtome and serially sectioned, transverse to the radial direction,to thicknesses of 60-80 pm from epicardium to endocardium'Each section was dehydrated with 100% alcohol and clearedwith methyl salicylate to render the myocardium transparentand the silicone elastomer-filled microvasculature visible inlight microscopy. The arterioles and venules were distinguished on the basis of their topology (19). The topology oftheirtenoles is treelike, whereas that of venules is rootlike

Morphometry of the venules was done by changing the focai.. . T°r . """'^ U1 "1C "i&Luiugicai section (optical

sectioning). An image-processing system described previously(19) was used to measure the dimensions. The venules wereviewed with an inverted light microscope (Olympus, opticalresolution 0.6 urn at x600) and displayed on a color videomonitor (Sony Trinitron) through a television camera (COHUsolid-state camera). The image was grabbed by the computersoitware and analyzed with a digitizing system

The widest width ofthe vessel seen as the cast was rotatedaround its longitudinal axis was measured between longitudinal tangents to the vessel cast. This width is called the majoraxis for the purpose of characterizing the vessel size in thedetermination of order numbers. The narrowest width ofthecast, on the other hand, is called the minor axis. If the crosssection of the vessel perpendicular to the longitudinal axis wasconsidered as an ellipse, then the major and minor axescharacterize the venous cross section.

For the study of the larger veins of order 4 and up theB l a s t o m e r - n e r f u s e H s n H . h c r - A ^ ^ A „ „ ; ■ , ' . . .30% KOH solution for several days. After the tissue was

les are assigned order 1. The smallest venules draining thecapillaries are assigned order -1. When two arterioles of order1 meet, the confluent vessel is order 2 if its diameter exceedsthe diameters of order 1 vessels by an amount specified by a setof formulas to be given below or remains as order 1 if thediameter of the confluent is not larger than the amountspecified by the formulas. When an order 2 artery meetsanother order 1 artery, the confluent is order 3 if its diameteris larger by an amount specified by the formulas or remains asorder 2 if its diameter does not increase sufficiently. This

increasing diameter and assigned orders 1, 2, 3,..., n,....Similarly, the veins are assigned orders -1,-2, -3,'..., -n'....A system of using positive integers to identify the ordernumbers of arteries and negative integers to identify the ordernumber of veins is convenient when we compare the morphology of the veins with that of the arteries.

Now let us explain the diameter-based ordering formulas. Asystem of assigning order numbers to the branches of a treewas first introduced by Horton (12) and used by him to studythe rivulets and rivers in geography. Horton's system wasmodified by Strahler (29). Horsfield and Gordon (11) and Yenetal. (36) used the Strahler system to study the morphology ofthe pulmonary veins in humans and cats, respectively Morphometry have used the Strahler system in different fieldsfor various applications (3, 6, 7, 9,10, 25, 26, 31, 34, 35). In ourstudy of the coronary blood vessels of the heart (18, 19) werecognized that the branching pattern ofthe coronary'vascular

parameter of the branching pattern from the hemodynamic

lcunes Lot any order of a large Strahler tree are stochastic;tree major coronary arteries 1-e> they have a mean and a standard deviation (SD). Let the

sketched according to cnevenous tree was reconstructed completely wilh ,f y p c o n l U ~ i - . . . c i • . . _ . . _■ '

of each vessel segment measured and recorded"1'To measure the minor axis ofthe lumen ofthe cast venous

vessels, two protocols were used. We either cut a cast vesseltransverse to its longitudinal axis and obtained its crosssection for measurement or rotated the vessel specimen in theFetr, dish on the microscope stage to visualize the minor axisof the vessel. The two methods gave similar results. Howeverthis process was very time consuming; hence we measuredonly a sample of vessel segments of each order to obtain amean value ofthe ratio ofthe major to minor axis for variousorders of vessels.

