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Morphometry of the dog pulmonary venous tree o
R. Z. GAN, Y. TIAN, R. T. YEN, AND G. S. KASSABDepartment of Biomedical Engineering, Memphis State University, Memphis Tennessee 181V- cdDepar tment o f AMES-Btoeng ineer tng , Untvers i ty o f Ca l i fo rn ta , San S^«TS^W
Gan, R. Z., Y. Tian, R. T. Yen, and G. S. Kassab. Morphometry' of the dog pulmonary venous tree. J. Appl. Physiol 75(1)-432-440, 1993.-The biophysical approach to the study ofblood flow in the pulmonary vasculature requires a detaileddescription of vascular geometry and branching pattern. Thedescription ol the pulmonary venous morphometry in the do^is the focus of this paper. Silicone elastomer casts of a dog lun*were made and were used to measure the diameters, lengthsand branching pattern of the pulmonary venous vasculature'I he anatomic data are presented statistically with a diameter-defined St rahler ordering scheme, a rule for assigning the ordernumbers of the vessels on the basis of a diameter criterion Theasymmetric branching pattern ofthe pulmonary venous vasculature is described with a connectivity matrix. Results showthat for the dog's right pulmonary venous tree 1) a total of 11orders of vessels lay between the left atrium and the capillarybed; 2) the average ratios ofthe diameter, length, and numberof branches of successive orders of veins were 1.701, 1 556 and3.762, respectively; and 3) a fractal description ofthe tree geometry resulted in diameter and length fractal dimensions of 2 49and 2.99, respectively. The morphometric data were used tocompute the cross-sectional area, vascular volume, andPoiseuillean resistance in the venous vessels.
vasculature; connectivity matrix; fractal dimension; diameter-defined St rahler system
THE biophysical approach to pulmonary blood circulation is incomplete without quantitative information onthe branching pattern ofthe vascular tree and the geometry (diameters and lengths) of the vessels. In the pastSinghal et al. (16) and Horsfield et al. (8,10) have studiedthe morphometry ofthe human arterial and venous treesin the lung. Using resin casts of human vascular treesthey measured the diameters, lengths, and order of allbranches of blood vessels in the range of 13 ^m to 3 cmtor arterial vessels and 13 urn to 1.4 cm for venous vessels.
The only other species besides humans for which detailed morphometric data for the pulmonary circulationhave been published is the cat. Yen et al. (28, 29) madesilicone elastomer casts of the cat arterial and venoustrees to measure the diameters, lengths, and number ofbranches of arteries and veins. Arterioles and venuleswere measured by optical sectioning of thick histologicalsections. The alveolar capillary network in humans and
^^investigated by Weibel (21) and Fun8 and SobinU, 3). I he structure of the pulmonary alveolar networkcannot easily be modeled by a system of branching andinterconnecting tubes. Instead of large meshworks madeof very short segments, it is more suitable to consider4 3 2 0 1 6 1 - 7 5 6 7 / 9 3 $ 2 . 0 0 C o p y r i g h t c 1 9 9 3
the alveolar wall vascular space as a sheet of blood flowing between two membranes held apart by numerousposts (3).
In addition to the morphological data, the elasticity ofcat pulmonary arteries and veins has been presented bvYen et a (24, 26). The elasticity of arterioles, venulesand capillaries has been given by Sobin et al. (17 18)'Based on a complete set of data on the morphometry ofthe cat pulmonary vascular trees and the elasticity ofthese blood vessels, a theoretical analysis of the bloodflow in the cat lung was presented by Zhuang et al. (30)and Fung et al. (4). However, the bulk of experimentalresults on pulmonary blood flow in the literature available for comparison to the theoretical model is from dogsand humans. Therefore, there is an urgent need to develop a morphometric data base for the dog lun<r
This paper reports the detailed morphometric data obtained from measurement of silicone elastomer casts of adog pulmonary venous tree in our laboratory. The morphometric data for the dog arterial tree are reported separately. With these data a fractal description ofthe pulmonary vascular tree is presented and the Poiseuillean resistance, total cross-sectional area, and blood volume in thevenous vessels were computed.
