Morphometry of the human pulmonary vasculature
W. HUANG, R. T. YEN, M. MCLAURINE, AND G. BLEDSOE
Department of Biomedical Engineering, The University of Memphis, Memphis, Tennessee 38152
pulmonary artery; pulmonary vein; diameter-defined Strahler
system; connectivity matrix; vessel element; vessel segment;
fractal dimension
of the morphometry of pulmonary
vasculature has been summarized by Miller (20). In
1963, Weibel (26) published his monumental work.
Cumming et al. (1), Singhal et al. (22), Horsfield (7–9),
and Horsfield and Gorden (11) then used Strahler’s (25)
system to study the morphology of the human pulmonary arterial and venous trees. Using resin casts and
vascular injections of human vascular trees, they measured the diameter, length, and order of all branches of
blood vessels in the range of 13 µm–3 cm for arterial
vessels and 13 µm–1.4 cm for venous vessels. Using the
silicone elastomer casting method, Yen and Sobin (28)
reported the diameter data in the range of 18.7–1,785
µm for human pulmonary arteries and 18.9–58.9 µm
for human pulmonary veins; similar results were reported for the cat (29, 30). However, major gaps in
knowledge remain, and a more complete set of morphometric data for the human lung is needed for medical
applications.
Horsfield (7–9) encountered a number of difficulties
with Strahler’s idea and introduced several improvements. In detail, there are already three versions of
Strahler’s scheme (8). Yet, some major difficulties remain: 1) all vessels of the same order are treated as
parallel, despite the fact that some are connected in
series, and the series-parallel feature is not given a
quantitative expression; 2) the range of diameters of
the vessels in successive orders has extremely wide
overlaps; 3) the connectivity of asymmetric branching
has not found a mathematical expression; and 4) the
long main pulmonary artery, which is tapered, has to be
THE EARLY HISTORY
given a single order number. Obviously, these difficulties would create dilemmas when one applies morphometric data to hemodynamic circuits. Recently, three
innovations (12–14) were introduced to ameliorate
these difficulties: 1) a new criterion based on the vessel
diameter change at points of bifurcation was adopted in
Strahler’s ordering system; 2) the concept of segment
and element was used to express the series-parallel
feature of blood vessels; and 3) the connectivity matrix
was introduced to describe the connectivity of blood
vessels among different orders. The details of these
three innovations have been discussed by Jiang et al.
(12) in their study of the rat pulmonary arterial tree.
The new method is called the diameter-defined Strahler
ordering system. Gan et al. (5) used the new system to
describe the dog pulmonary venous tree.
The objective of this study is to describe the morphometry of human pulmonary vasculature by including
innovations 1–3. The data are intended to provide a
basis for formulation of hemodynamic circuits for the
analysis of blood flow in the human lung. Although
hemodynamic analysis is not presented, the data should
help interpret clinical observations. Much literature is
available on clinical investigations of the human lung.
Additional theoretical analysis supported by morphometric data to the clinical researchers’ chest of tools will
obviously be helpful.
METHODS
Specimen Preparation
This study was carried out on two postmortem human
lungs (Table 1). In both cases, the cause of death was
accidental and did not involve the lung. The pulmonary
arterial cast was obtained by antegrade perfusion in the left
lung of a 44-yr-old man, and the pulmonary venous cast was
obtained by retrograde perfusion in the right lung of a
24-yr-old man with the silicone elastomer casting technique,
which was introduced by Sobin (23). This technique has been
used in the study of the morphology of the pulmonary
vascular trees of cats (24, 29, 30), dogs (5), and rats (12). The
silicone elastomer used in the present study was the same as
that used in the earlier studies; it is a clear and colorless fluid
that can pass through the capillary bed (24) and has a low
viscosity (4), a low surface tension (24), and a negligible
volume change on catalysis (24). It is also nontoxic to the
endothelium (23).