From the data on major axis of each segment and the sketch

^Lll00™?:!,! ee' we, assigned order numbers to the venoussystem

*rs...,D_n-A_n,D_n,D-n + A_n,D^n+i, D-i, D_, + A_,, Dj - Als A,, D, + Alv..,

'"••- We now want to redivide the real axisecutive order numbers. For this purpose

define veins of order -n to be those veins with diameters inthe interval bounded by

[(£_„_! + A_n_{) + (D_„ - A_„)] /2 ( la)on the left, and

l(D-n + A_B) + (D_a+1 - An+1)I/2 (lb)on the right. Similarly, an artery of order n is bounded by

K A , - I + A n _ 1 ) + ( D n - A „ ) ] / 2 ( l c )on the left, and

[ C D „ + A J + ( D „ + 1 - A n + 1 ) ] / 2 ( i d )on the right. We have used Eqs. lc and Id in Ref. 19. We shall

- Eqs. la and lb in this paper. The virtue and weakness ofi approach have been discussed in Ref. 16.

Ab second innovation introduced in Ref. 19 is to combine•'essel segments of a given order but connected in series intonents. The third innovation of Ref. 19 is to describenmetnc branching by a connectivity matrix C(m,n), whose

-ment in row m and column n is the ratio ofthe total numberelements of order m, which spring directly from parent

H2102 MORPHOMETRY OF CORONARY VEINS

, j

Fig- 1. A: photomicrograph of a venule in leftepicardial surface. It shows a large venule drainLV of pig. An 80-^m-thick section taken at -draining into a small vein; 1 cm = 42 ^m

71 p.m. B: photomicrograph of a venule insurface. It shows several adjacent venules

MORPHOMETRY OF CORONARY VEINS H2103

Fig. 2. Cast of coronary sinus veins taken through astereo-dissecting microscope.

elements of order n divided by the total number of elements oforder n.

RESULTS

The branching pattern ofthe venules is very differentfrom that ofthe arterioles. The smallest venules initially

he in the direction of the capillaries they drain, thenusually break away from their original directions, andrun obliquely toward larger veins. The characteristicbranching pattern of endocardial venules has beenreferred to as the turnip root pattern, ginger rootpattern, or as fingers collecting into a hand (see Ref. 1

Fig. 3. Schematic reconstruction ofsome coronary venous patterns.

H2104 MORPHOMETRY OF CORONARY VEINS

Table 1. Major axis, major-to-minor axis ratio, and lengths of vessel segments and elementsin each order of vessels in pig coronary sinusal veins

Segments

- 1 1 0 . 8 ± 1 . 7 8 7 2 0 . 0 5 1 + 0 . 0 4 1 2 5 1- 2 1 7 . 6 ± 3 . 0 6 2 4 0 . 0 5 6 ± 0 . 0 4 1 3 1 3- 3 3 0 . 0 ± 4 . 3 4 2 5 0 . 0 6 3 ± 0 . 0 4 3 2 6 3- 4 5 5 . 5 ± 1 3 . 5 1 3 , 7 1 9 0 . 2 2 3 ± 0 . 1 7 9 1 , 9 4 8- 5 1 1 7 ± 2 5 . 1 1 0 , 2 3 8 0 . 3 0 2 ± 0 . 2 3 9 6 , 4 0 8- 6 2 0 6 ± 4 2 . 4 6 , 6 1 1 0 . 3 6 7 ± 0 . 2 9 7 6 , 1 7 5- 7 3 2 1 ± 6 3 . 0 3 , 5 7 2 0 . 4 4 7 ± 0 . 3 7 6 3 , 4 3 5- 8 4 8 7 ± 9 7 . 5 1 , 7 8 5 0 . 5 9 1 ± 0 . 4 7 4 1 , 7 0 5- 9 7 7 0 ± 1 5 4 6 7 5 0 . 8 5 7 ± 0 . 7 4 3 6 6 2

- 1 0 1 , 1 9 2 ± 1 8 5 2 1 3 1 . 2 6 ± 0 . 9 6 7 2 0 0- U 1 , 9 9 9 ± 7 3 1 1 1 7 1 . 7 2 ± 1 . 3 7 1 0 6- 1 2 5 , 9 1 9 ± 2 , 3 5 3 2 4 2 . 8 4 ± 1 . 4 8 2 3

D, major axis (means ±SD); L, length of segment (means ±SD)-(means±SD).