METHODS
Preparation of pulmonary venous tree casts. The lung ofa mongrel dog weighing 25 kg was prepared by the samemethod reported earlier by Yen et al. (28). Briefly afterthe dog was anesthetized with a pentobarbital sodiumsolution (30 mg/kg iv), the trachea was cannulated andthe lung was ventilated. After a midline incision, theheart and lungs were exposed and the airway pressurewas held at a constant pressure of 10 cmH20 above thepleural (atmospheric) pressure. Through a cannula inserted into the main pulmonary artery, the pulmonaryblood vessels were perfused with a low-viscosity (20 cp)silicone elastomer freshly catalyzed with 3% tin octate(stannous 2-ethyl hexoate) and 5% ethyl silicate Theelastomer was drained from a cannula inserted into theleft atria After perfusion at a pressure of 34 cmH20 for20 min, the atrial cannula was closed and the perfusionpressure was lowered and maintained at 3 cmH20. In thiscondition, the fluid pressure in the capillary blood vesselswas lower than the airway pressure and hence the capillary vessels were collapsed, separating the arterial andvenous trees. After 2-3 h the silicone elastomer was completely hardened, and the lung was frozen for 2 wk toincrease the strength ofthe elastomer. The lung was thenthe American Physiological Society
MORPHOMETRY OF PULMONARY VENOUS TREE
1 7 1 0(29./67,
l 32, 66) i t f .V " ! ^S Y , 6 3 . 1 8 5
l24,\86)
FIG. 1. Sketch of pulmonary subtree with diameter and length measurements in If) «m«nit. <3;„„iQ ,
corroded with a 10% KOH solution for several days. Ondissolution the capillary vessels disappeared, and casts ofthe arterial and venous trees were obtained.
icfi vessel
dicular to the longitudinal axis ofthe vesstance equal to that from one side of theother, a measure of the vessel diameter
1 was cali-lits. In the
Ms millions of
upper, middle, and lower lobes was trimmed into severalmain trees. Small subtrees, with the largest branch having a diameter of 600-800 jum, were cut o^"-^from the main tree. Some small trees wei.selected as samples to be measured in detdirThTrestofthe small trees were further pruned to remove sub-subtrees with a diameter of <200 „m. Each tree was thenviewed under the dissection microscope for measurement. The remaining main trees were also pruned andmeasured. The diameter was measured at the midpointof each vessel segment from the grabbed image. Thelength was measured between bifurcation points alon*the centerhne. The branching pattern of each pulmonary"venous tree was sketched with its measured diametersand lengths. Figure 1 shows an example of a reconstructed pulmonary venous subtree.
Assignment of order numbers to venous branches. Twoof the best-known mathematical models of treelikebranching systems in physiology are Weibel's generations model (21) and Strahler's orders model (29). Wei-bel (21) grouped the human airways in generations byassuming a symmetrical branching pattern. The tracheaI S d * 3 c ; , 0 ' n ; : i t c i r l l t Q n n i . n t m » C \ 4 . U - _ * _ 1 . j 1 i / . . .chi'° 1U" x> """ o" un uowii unetree. ine generationscheme was also used for the pulmonary arterial and ve-
434MORPHOMETRY OF PULMONARY VENOUS TREE
Dn.min
and- [U>«-i+ «)„.,) + (Z)„ - 5DJJ/2
(/)
Di.max = KD„ + SDJ + (Dn+ .-5D„ + ,)]/2 (2)
;lnd A,.maI are the minimum and maximum
•en,™': H^B&SISS&a?' ;" Sf W« •**• sy,stage 2. D: diamettr-dSWI S" n , ' s'Sge '•C: 'Slril,ller °"fers.designated order fo el^ """ "" EilCh 'l'""1'" "•>—„,.,BD. DE. and DP. ~2ZSSS&!SS?JSSr * *C-
ir the bronchial tree and the pulmonarv arteries and
S^S thn generati,°nS apProach' StrahTeroraers Begin at the smallest vessels and count upward Inhis method, the smallest vessels are defined as order1
two order 1 vessels join to form a larger vessel of order 2*and so on toward the main trunk. Strahler- me hod has
teantSSwa SS vers -The fi™-i&ja) is called Horsfield s method of ordering which
conventional Strahler system. When two vess'efs o d,/
ma n "he*" ""*'' ^ °r<ier of the ™othe vessel rt"fter all theXf S* °f *! ,arger dau»h<" vessel.