The casting procedure is briefly described below. The lung
was placed in cold saline solution during cannulation and
perfusion. The trachea was cannulated after gentle suction to
remove retained secretions. The airway pressure was then
held constant at 10 cmH2O above the pleural pressure (Ppl),
which was atmospheric. The pulmonary artery and pulmonary veins were cannulated. The lung was initially inflated to
20 cmH2O and cyclically inflated from 5 to 15 cmH2O with
periodic inflation to remove areas of superficial atelectasis.
After the lung was well inflated, a small amount of noncatalyzed fluid silicone elastomer with a low (20 cP) viscosity
(Microfil CP-101, Flow Tech, Boulder, CO) was perfused from
0161-7567/96 $5.00 Copyright r 1996 the American Physiological Society
2123
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Huang, W., R. T. Yen, M. McLaurine, and G. Bledsoe.
Morphometry of the human pulmonary vasculature. J. Appl.
Physiol. 81(5): 2123–2133, 1996.—The morphometric data on
the branching pattern and vascular geometry of the human
pulmonary arterial and venous trees are presented. Arterial
and venous casts were prepared by the silicone elastomer
casting method. Three recent innovations are used to describe
the vascular geometry: the diameter-defined Strahler ordering model is used to assign branching orders, the connectivity
matrix is used to describe the connection of blood vessels from
one order to another, and a distinction between vessel segments and vessel elements is used to express the seriesparallel feature of the pulmonary vessels. A total of 15 orders
of arteries were found between the main pulmonary artery
and the capillaries in the left lung and a total of 15 orders of
veins between the capillaries and the left atrium in the right
lung. The elemental and segmental data are presented. The
morphometric data are then used to compute the total
cross-sectional areas, blood volumes, and fractal dimensions
in the pulmonary arterial and venous trees.
2124
MORPHOMETRY OF PULMONARY VASCULATURE
Table 1. Specimen information
Specimen
No.
Age, y
Sex
Hours
Postmortem
Preparation
Perfusion
Direction
Body
Weight, kg
Body
Height, cm
1
2
44
24
Male
Male
18
6
Left lung
Right lung
Antegrade
Retrograde
95
128
185
180
D/D10 5 A(P v 2 PA) 1 B
for 210 , Pv 2 PA , 10 cmH2O
(1)
where A and B are constants that vary with D10 and Pv 2 Ppl.
According to fact 1, the capillaries are collapsed and dissolved. Lamm et al. (18) showed that the alveolar corner
vessels (those at the junctions of interalveolar septa) will also
collapse when Pv 2 PA is less than 28 to 216 cmH2O. At our
experimental condition Pv 2 PA 5 27 cmH2O, few corner
vessels were seen. According to fact 2, the relative sizes
(ratios of sizes) of the vessels of successive orders will remain
approximately the same whether the ratio is measured at 27
cmH2O (as in our preparation) or at 10 cmH2O (at lower end of
in vivo values). Because the diameter-defined Strahler ordering method depends only on the ratio of the vessels of
successive orders at points of bifurcation, our method of
preparation will not affect the assignment of order numbers
to vessels. The uncertainty is that the range 210 cmH2O ,
Pv 2 PA , 10 cmH2O is that of the cat (27), and the exact
range for humans is unknown. The possible species difference
must be checked in the future.
Morphometric Measurement of the Polymer Cast
of the Vasculature
The pulmonary vascular casts were dissected and viewed
with a zoom stereomicroscope (model SZH, Olympus). An
image-analysis system was set up to measure accurately the
size of the vessels. The system consists of a Zenith computer
with a DT2851 (Data Translation, Marlborough, MA), an
inverted light microscope (model SZH-ILLB, Olympus), a
video monitor (Sony Trinitron color video monitor), and a
television camera (Cohu solid-state camera). The solid casts
were viewed with the inverted light microscope and displayed
on the video monitor through the television camera. The
image was analyzed with the software package Optimas
(BioScan). The Optimas computing program focuses on the
image of a blood vessel chosen by the operator. By photo
density contrast, the computer draws the boundary contours
of the object. For diameter measurement, the program computes normal vectors to the contour, draws two neighboring
normals to define an area, measures the area, and computes a
width equal to the area divided by the length between
normals. We use the word ‘‘diameter’’ to indicate the computed width of the vessel. Three diameter measurements
were made along each vessel to obtain a mean diameter. A
section normal to the vessel contour can be drawn on the
screen. The centers of the normal sections are joined by the
operator, and the line is considered to be the centerline of the
vessel. This centerline is that of the two-dimensional image.