10.6 ±1.616.5 ±2.729.6 ±3.257.5 ±11.8117±18.7205 ± 25.6317 ±32.7488 ± 46.5773 ± 62.4

1,165 ±88.21,804 ±464

5,919

Elements

0.079 ±.0540.092 ±.0650.117±.0710.350 ±.2770.698 ±.649

1.26 ±1.132.08 ±1.873.62 ±3.196.07 ±5.1110.2 ±8.3225.8 ± 24.2

71.9

D/minor axis

n, no. of vessels measured; D/minor axis, major-to-minor axis ratio

beveral adjacent to distinguish sinusal from thebesian vessels. Hence it i

gvumcuico. luiuiuduuus, quaarincauons, ana quintifi-cations are more frequent in venules than arterioles.The branching ofthe whole arterial tree was found to be98% bifurcations and 2% trifurcations; that of venoustrees was found to be 86% bifurcations, 12.8% trifurcations, 1% quadrifications, and 0.2% quintifications. Thebranching pattern of the endocardial venules is, however, strictly treelike, lacking arcading.

The morphometric data of right and left ventricular(LV) venules, excluding the epicardial venules, werecombined, since no statistical differences were found inthe venular morphometric data of the two ventricles.Tables 1 and 2 show our experimental results of means ±SD of the major axes, major-to-minor axis ratios, andthe length for the segments and elements of sinusal andthebesian veins, respectively. Note that the data of thesmallest venules of orders -1, -2, and -3 were ob-

series. Hence in any order the length of elements is

Tables 1 and 2 show that there are a total of 12 orders ofveins in the sinusal system, whereas there are only 10orders of veins in the thebesian system.

Figure 4 shows the relationship between the meanvessel major axis and the order number for the elementsofthe sinusal and thebesian veins, respectively. Figure 5shows the relationship between the mean ratio of themajor to minor axis and the order number fitted by afourth-order polynomial. Figure 6 shows the relationship between the mean vessel element length and theorder number for the coronary sinusal and thebesianveins, respectively. The curves in Fig. 4 for the sinusaland thebesian veins are fitted by the equations

l o g i o D n = a + b { n ) ( 2 )where D is the length of the major axis, n is the order

Table 2. Major axis, major-to-minor axis ratio,and lengths of vessel segments and elementsin each order of vessels in pig thebesian veins

Segments

D/minor axis

Values for D and L are means ± SD and for D/minor axis are means ±

MORPHOMETRY OF CORONARY VEINS H2105

. 1 0 0 0

100000

- 1 2 - 1 0 - 6 - 4 - 2O r d e r

Fig. 4. Relation between average major axis of vessel elements insuccessive orders of vessels and order number of vessels in sinusal andthebesian veins of pig. Sinusal (□): a = 0.803, 6 = 0.236, R2 = 0.991.Thebesian (x): a = 0.795, b = 0.237, R2 = 0.995.

number, and a and b are two constants. Using theleast-squares method, we obtained the empirical constants a and b listed in Fig. 4. Equation 2 is known asHorton's law (11). The curve in Fig. 4 can be betterfitted by adding a trigonometric term (32)

log10 Dn = a + bin) + c cos [ir(n)/d] (3)where c and d are additional empirical constants. Thephysical meaning of the constant 6 is seen by applyingEq. 2 first to n then to n + 1 and subtracting

Hencelogio A*+i - logio A, = b

D{n + l)IDn = IO6Thus IO6 is the ratio of the major axis of the successivegeneration. The physical meaning of a is obtained bysetting n to be -1 in Eq. 2

a = \ o g 1 0 D _ 1 + b ( 6 )Hence a is the major axis intercept. The physicalmeaning of c is the amplitude of oscillatory deviationfrom Horton's law. d is the wavelength of oscillation interms of order numbers.