just one vessel branch (Fig 20 constitutingh1Im!Vi0U?Udi? °n PuIm°nary arteries and veins off t ; ^ f r f a n d c o - w o r k e r s ( 8 , T o L ° di en et al. us, 29) have shown that Strahler's methnd
tt:^te. aSy?metric bifurcations nicely Howeveran unsatisfactory feature of thp QtraMn, « u 'when t daugh vessel0 are nfsmafirlLTthTar5ent but are assigned two different order numbers or whena long vein has many small branches suchTha, the ma n
be In ePi?hrS T* bUt h aSSi«ned a fi«° order num
Dns> hA meter-def™e° Strahler ordering schemeIn the Dn?" '"troduced ^ Kassab et al. (12, 13)In the DDS system, vessel diameter is the man focus
where D«.<,]. £•","" ~ "".max "*•= me minimum and mari'mnmvalues, of the range of diameters around D , respectiveTmmMm
a r . j ' . ™ ? : r ■■ " * •■ » . « » s ismau cnanges in Dn after one or two iterations Finallv
noeodvXer ra"geS °f V6SSelS " SUCC6SSiVe £*«E5.
ordert'thePrdeoS tnnldy' "* haVe USed the DDS »>«hod to
method. The definition of vessels of 0X1 f 11 5description of pulmonary vZus vess Is or M and hu'mans by Yen etal (97 9«^ E»„« iL , . ? dCS ana nu"
present study. Thus, our measured diameters of nrHor 1vessels were greater than those measured by Yenet a
Counting of branches. Because a venous tree of the dL
3S55BS3SI
MORPHOMETRY OF PULMONARY VENOUS TREE
3 . , 2
- h o d t o c J * V & ^ J Z £ ^ £ > « " R a t i o n ' J d o n e f i r s t f o r „ary venous tree. nes m tne PuIm°- sively proceeds to n = n -1 '"'In general, vessels of order „ may snrW ^ *_ .... ^includes all the missing 8ubSS, t!' ~ J***?0"«w oi oraers n + 1 n + 9 t«*U 1- —. ves" UI mzact vessels in ordpr n (M \ ■ . uuraoei
ssels 3*1 Ti tow atthWhen ^ nUmber 0f Cr i t" ia "SSSKft ff lEl ^°rph°10^ 04). Th,m - 1. •. 2 and 7 Jh.T '., 6 numbers of orders geneity, self-similaritv «^w fge degree of hetero-- f o r d e r m c t r / a S f t C t T £ " ° ^ ^ S S W J W S r *
-nffnrk,«i "Iuerm tne missing subtrees of the mensinn ;« „.«*.i • , s l°'. **'• rne conceDt of di.