The intersection of the centerlines of two intersecting vessels
is the bifurcation point. The vessel segmental length was
obtained by measuring the length between two successive
bifurcation points along the centerline of a vessel on the
two-dimensional image. Tacitly, we assumed that the blood
vessels were round. Actually, the cross sections of the large
pulmonary veins were found to be noncircular, but in the
present study this matter was not pursued. An analysis of the
‘‘errors’’ caused by these projections is made by Yen et al. (30).
They found that if the diameter of a blood vessel with an
elliptical cross section was measured by projection from
arbitrary directions, the mean diameter of a random sampling of the projected width is quite close to the diameter of a
circular cylinder of the same circumference.
Because there are many branches in the human pulmonary
arterial and venous trees, it was impossible to examine,
measure, count, and list every branch. Therefore, pruning
and statistical methods were used to obtain representative
measurements (5, 12–15, 29, 30). The backbone of the left
pulmonary artery was sketched, and its segments were
measured. The subtrees arising from the backbone were
labeled, excised, and placed in separate dishes. To facilitate
the measurement, daughter trees with a diameter of 600–800
µm were trimmed from each subtree. Daughter trees were
randomly selected as statistical samples from each subtree
and measured in detail. In the statistical samples of the
daughter trees, branches with a diameter ,100 µm were
pruned. A small number of the branches with a diameter
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the pulmonary artery to the pulmonary veins to establish
vascular continuity across the lung. This step was followed by
perfusion with silicone elastomer freshly catalyzed with 3%
tin octoate (stannous 2-ethyl hexoate) and 5% ethyl silicate.
Before perfusion the catalyzed solution was well stirred, and
no air bubbles remained. The perfusion was carried out under
a pressure drop of 34 cmH2O from inlet (34 cmH2O) to outlet
(0 cmH2O) for 20 min, while the alveolar gas pressure was
maintained at 10 cmH2O and Ppl was zero (atmospheric).
Then the perfusion pressure was lowered and maintained at 3
cmH2O, and the left atrial cannula was closed. The time
course of hardening and the flow behavior of the catalyzed
silicone elastomer in the first 2 h are discussed in detail by
Fung et al. (Appendix in Ref. 4). The hardening was so slow
within 1 h after the flow stopped that it was possible for the
fluid to redistribute itself with the requirement of equilibrium
(4). Because the fluid pressure in the capillary blood vessels was
lower than the alveolar gas pressure under this condition, all the
capillary vessels were collapsed (4, 29, 30), separating the arterial
and venous trees. After 3 h the cast lung was moved to a
refrigerator and frozen for 2 wk to increase the strength of the
silicone rubber. Then the lung was carefully removed and suspended in a 10% KOH solution for 2 wk to dissolve the lung
tissue. Next, the cast was washed several times with water to
remove any remaining tissue. The pulmonary arterial and
venous trees were gently separated. Dimensional measurements were carried out on the pulmonary arterial cast of a left
lung and the pulmonary venous cast of a right lung.