The mean element length also obeys Horton's law butwith a discontinuity in the slope at order 3 (see Fig. 6).

1.0- 1 2 - 1 0 - 8 - 6 - 4 - 2

O r d e rFig. 5. Relation between average major-to-minor axis ratio of vesselsin successive orders of vessels and order number of vessels in coronaryveins of pig.

- 1 2 - 1 0 - 8 - 6 - 4 - 2O r d e r

Fig. 6. Relation between average lengths of vessel elements in successive orders of vessels and order number of vessels in sinusal andthebesian veins of pig. Sinusal (G): a = 1.80,6 = 0.085, R2 = 0.984, n =-1 to -3; a = 1.44, b = 0.271, R2 = 0.986, n = greater than -3.Thebesian (x): a = 1.80, 6 = 0.085, R2 = 0.984, n = -1 to -3; a =1.68, b = 0.236, R2 = 0.992, n = greater than -3.

The equation for the element length (L„) can be written

l o g 1 0 L „ = a + b i n ) ( 7 )with a and 6 for orders -1, -2, and -3 being differentfrom a and b for orders -4, -5, and above of sinusal andthebesian veins as listed in Fig. 6. Also listed are thecorrelation coefficients of the fitted curves. It is seenthat the correlation is excellent and the element majoraxis ratios of sinusal and thebesian veins are 1.69 and1.65, respectively, whereas the element length ratios ofsinusal and thebesian veins are 1.77 and 1.61, respectively, for veins of orders -4, -5, and above but 1.18 and1.18 for orders -1, -2, and -3.

The "parallel-series" feature of the network of thevessel segments of any given order is characterized bythe ratio of the number of segments divided by thenumber of elements in a given order (designated asS/E). This numbers ratio is presented in Table 3, plottedagainst the order number n for each system and fitted bya fifth-order polynomial. The fitting is shown in Fig. 7.S/E has the physical meaning ofthe average number of

Table 3. Segment-to-element numbers ratio foreach order of vessels in sinusal and coronarythebesian veins of pig

Sinusal Veins Thebesian VeinsOrder

S/E n S/E n

-1 1.77±0.90 121 1.77 ±0.90 121- 2 1.83 ±1.0 115 1.83 ±1.0 115- 3 1.91 ±1.0 88 1.91 ±1.0 88- 4 1.87 ±1.2 605 1.88±1.3 518- 5 2.33 ±1.7 2,382 2.36±1.6 686- 6 3.47 ±2.7 1,912 3.67 ±2.8 327- 7 4.71 ±3.8 786 5.41 ±3.9 123- 8 6.16±5.0 286 6.05 ±3.8 42- 9 7.63 ±5.3 98 7.14±6.1 22

- 1 0 7.66 ±5.3 38 8.5 ±4.5 8-11 10.5 ±9.0 13- 1 2 24 1

Values are means ± SD. S/E, series-to-element numbers ratio; n,no. of observations.

H2106 MORPHOMETRY OF CORONARY VEINS

« 3 0 t

E * 2 5 ?<U CC

m «,201--. *-J2 J2l5:c EE z 10 : ID)Q ) 5 :

W

0- 1 2 - 1 0 - 4 - 28 - 6

Orde rFig. 7. Relation between average number of segments in series insuccessive orders of vessels and order number of vessels in sinusal (□)and thebesian (x) veins of pig.

vessel segments in series. It is seen that the largestorders have more segments in series. The connectivity ofblood vessels of one order to another is given by theconnectivity matrix [m,n]. The connectivity matrixes ofsinusal and thebesian veins are given in Tables 4 and 5,respectively, which list values as means ± SE.