'■■1 * - . . . ^ „ , S ^ S t S & s i S S r l -tal description ofthe do*? n U? 't0 make a frac"
MORPHOMETRY OF PULMONARY VENOUS TREETABLE 1. Experimental measurements of diameter andlength of vessels in each order of dog venous tree
Diameter, nm
rder Means ± SDNo. of vesse
measured
11 8,450±250 210 4,480±235 69 2,775±354 208 1,826+137 287 1.167±113 556 737±101 1685 441±70 3204 251±39 4913 142±24 9792 78±14 1,1221 29±12 2,407
Length, mm
Order Means ± SD measured ' Means t SD "t^Sf
3 4 . 4 0 io 2 0 . 5 1 6 ± 9 . 8 0 5 f i
™ 1 4 . 7 6 5 ± 7 . 9 8 9 1 3■8 1 2 . 6 5 3 + 6 . 6 2 4 1 9
f i l , J S S J ? i l 8 . 1 4 8 ± 5 . 3 6 4 3 86 . 3 , ± 1 0 1 1 6 8 5 . 1 9 0 ± 3 . 5 0 1 ?5 4 4 1 ± 7 0 3 2 0 2 . 9 7 5 ± 1 . 8 5 4 1 0 9
2 5 1 ± 3 9 4 9 1 1 . 7 6 7 , 1 . 2 0 1 2 1 23 1 4 2 ± 2 4 9 7 9 1 . 0 9 3 ± 0 . 8 2 1 9 3 72 7 8 ± 1 4 1 , 1 2 2 0 . 6 8 3 ± 0 . 4 3 4 2 2 9
2 9 ± 1 2 2 , 4 0 7 0 . 4 0 1 ± 0 , 2 4 8 j - l SValues were obtained at pleural pressure = 0 cmH.,0 (atmospheric)
airway pressure = 10 cmH20. and pulmonary venous pressure "3
Mean t S.D.
1 _ D = log (total size of object for order n)log(average size of object for order n) (4)
The fractal dimension is then obtained from plotting onlogarithmic scales for both axes, the total size versus theaverage size of the object.
RESULTSOur data of the pulmonary venous tree in the do*'s
right lung show that there is a total of 11 orders of pulmonary veins lying between the capillaries and the leftatrium. Table 1 shows our experimental results of themeans ± SD of the diameters and lengths of vesselbranches as well as the number of vessels measured,irom these data, the relationship between the mean diameter of vessels in each order and the order number isshown in Fig. 4, and the relationship between the mean
5 - Mean ± S.D.
9 1 0 1 1
FIG. 4. Average diameter of vessels in each order vs. order number
FIG. 5. Average length of vessels in each order vs. order number Ifv
3KR n0?Q9nh^ °f'e,ngth ^ X repr6SentS °rder numb". thenylU.oJUb + 0.1920x describes results.
values of vessel length and the order number is depictedin big. 5.
In Fig. 4, if y represents the logarithm ofthe diameterand x represents the order number, then the least-squares fit regression line is y = 1.4128 + 0.2307* Theslope gives an average diameter ratio of 1.701. Similarlyin F ig. 5, if y represents the logarithm ofthe vessel lengthand x represents the order number, then the regression
te 1S K r«?"5206 + °-1920x> y[eldinS an average lengthratio of 1.556.The connectivity of blood vessels of one order to an
other is expressed in the connectivity matrix for whichthe component C(n,m) is the ratio ofthe total number ofvessels of order n sprung off from parent vessels of orderm to the total number of vessels of order m. To experimentally obtain the matrix for the average branchingpattern in the venous vasculature, we examined thewhole tree for all the branches ofthe casts of orders 1-11.
As a simple numerical example, the tree in Fig 2D isused to calculate the element C(l,m), where m = 1 2 3and 4. There are three vessels of order 2 that give rise to'eight vessels of order 1 and one vessel of order 3 thatgives rise to one order 1 vessel. Thus, C(l,2) = 8/3 or 2 67the average ofthe three ratios of 3, 3, and 2, over all order2 vessels with a standard deviation of 0.47; likewise,MMJ - 1/1 or 1. Because there is no order 1 vessel risingfrom order 1 or 4 C(l,l) = 0 and C(l,4) = 0. By similarreasoning the other components of C(n,m) and theirstandard deviations can be derived.