Our method of preparation relies on two facts: 1) the
pulmonary capillaries collapse when the pulmonary capillary
blood pressure is lower than the alveolar gas pressure by $1
cmH2O (27); and 2) the pulmonary arteries and veins do not
collapse when the pulmonary blood pressure falls below the
alveolar gas pressure (4). In fact, according to Yen and
Foppiano (27), for the cat, the slope of the vessel diameterpressure difference DP 5 Pv 2 PA (where Pv is the blood
pressure and PA is the alveolar gas pressure) does not change
in the range of 210 to 110 cmH2O. The slope of the
normalized vessel diameter D/D10 (vessel diameter divided by
diameter at Pv 2 Ppl 5 10 cmH2O) depends somewhat on the
Ppl and vessel diameter. The relationship can be expressed as
MORPHOMETRY OF PULMONARY VASCULATURE
2125
,100 µm from each lobe of the lung were randomly selected
as small sample trees and measured in detail. This process
was continued until the entire tree cast was sketched and the
morphometric measurements were made. The same process
was used to obtain measurements in the venous tree. With
the morphometric data on the vascular geometry and branching pattern from the backbones, subtrees, daughter trees, and
small sample trees, the left pulmonary arterial tree and the
right venous tree were reconstructed.
Data Analysis
Dn, left 5 [(Dn21 1 SDn21) 1 (Dn 2 SDn)]/2
(2)
and smaller than
Dn, right 5 [(Dn 1 SDn) 1 (Dn11 2 SDn11)]/2
(3)
This test is made at each point of bifurcation.
A process of iteration is used to determine the order
number of the vessels. In this study the pulmonary vascular
Fig. 1. Illustration of diameter-defined Strahler ordering system.
Vessel order numbers are determined by their connection and
diameters. Arteries with smallest diameters are of order 1. A segment
is a vessel between 2 successive points of bifurcation. When 2
segments meet, order number of confluent vessel is increased by 1 if
and only if its diameter is larger than either of the 2 segments by a
certain amount specified by Eqs. 2 and 3. Otherwise, order number of
confluent segment is not increased. In Horsfield’s Strahler system,
diameter test is not applied, resulting in differences illustrated in the
3 paired inset boxes. Consequences of these differences are discussed
in DISCUSSION. Horsfield and diameter-defined systems apply to
arterial and venous trees only. They are not applicable to capillary
network, topology of which is not treelike. Each group of segments of
the same order connected in series is lumped together and called an
element.
trees initially were assigned by the diameter ranges of orders
1–3 from Yen and Sobin (28) according to Strahler’s ordering
system. Yen and Sobin measured the diameters and branching orders of the pulmonary microvasculatures with diameters ,100 µm from the histological preparations of postmortem human lung prepared by perfusion with a silicone
elastomer. Then, by the same Horsfield method, the order
numbers of the vessels of orders 4–15 in the pulmonary
arterial and venous trees were assigned. The mean and
standard deviation of the diameters were calculated.
Once the initial mean and standard deviation of the
diameters are obtained, we use Eqs. 2 and 3 to decide the
order number of every vessel. Some changes will occur in
the order number of some vessels in this process. After the
change, new values of Dn and SDn are computed and used to
reset the diameter ranges for order n. The process is repeated
until the changes of Dn and SDn between successive iterations
became ,1%, at which convergence is considered achieved.
By using the diameter criteria, the final diameter ranges of
successive orders do not overlap and standard deviations are
relatively small (Table 2).
Vessel segment and vessel element. In the first use of
Strahler’s ordering system by Horsfield, no distinction is
made between series and parallel vessels of the same order,
nor is it possible to express the series-parallel features in the
circuit representing the pulmonary vasculature. To obtain a
correct circuit model, Kassab et al. (14, 15) defined every
vessel between two successive bifurcation points as a segment
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Assignment of order number to branches of the pulmonary
vascular tree according to diameter. We use the diameterdefined Strahler ordering system to describe the branching
pattern of the pulmonary arterial and venous trees. This
system modifies Horsfield’s Strahler system with the addition
of a step to be described below.
Let us first explain the original Strahler ordering system.
In the Strahler ordering system, the smallest noncapillary
blood vessel is defined as of order 1. When two vessels of the
same order meet, the order number of the confluent vessel is
increased by 1. When a vessel of order n meets another vessel
of order , n, the order number of the confluent vessel remains
n. Strahler’s ordering system deals with the asymmetric
bifurcations nicely; however, in the studies of the human and
cat pulmonary vasculature, investigators (1, 7, 11, 22, 29, 30)
found very large overlaps of the diameters in the successive
orders of vessels by Strahler’s ordering system caused by the
rather indiscriminating assignment of order numbers in very
large trees. Additionally, other difficulties mentioned in the
introduction were encountered.