The total number of elements of each order of thecoronary sinusal veins can be computed from information on the number of intact and cut elements and theconnectivity matrix by a method given in Ref. 19. Whenthis method was applied to the thebesian veins, weobtain the results, mean values ± propagated errors ofthe mean, as presented in Table 6. If the total number ofvenous segments of each order is desired, one need onlyto multiply the number of elements by the ratio of thenumber of segments to the number of elements presented in Table 3.

When the number of elements of veins of order n, Nn,is plotted against the order number on a semilog paperas in Fig. 8, it is seen that the data can be fitted by astraight line. Thus the regression line is

log10iV„ = a - bin) (8)The constants a and b obtained by least-squares fittingof data are presented in Fig. 8. The mean ratio of thenumbers of vessels of successive orders is called the"branching ratio." The branching ratio is given by theantilog of the absolute value of the slope of the lines ofFig. 8. The element- branching ratios of sinusal andthebesian veins are 3.37And 3.05, respectively.

The data on the major axis, -the ratio of the major tominor axis length, and the lengths and number ofelements can be used to compute the total cross-sectional area (CSA) and the blood volume in thecoronary veins of each order. If the venous cross sectionis assumed to be elliptical, then An is equal to theproduct of the area of each elliptical element and thetotal number of elements (see appendix for derivation)

A„ = [(irMXDjj Ay/(major/minor axis)n i9)The total blood volume in all elements of a given order,

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H2108 MORPHOMETRY OF CORONARY VEINS

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the feeding and draining vessels of the arcades can beexpressed in terms of a tree/arcade [m,n] connectivitymatrix. Table 8 shows the tree/arcade [m,n] connectivity matrix for the venous vessels. The mth row is theorder of the vessels draining the arcade, and the nthcolumn is the order of the tree vessels feeding thearcade.

There also exists anastomoses between different veins.Figure 11B shows an example of an epicardial anastomoses between two different veins. Venous anastomoses,however, are not restricted to the epicardial surface.They may be endocardial, connecting sinusal to thebesian veins. Anastomotic veins may have multiple feedingvessels but only two draining vessels. Table 9 shows themorphometric data and the connectivity matrix of theanastomoses found as a function ofthe draining vessels.Means ± SD of the anastomotic vessel diameter andlength are shown. Also shown is the tree/anastomoses[m,n,f] connectivity matrix. The anastomoses connectthe trunks of two major trees together. There are,however, many trees feeding into the length of theanastomoses. The tree/anastomoses [m,n,f] connectivity matrix shows that for a given orders m and ndraining the anastomoses, the order of tree vessels ffeeding into the anastomoses.DISCUSSION

The coronary venous system has a treelike branchingpattern except at the epicardial surface, where arcadesare found connecting the sinusal veins, and at theendocardial surface, where arcades are found connecting thebesian veins. Tables 1-6 describe the morphometry and connectivity of the treelike sinusal and thebesian veins. It is interesting to compare the morphometricdata of the first several orders of veins with those of thearteries (19). The major axes ofthe first several orders ofveins are larger than the diameters ofthe correspondingarteries, the lengths are shorter, and the numbers aregreater. These features imply that the venous system isa lower resistance system. Hence our morphometricdata are in agreement with the epicardial micropressuremeasurements, which have shown that the pressuredrop over the first several orders of veins is smaller thanthat ofthe corresponding arteries (4, 5, 15, 30).