The experimentally obtained connectivity matrix ofthe pulmonary venous tree ofthe dog is shown in Table 2In addition to displaying the global branching pattern ofthe whole vascular tree, the connectivity matrix providesample information for calculating the number ofbranches in each order. For instance, the last columnshows that the largest (order 11) vessels in the pulmo-
table 2. Connectivity matrixMORPHOMETRY OF PULMONARY VENOUS TREE
1 0.690*1.207 2953±1.758 1.517±1.943 1.072±1.740 1.010±1.783 .- 0 .263±0 .o l8 1 .716±1 .504 1 .478±1.503 1 .559±1 536 «»•«• -
J 0 . 6 5 9 ± 0 . 8 3 8 1 . 7 7 0 ± 1 . 1 2 0 1 . 5 3 9 ± L 5 2 6 \5 0 . 1 9 1 ± 0 . 4 4 0 1 . 7 9 4 ± 1 . 1 1 1 L _ .3 0 . 2 1 6 ± 0 . 4 3 6 2 . 0 4 87 0 . 4 1 5 :
Values are means ± SD. Element (n, m) in row n and coluibranches of order m to total number of branches SSm in ^VZous^
1 . 0 9 1 ± 1 . 2 2 1 0 0:i.on 1.727±1.489 0.4±0.548 0
'■" - - T- l ' o69 L727± l -009 0 .4±0 .548 00.143±0.3oo 1.471±0.800 1.636±0.809 0.8±0 837 00.235±0.397 1.545±0.522 0.8±0.837 o
0.091±0.301 2.2±0.837 00 6
nary veins are connected to six vessels of order 10 twovessels of order 8, and no vessels ofthe other orders The
T^aTI ?WS that n°ne of the order 9 vessels is'cor(9 o v°vhe aogoSt VeSSGl °f 0rder U> 26 vesse*s of ord.(2.2 X Nl0 = 2.2 X 12 = 26, where 12 is the numb
essels of order 10 from Table 3) are connected to ve:oi order 10, and three vessels of nrdpr Q (c\ not v ^^
%T }e 3) are connected to vessels of order 9lhe total number of branches of each order in the venous tree can be computed from the information given inrabies 2 and 3 by using Eg. 3. The computation belrom the largest vessels of order 11, i.e., n = 11 in3, and then proceeds successively to n = W9 l 1 Ml.results are shown in Table 3. The number'of intact vessels and cut-off or broken-off vessels and the total number of branches in each order are also listed in Table 3
.Jhnrd! 10f.uiP b!tWGen the number of branches ineach order and the order number is shown in Fig 6 F—the logarithm of the number of branches (y) in each oiuis plotted against the order number (x). The regressionfine is y = 6.5602 - 0.5754*, yielding an average branchmg ratio of 3.762 over the 11 orders
JS ? ?l°tS the total diameter ^suiting from thecollection of diameters-of all vessels in a given order vs.
ameter obtained from the regression line is 2.49.