To ameliorate these difficulties, Kassab et al. (14, 15)
developed the diameter-defined Strahler ordering system. In
this system, a new rule is added: when a vessel of order n
meets another vessel of order # n, the confluent vessel is
called of order n 1 1 only if its diameter is larger by a certain
amount, which is determined by the statistical distribution of
the diameters of each order, as discussed below. Figure 1
shows a scheme of the diameter-defined Strahler ordering
system.
Strahler’s system is applicable only to treelike structures.
Pulmonary capillary blood vessels are not treelike in topology.
Fung and Sobin (3) and Sobin et al. (24) showed that the
pulmonary capillaries can be described by a sheet-flow model.
We designate an order number of 0 to the pulmonary capillaries. Strahler’s system begins with arterioles and venules. The
smallest arterioles supplying the capillaries are assigned an
order number of 1. The smallest venules draining the capillaries are assigned an order number of 21. Positive integers
identify arteries; negative integers identify veins. This system has been used by Kassab et al. (14) for the pig coronary
system. Use of positive and negative integers to differentiate
arterial and venous trees is convenient in morphometry.
The mean and standard deviation of the diameters of the
pulmonary arteries of an arbitrary order n are denoted by Dn
and SDn, respectively. There are vessels of all sizes between
Dn 2 1 and Dn and between Dn and Dn 1 1. We shall designate a
vessel to be of the order of n if its diameter is greater than
2126
MORPHOMETRY OF PULMONARY VASCULATURE
Table 2. Diameter and length of elements of
pulmonary arteries and veins of two human lungs
n
Diameter, mm
115
114
113
112
111
110
19
18
17
16
15
14
13
12
11
2
5
16
36
84
142
182
227
315
315
180
50
48
81
113
14.80 6 2.10
7.34 6 1.14
4.16 6 0.60
2.71 6 0.35
1.75 6 0.19
1.16 6 0.10
0.77 6 0.07
0.51 6 0.04
0.34 6 0.06
0.22 6 0.02
0.15 6 0.02
0.097 6 0.012
0.056 6 0.005
0.036 6 0.005
0.020 6 0.003
21
22
23
24
25
26
27
28
29
210
211
212
213
214
215
36
31
38
195
150
81
50
33
64
79
42
25
9
4
2
0.018 6 0.002
0.031 6 0.005
0.067 6 0.010
0.13 6 0.02
0.23 6 0.03
0.38 6 0.04
0.62 6 0.06
0.90 6 0.07
1.42 6 0.15
1.99 6 0.21
2.88 6 0.21
4.00 6 0.33
5.86 6 0.38
8.65 6 0.76
12.97 6 1.70
Length, mm
Arteries
SEC(m,n) 5
25.30 6 28.15
35.69 6 10.28
25.97 6 14.19
18.07 6 11.65
12.35 6 6.77
6.58 6 4.26
3.73 6 2.43
2.81 6 1.77
1.92 6 1.22
1.08 6 0.65
0.68 6 0.36
0.45 6 0.25
0.36 6 0.20
0.26 6 0.12
0.22 6 0.08
(5)
ŒNm(n)
where SDC(m,n) is the standard deviation of C(m,n) and Nm(n) is
the number of observations of vessels of order m connected to
vessels of order n. For a tree with k orders, the connectivity
matrix is a k 3 k upper triangular matrix.