It is well known that the total coronary CSA and bloodvolume vary through the cardiac cycle (20). For amaximally vasodilated relaxed myocardium, the totalCSA and blood volume of coronary blood vessels of eachorder are shown in Figs. 9 and 10, respectively, in whichthe arterial inlet pressure was 80 mmHg and venousoutlet pressure was zero (atmospheric). It can be seenthat the total CSA of various orders of veins are largerthan those ofthe corresponding orders of arteries (19).The accumulative arterial volume (RCA, LAD, and LCX)is 3.0 ml (19), whereas the accumulative venous volume(sinusal and thebesian trees, arcades, and anastomoses)is 4.1 ml. Our venous volume calculations show that indiastole the thebesian veins contain 5% of the volume ofsinusal veins. These values are in agreement with ourexperimental measurements of the volumes of the venous casts. The cast volume of the entire coronaryvasculature was also measured. We found a total coronary vascular volume of 10.4 ml for a 150-g heart (85 gLV). Hence the capillary blood volume is 3.3 ml, obtained by subtracting the arterial and venous volumesfrom the whole coronary vascular volume. The massnormalized coronary blood volume in the arterial, capillary, and venous compartments is 3.5, 3.8, and 4.9ml/100 g LV, respectively. The total mass normalizedcoronary blood volume is therefore 12.2 ml/,100 g LV.The mean blood volumes reported for the coronaryvasculature range from 4.8 to 14 ml/100 g LV forvarious species [see critical review by Spaan (28)]. Thelower values are underestimates, since those heartswere blotted before determination of blood volume,implying that some of the blood volume from the largerarteries and veins was lost.

The pig heart has arcading veins at the epicardial andendocardial surfaces but no arcading arteries. In otherorgans (13, 14, 33) arcading veins are accompanied byarcading arteries. The dog heart, for instance, doescontain arcading arteries along with arcading veins atthe epicardial surface (unpublished observations). Themorphometry and connectivity ofthe pig venous arcadesare shown in Tables 7 and 8, respectively. The arcadediameters and lengths are found to obey Horton's law inrelation with the order numbers. A system obeying sucha law is said to be "fractal." We have shown in Eqs. 2, 7,and 8 and Figs. 4, 6, and 8 that the coronary venoustrees of the pig heart have the characteristic of being

.001 -.- 1 2 - 1 0 - 8 - 6 - 4 - 2

O r d e rFig. 10. Relation between calculated total blood volume of coronaryvenous system in each order and order number of vessels in pig.

MORPHOMETRY OF CORONARY VEINS H2109

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H2110 MORPHOMETRY OF CORONARY VEINS

Table 7. Diameters and lengths of venous arcades ineach order of draining vessels in pig

Order D, jim L, mm n

- 3 21.2 0.341 1- 4 38.9 ±13.6 0.568 ±0.233 11- 5 60.6 ±18.4 0.645 ±0.504 18- 6 117±37.1 1.37 ±1.24 47- 7 180 ±57.2 2.30 ±2.08 67- 8 266 ±86.2 2.95 ±2.68 29- 9 351 ±82.1 3.98 ±3.61 13

- 1 0 460 ±159 9.02 ±7.95 10-11 728 15.5 3

Values are means ± SD. D, diameter of venous arcades; L, length ofvenous arcades; n, no. of arcades measured. CD O COf o mco io . •

fractal. In Ref. 19 we have shown that the coronaryarteries ofthe pig are fractal also.

There are other interesting features in the detailedgeometry of the coronary veins. Figure 3 shows somefeatures not found in arteries. Some parts of the coronary veins look like a sinus, which is a local dilation ofthe vessel without branching; other parts look like asinusoid, which has dense networks of vessels draininginto it.

The thebesian and sinusal veins communicate throughvenous anastomoses, whose morphometric and connectivity data are given in Table 9. These anastomoses arevery important when considering the clinical applicationof retrograde perfusion. Retrograde perfusion throughthe coronary veins, when the coronary arteries arestenosed, results in less capillary flow than expected dueto loss of flow through the thebesian vessels. We havepreviously demonstrated the relationship between theanatomy of the coronary venous system, myocardialfunction, and transmural blood flow during coronaryvenous retroperfusion in pigs (22). Briefly, retrogradeperfusion through the coronary sinus or left anteriordescending vein drains into the chambers of the heartvia anastomoses of the thebesians with the sinusalveins. At low retrograde perfusion pressures, most ofthe blood is shunted from the capillaries through thelower resistance anastomoses. As the retrograde pressure is increased, the epicardial capillaries begin to fillfirst while the endocardial flow is still shunted throughthebesians. As the retrograde pressure is increasedfurther, transmural filling of the capillaries occurs witheventually all the capillaries throughout the thickness ofthe heart becoming filled at higher pressures. At thosepressures, however, small venules "tear" and microhem-orrhage can occur.