TABLE 3. Computation of total numhor -' brandsin each order of dog's venous tree
Figure 8 plots the total branch length resulting fromthe summation of the lengths of all vessels in a given-der vs. the average branch length for the same order
asels on a log-log scale. Each point on the graph corresponds to a specific branching order. The slope of the
iSTm^ *' GS ^ fraCtal dimensi°n of the branchlength of the dog s pulmonary venous tree as 9 99
Ji1™66" ^°m ^ ? and 8 that the Pu^onary venoussystem is a fractal structure that provides a mechanismfor converting an alveolar-capillary surface area of dimension two into a volume of dimension three. The esti-
Z ehd™alU|f1 °f 2f ^^u dimensi0n for vesseI diameterand branch length may be useful for developing struc-
Zi!ZTng frf°nS- H°WeVer' °Ur aPProach forractal analysis of pulmonary vasculature is derived fromthe morphometric measurement of one dog's lung castswhich is not sufficient to complete the study of fractalgeometry for the pulmonary vascular tree of the dog
No. of Intact No. of Cut• _ V ^ . V e s s e l , B - n g & S f y — f c . C o r ^ e ^ o t l ,
14584 842101 181 3,1074
32
205215
258717
10,11177,478199 903 213,540596 1,046 2,474,291
197989
3,38910,57478,410
214,6422,475,933
ORDERFIG. 6. Number of branches in each ord
represents logarithm of number of branch-
MORPHOMETRY OF PULMONARY VENOUS TREEHoweve r, ou r p resen t r esu l t s se r ve as a f ounda t i on f o r A = ( i r / 4 )n2No t h e r m o r p h o m e t r i c r e s u l t s t o f o l l o w . " K * / * ) I J * N m ( i
On the basis of the morphometric data of the pulmo- ^f® °n u the averaSe diameter of vessels and N is thenary venous vasculature, the total cross-sectional area ™tal number of vessels of order n. The results shown inand vascular volume in every order can be computed fpu afe Wlth an exP°nential function,along the venous pathway. Figure 9 shows the variation t0tal bl°°d volume of order n (V„) is given byo f t h e t o t a l c r o s s - s e c t i o n a l a r e a w i t h t h e o r d e r n u m b e r i n V = A L i a \t h e v e n o u s t r e e . T h e c r o s s - s e c t i o n a l a r e a o f o r d e r n ( A ) " K }is defined as the product of the area of each vessel and °rt h e t o t a l n u m b e r o f v e s s e l s V „ = ( i r / i ) D l L n N n { 6 b ) ,
5 w h e r e L n i s t h e a v e r a g e l e n g t h o f v e s s e l s o f o r d e r n T h e1 0 1 B D ? 4 Q c a l c u l a t e d v e n o u s b l o o d v o l u m e s a r e p l o t t e d o n a l o g -F " anthmic scale against thei r order number in Fig 10 I t is
seen that the data may be fitted by a straight line' If yrepresents ^e logarithm of the blood volume and x rep-
AVERAGE DIAMETER OF VESSELS (mm)FIG. 7. Fractal dimension (DK) for vessel diameter (total vs average
vessel diameter) was computed from points. Each point corresponds tospecific branching order.
0 1 2 3 9 10 11 12
D =2.99F
fig. 9. Computed cross-sectional area of venous branches in eachorder vs order number. Uy represents cross-sectional area and x represents order number, then v = 19.098 X UT*»*X describes results
AVERAGE BRANCH LENGTH (mm)FIG. 8. DF for branch length (total vs. average branch length) was
computed from points. Each point corresponds to specific branching
1 0 11 1 2
-0.2398 + 0.0780x describes results.
table 4. Comparison of diameter, length, andbranching ratios and number of orders in pulmonaryvenous tree of dog, cat, and human Pulmonafy
MORPHOMETRY OF PULMONARY VENOUS TREEr , l e n g t h , a n d , - _ n . _ n _ ,-ders in pulmonary g °n ~ (log ^ " loS *>*) + « log DR (U), a n d '
Human 1.682 (7-14)1.493 (1-6)
1.5561.533 (4-10)2.402 (1-3)1.679 (7-
tively.