In addition to expressing the branching pattern for the
whole vascular tree, the connectivity matrix was used to
calculate the total number of elements in each order in this
study. The elements of order n, n 2 1,...,1 spring directly from
the elements of order n. Therefore, when the number of
elements of order n, Nn, is known, the total numbers of
elements of order n, n 21,..., are C(n,n)Nn, C(n 2 1,n)Nn,...,
respectively. Considering all the vessels in a tree, we see that
the total number of elements of order m is given by
Veins
0.13 6 0.07
0.21 6 0.15
0.38 6 0.24
1.06 6 0.54
1.50 6 0.85
2.92 6 1.91
4.79 6 3.64
6.78 6 6.32
11.24 6 6.91
14.78 6 7.71
17.90 6 10.92
26.49 6 13.11
19.49 6 11.89
34.99 6 16.97
35.68 6 7.36
k
Nm 5
o C(m, n)N
(6)
n
n5m
where Nm and Nn are the total numbers of elements of orders
m and n, respectively. The summation is from n 5 m to the
highest order of the tree, n 5 k.
Many branches are missing, because the tree was pruned
for practical counting or broken off, but the stubs were
recognizable. The number of missing branches must be
considered while the total number of elements is counted. The
extrapolated number of the elements of each order in the
missing subtrees is calculated from the number of cut-off and
broken-off subtrees and the connectivity matrix by the following equation
Values are means 6 SD; n, no. of elements.
and combined those vessels of the same order connected in
series as an element. Statistical data are obtained for elements and segments. Flow circuits are built of elements.
In this study we used the same terminology. The measurement of diameter and length were made in the twodimensional grabbing images. We found that the diameter of
each segment is constant, and we measured it at the midpoint
of each segment. The segment length is the distance between
bifurcation points along the centerline of the segment. The
diameter of an element was computed as the average of the
diameters of the segments that make up the element. The
length of an element was obtained by adding the lengths of
the segments within the element. The relationship between
the total number of segments of order n, S(n), and the total
number of elements of the same order, E(n), can be described
by
S(n) 5 E(n) 3 R(n)
SDC(m,n)
k
N8m 5
o C(m, n)[N8 1 N
n
n,cut]
(7)
n5m
where N8m and N8n are the extrapolated number of elements of
orders m and n, respectively, in the missing subtrees, Nn,cut is
the number of cut-off elements (pruned and broken-off elements) in order n, and n $ m with an upper limit of k. The
calculation starts from n 5 k, which is the highest order, and
then proceeds down to n 5 1, successively. Therefore, this
process includes all the missing subtrees.
The number of cut-off elements in order n, Nn,cut, and the
number of intact elements in order n, Nn,in, are counted from
the cast tree directly. The corrected total number of elements
of each order is the sum of the number of intact elements, the
number of cut-off elements, and the extrapolated number of
elements.
(4)
where R(n) is the segment-to-element ratio in order n.
Connectivity matrix. We used the connectivity matrix to
express how blood vessels of one order are connected to
vessels of another order. Blood vessels of order n not only
arise from vessels of order n 1 1 but also originate from
vessels of order n, n 1 2, n 1 3,.... There was no quantitative
expression for the connectivity feature of blood vessels of one
order to another until the connectivity matrix was first
developed and used by Kassab et al. (14, 15). In the connectivity matrix, each component in the mth row and the nth
column, designated as C(m,n), is expressed as mean 6 SE.
RESULTS
This study shows that there are 15 orders of pulmonary arteries between the main pulmonary artery and
the capillaries in the left lung of one man and 15 orders
of veins between the capillaries and the left atrium in
the right lung of another man.
The mean and standard deviation of the diameters
and lengths of the elements in each order are listed in
Table 2. Two significant digits after the decimal point
for diameters and lengths are justifiable in Table 2. The
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Order
The mean value is the ratio of the total number of elements of
order m sprung from parent elements of order n divided by
the total number of elements of order n. The standard error of
C(m,n), SEC(m,n), is obtained by
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MORPHOMETRY OF PULMONARY VASCULATURE
Fig. 3. Relation between order number and mean and standard
deviation of length of arterial and venous elements of each order. If y
represents logarithm of length and x represents order number,
regression line is y 5 20.95 1 0.17x for arterial tree and y 5 20.79 1
0.18x for venous tree.
pulmonary arteries and y 5 7.66 2 0.52x for pulmonary
veins. The antilog of the absolute value of slope yields
an average branching ratio of 3.36 for all orders of
pulmonary arteries and 3.33 for veins. The branching
ratio is a ratio of the corrected total number of elements
of order n to that of order n 1 1.