Finally, our choice of the characteristic dimension todescribe the size of the vessel must be discussed. Asmentioned before, the shape of the normal cross sectionof the coronary vein is neither circular nor elliptical butsomewhat irregular. When the plastic cast of a vein isexamined while rotating it about its longitudinal axis,one sees that there is one position at which the width ismaximum. We call that width the "major axis" of thevessel, as a short hand ofthe phrase "the length ofthemajor axis of an ellipse that approximate the normal

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MORPHOMETRY OF CORONARY VEINS H2111

Table 9. Morphometry and connectivity oftree Ianastomoses in veins of pig

Order ofDrainers

D, L,mm

Order of Feeders

-1 - 2 - 3 - 4 - 5 - 6 - 7 - 8 -.9 ^SUH - 11 -12

- 6 , - 6 120 2.04 0 1 1 5 1.5 2.5 0 0 0 0 0 0- 7 , - 8 234 4.94 0 1 2 4 2.5 1.5 0.5 0 0 0 0 0- 7 , - 1 0 249 3.35 0 0 0 0 0 1 1 1 0 1 0 0

- 8 , - 8 274 5.90 0 1 2 7 8 3 0 0 0 0 0 0

- 8 , - 1 0 228 4.34 1 1 0 5 5 1 2 1 0 1 0 0

- 11 , - 1 2 655 26.4 0 0 3 15 16 5 7 6 0 1 0 1

- 1 2 , - 1 2 741 41.6 0 0 0.5 6 10 5.5 4.5 2.5 2 2 1 2

Values are means. D, anastomoses diameter; L, anastomoses length.

cross section of a vessel by a plane perpendicular to thelongitudinal axis."

From the point of view of hemodynamics, it can beargued that if one approximates the venous cross sectionas an ellipse, then the length of the minor axis isimportant. We have acknowledged this and presenteddata on the ratio ofthe major to minor axis (Table 1) butnoted that the measurement of the minor axis was timeconsuming. Other important parameters of the normalsection are the CSA, the length ofthe circumference, theresistance to Poiseuillian flow, or the coefficient of theratio of flow divided by pressure gradient. These param-

axes as given in appendix.The morphometric data on the coronary venous sys-

description of the coronary vascular circuit in diastole.Hemodynamic analysis can now be done for a specificcircuit or for a selected set of special circuits, all of whichare consistent with the morphometric data measured.

APPENDIX

Relationship Between Geometric and HemodynamicParameters of an Elliptical Tube and Lengths of ItsMajor and Minor Axes

If the normal cross section of a vein is approximatedby an ellipse, then, relative to a set of rectangularCartesian coordinates x and y with origin located at the

a _ _ . a i . _ a _ • _ a - - t a l . - _ u : : /■■

semimajor axis a and a semiminor axis b arex = a cos 0; y = b sin 9

The CSA isarea = 1/2 $■ ixdy + ydx)

= 1/2 f2w iab cos28 + ab sin20)d6Jo

= tt abAn equivalent circle of radius Re will have the same area if

■T r i ? 2 = T T a 6 ( 2

Re = iab)112 = a(b/a)m

Table 1 shows that (a/6) varies from 1.25 to 1.96 for vesselsbetween orders -1 and -12. Hence Re varies between 0.714aand 0.895a.