log Ln - (log Lx - log LR) + n log LR (12)ct forms of fifc U and 72 for the dog's venousillustrated in Figs. 4 and 5. The diameter anditios were calculated as 1.701 and l.SsTrespec
1 . 4 9 3 ( 1 - 6 1 . 4 7 fi M .DR, diameter ratio; LR, length ratio; BR, branchine r
ratios are calculated from total number of orders exce
SSSESSSSSSgiiisF
3r 2 branches in humane ««~ u_^_. ' ' v 'The morphometric data ofthe pulmonauvide three system ratios: the diametern c h i n o - r a t i n e T K Q „ , . - / • ? its, which was pointed out by
3, in response to this interpreta-e has 12 DDS orders compared
b e r * ~ * " " — " • " * " W 1 w w o r d e r n u m -
K ^ N J B U ) 1 - " ( 7 )«r?? iSiue 3Verage branching ratio for the systemMM£" ^^ ^ l0garithm °f ^ sidfsTf
log Nn = (log ^ + log BR) - n log BR (8)By represents the logarithm ofthe number of branchesand x represents the order number then Fn lil+Ztleast-squares fit regression line for the plotting of ttnumber of branches against the order number The antiog of the absolute value ofthe slope o"the^regressionline is defined as the average branching ratio For hepulmonary venous tree, Eq. 8 is presented in Pig" 6 ash2 7 °'5.754x and the branchins «*io ksmHorton s law is statistical in nature and provides in-
formation on average-properties. It can also be applied tomeasurements of vessel diameter and branch length forLn<T m°nary ™ous tree- Becau*e the diameter andAhZSZ Wlth f'thG 3Verage *2S?5 vessels (A,) and the average length of branches (L ) in eacho r d e r a r e p r e s e n t e d a s n
A . - A C D R ) — 1 ( Q )and
A ^ L ^ L R ) " - ' ( i 0 )vhere DR and LR are the average diameter and lonarh
0 1 2 3 4 5 6 7
ORDER9 1 0 1 1 1 2
each'order ^rd:' SSTkLS^T °f ~«5 bra"cheS inbranching order. 8Ch point corresP°nds to specific
440 MORPHOMETRY OF PULMONARY VENOUS TREE
orAP„ = (128MnLn/^X)Q„
AP„ = R„Q„
(23)
(14)where AP„ is the pressure drop at order n, nn is the apparent viscosity of the blood, Q„ is the total flow in order n,and R„ is defined as Poiseuillean resistance of order nand is given by the terms within parentheses in Eq. 13.Obviously, the Poiseuillean resistance considers only themorphometric aspects of the vessels.
To calculate the Poiseuillean resistance in each order,the apparent viscosity values of blood should be estimated first. According to the results determined by Yenand Fung (25) in model experiments and calculations, anapparent viscosity of 4.0 cp is assumed in large vessels.The apparent viscosity of blood in small vessels of orders1-3 is obtained by interpolation as 2.5, 3.0, and 3.5 cp,respectively.
With these data the Poiseuillean resistance fromorders 1-11 in the dog's venous tree was computed, andthe results are shown in Fig. 11. It is seen that most oftheresistance occurs in the small vessels of diameters <100nm and that the resistance in the small vessels is 55%of the cumulative Poiseuillean resistance in the venoussystem.
In addition to the Poiseuillean resistance, the bloodvolume distribution within the venous bed provides afunctional analysis for pulmonary circulation. The results of Fig. 10 show that blood volume increases withorder number, and the average vascular volume in ordern (V„) is presented as
V„ = V^VR)' (15)where VR is the average volume ratio. The best fit relationship between the average blood volume and the ordernumber is also illustrated in Fig. 10. The equations resultin the vascular volume ratio of 1.197 for the pulmonaryvenous tree of the dog. The cumulative venous volume is22 ml in the dog's right lung, where 50% of the totalvenous volume resides in the largest three orders of thevenous tree.
Results obtained from measurements of vascular volume in the lung (6, 9, 10, 20) are quite variable. Thesewide variations may be related to the different physiological states and experimental conditions for data measurement. Horsfield (9) studied the cast model ofthe humanpulmonary venous tree and showed that the total venousvolume was 73 ml, compared with an arterial volume of86 ml (excluding the main pulmonary artery). Comparedwith this result, the 22-ml volume of blood in the venoustree in the dog's right lung is reasonable.
Address for reprint requests: R. T. Yen, Dept. of Biomedical Engineering, Memphis State Univ., Memphis, TN 38152.Received 27 April 1992; accepted in final form 22 February 1993.
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