Data on the segmental diameter, length, and segment-to-element ratio of each order in the pulmonary
arterial and venous trees are listed in Table 6. Two
digits after the decimal point in Table 6 are significant.
The average cross-sectional area of vessel elements
of order n, an, is calculated as
an 5
p
4
D2n
where Dn is the average diameter of elements in order
n. The total cross-sectional area of vessel elements of
order n, An, is calculated from the formula
An 5 anNn 5
Fig. 2. Relation between order number and mean and standard
deviation of diameters of arterial and venous elements of each order.
If y represents logarithm of diameter and x represents order number,
regression line is y 5 21.84 1 0.19x for pulmonary arteries and y 5
21.74 1 0.20x for pulmonary veins.
(8)
p
4
D2nNn
(9)
where Nn is the number of elements of order n. Figure
5A illustrates the distribution of the total crosssectional area of elements with the order number in the
left pulmonary arterial tree of a 44-yr-old man and the
right pulmonary venous tree of a 24-yr-old man. For
comparison, the total cross-sectional area distribution
of arteries and veins in the whole lung from data of
Horsfield (7) and Horsfield and Gorden (11) is shown in
Fig. 5B. Our data on total cross-sectional areas of
arteries and veins are larger than the respective areas
given by Horsfield and Gorden. The total crosssectional areas are related to the total number of
branches. According to our data the total number of
branches in small vessels is greater than that reported
by Horsfield and Gorden. Additionally, the total cross-
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diameters of the elements for arterial and venous trees
are plotted in logarithmic scale against the order
number in Fig. 2. Straight regression lines were determined by the least squares method. If y represents the
logarithm of the diameter and x represents the order
number, the regression lines for pulmonary arterial
and venous trees are y 5 21.84 1 0.19x and y 5
21.74 1 0.20x, respectively. The antilog of the slope
gives an average diameter ratio of 1.56 for all pulmonary arteries and 1.58 for all veins. The ratio of the
diameter of the elements of order n to that of order n 1
1 is called the diameter ratio. The relationship between
the logarithm of element length and the order number
is depicted in Fig. 3. Similarly, the least squares fit
regression line for pulmonary arteries is y 5 20.95 1
0.17x and that for veins is y 5 20.79 1 0.18x. The
antilog of the slope yields an average length ratio of
1.49 for all pulmonary arteries and 1.50 for all veins.
The length ratio is defined as a ratio of the length of the
elements of order n to that of n 1 1.
The connectivity matrices of pulmonary arteries and
veins are presented in Tables 3 and 4, respectively. The
corrected total number of elements of each order in the
pulmonary arterial tree of the left lung of one man and
in the pulmonary venous tree of the right lung of
another man is computed from the connectivity matrices given in Tables 3 and 4 starting from order 15 for
arteries and veins and extrapolating downward. The
results are shown in Table 5. The total number of intact
elements and the total number of cut-off subtrees in
each order are listed in columns 2 and 3, respectively, of
Table 5. The values in column 4 were computed by Eq.