The circumferential length of the ellipse iscircumference = f (dc2 + dy2)1/2

= f 2tt (a2 sin28 + b2 cos29)1/2 d8• ' O

which is an elliptical integral involving a and b. It may beargued that the cross-sectional shape of a vein is sensitive tointernal pressure, especially if the transmural pressure isnegative, when the compression of the wall causes elasticremains constant in the buckling process. Hence the circumferential length is a more stable parameter than the major andminor axes a and b. It is, however, difficult to measure this

and any irregularity. We do not think that it is practical tomeasure the circumferential length of the normal section ofthe coronary veins.

The parameters relevant to the flow can be derived from theNavier-Stokes equation. For a steady longitudinal flow of aNewtonian viscous fluid in a long cylindrical tube of ellipticalcross section subjected to a constant pressure gradient. Inanalogy to the exact solution of flow in a circular cylinder, thevelocity profile (u)

u = 2 U [ l - ( x / a ) 2 - i y / b ) 2 ] U 5 )satisfies the Navier-Stokes equation and the boundary condition that u is zero on the elliptical wall described by Eq. 11. U isthe mean velocity over this section. With Eq. 15, the Navier-Stokes equation yields

dP/dx = - 4u.(7[a2 + 62)/(a2 b2)} U6)where p. is the coefficient of viscosity ofthe fluid, x is the lengthalong the longitudinal axis of the tube, and dP/dx is the,pressure gradient. Then the volume rate of flow (Q) is

Q = area U = -nab U = - ir/4ji[(a3fc3)/(a2 + 62)]dP/dx U7)The conductance is given by the coefficient

conductance = -rr/4pX[(a3 63)/(a2 + ft2)] U8)where L is the length of the tube. The resistance to flow isgiven by the inverse of conductance

resistance = 4pXM(a2 + 62)/(a3 63)] (19)Equations 15-19 show that a and the ratio b/a are the mostimportant parameters of venous blood flow in which theWomersley number is < 1.

H2112 MORPHOMETRY OF CORONARY VEINS

Finally, if the cross section is very narrow, the normal crosssection may be better approximated as a rectangular slit ratherthan an ellipse. If h represents the thickness ofthe slit and thetube is rigid, the dP/dx is related to the mean flow velocity bythe equation

d P / d x = - v . U h ~ 2 F ( 2 0 )in which F is a number that depends on the structure of theslit, red cell dimensions, and hematocrit. In this case theelasticity of the tube becomes very important to hemodynamics. Details of this type of analysis is given by Fung (Ref. 8, p.310-332). In the present paper we found the pig coronaryveins not to be very narrow; alb lies in the range of 1.25-1.96(see Table 1). However, it is conceivable that the cross sectionof the coronary veins may become very narrow at certainplaces in systolic condition due to contraction of the heart,muscle force, and ventricular pressure. At the sluicing gates ofthe "waterfall phenomenon" in coronary blood flow, the crosssection is believed to be very narrow. The waterfall theory isdiscussed at length by Spaan (27).

In summary, we can say that the length of semimajor axis ais significant in hemodynamics. The length of the semiminoraxis b is also important, but not necessarily much more soexcept in buckled veins as in waterfall phenomenon. Hence inthis article we present the morphometric data of a and theratio alb. To base the order numbering system on a, however,is because a is easier to measure than b, and the data on a aremore accurate.

We thank Vincent Dickow, Meena Joshi, Karin Obergfell, TinaPatela, and Morris Yen for excellent technical expertise and effort incollecting data on the venous vessels.

This research is supported by National Heart, Lung, and BloodInstitute Training Grants HL-07089 and HL-43026, the AmericanHeart Association, California Affiliate, through Postdoctoral Fellowship 90-51 (to G. S. Kassab), and National Science Foundation GrantBCS-89-17576.

Address for reprint requests: G. S. Kassab, Center for BiomedicalEngineering, University of California, San Diego, 9500 Gilman Dr., LaJolla, CA 92093-0412.Received 5 November 1993; accepted in final form 10 May 1994.

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