7. The last column shows the corrected total number of
elements in each order, which is the sum of numbers in
the previous three columns of the given order. If the
logarithm of the corrected total number of elements (y)
in each order is plotted against the order number (x) as
displayed in Fig. 4, the relationship between the corrected total number of elements and the order number
is given by a regression line y 5 8.41 2 0.53x for
2128
MORPHOMETRY OF PULMONARY VASCULATURE
Table 3. Connectivity matrix of elements of pulmonary arteries of a human left lung
1
1
2
3
4
0.17
60.05
2.54
60.16
0.23
60.06
1.11
60.18
1.67
60.16
0.26
60.08
0.38
60.20
0.97
60.14
1.44
60.12
0.18
60.08
2
3
4
5
6
7
8
9
10
11
12
13
14
15
0.02
0.01
0.02
0
0.02
0
0
0
0
0
0
0.17
60.07
0.63
60.08
1.96
60.08
0.06
60.04
0.09
60.03
0.28
60.06
1.50
60.09
1.81
60.08
0.12
60.02
0.06
60.03
0.20
60.05
0.93
60.11
1.10
60.06
2.54
60.09
0.25
60.03
0.04
0.02
0
0
0
0
0
0
0.19
60.04
1.07
60.14
0.81
60.09
1.16
60.10
2.52
60.10
0.28
60.04
0.11
60.06
0.65
60.14
0.83
60.11
0.83
60.10
1.08
60.11
2.06
60.08
0.21
60.03
0.03
0
0
0
0
0
0.03
0
0
0
0
0.02
0
0
0
0.05
0
0
0.43
60.18
0.43
60.18
0.81
60.25
1.48
60.29
0.95
60.07
2.57
60.19
0.19
0
0
0
0
0.33
0
0
0
0.33
0
6
7
8
9
10
0.16
60.04
0.45
60.09
0.93
60.11
1.08
60.13
1.26
60.08
2.38
60.08
0.21
60.02
11
0.12
60.08
0.71
60.13
1.49
60.15
1.63
60.16
1.47
60.12
2.52
60.19
0.25
60.03
12
0.20
60.02
0.94
60.13
1.45
60.20
0.88
60.13
1.35
60.23
2.43
60.15
0.14
13
14
1.00
60.24
2.83
60.40
0
15
0.50
1.50
3.50
60.50
0.50
Values are means 6 SE. Each entry is ratio of total no. of elements of order m produced from a parent element of order n divided by total no.
of elements of order n.
Table 4. Connectivity matrix of elements of pulmonary veins of a human right lung
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
1
2
3
0.38
60.10
2.59
60.19
0.53
60.12
1.24
60.22
1.80
60.18
0.28
60.08
4
5
6
7
8
9
10
11
12
13
14
15
0.01
0
0
0
0
0
0
0
0
0
0
0
0.03
0.01
0
0
0
0
0
0
0
0
0
1.13
60.11
3.16
60.12
0.31
60.03
0.82
60.09
2.15
60.21
2.64
60.14
0.25
60.05
0.06
0.02
0
0
0
0
0
0
1.03
60.32
1.64
60.34
1.73
60.32
1.67
60.15
0.12
0.36
60.08
1.76
60.17
1.86
60.23
1.57
60.13
1.73
60.10
0.10
0.34
60.06
1.39
60.14
2.07
60.22
1.39
60.13
0.69
60.06
2.13
60.10
0.05
0.24
60.11
1.24
60.21
1.78
60.26
1.93
60.28
0.66
60.13
0.83
60.13
1.66
60.09
0
0.12
0
0
0
0.68
60.17
1.52
60.20
0.88
60.21
1.24
60.24
0.84
60.21
0.88
60.09
1.72
60.15
0.04
0
0
0
0
0
0
0.11
0.25
0
0.22
0
0
0.33
60.24
0.11
0
0
0.75
60.35
0.25
0
1.50
0
2.50
60.65
0
0
0.20
60.05
3.23
60.10
0.40
60.04
0.42
60.15
1.68
60.27
1.74
60.20
2.30
60.19
0.24
60.04
0.78
60.14
2.11
60.20
0.11
0
2
0
Values are means 6 SE. Each entry is ratio of total no. of elements of order m produced from a parent element of order n divided by total no.
of elements of order n.
Downloaded from http://jap.physiology.org/ by 10.220.32.247 on June 18, 2017
5
2129
MORPHOMETRY OF PULMONARY VASCULATURE
Table 5. Computation of corrected total number of
elements in pulmonary arterial tree of left lung and
pulmonary venous tree of right lung
Order
No.
Intact
No.
Cut
No. of
Extrapolations
Corrected
Total
Arteries
2
6
21
49
99
153
132
134
179
173
102
34
27
35
69
0
1
15
9
52
223
464
612
716
776
719
858
233
105
81
0
0
7
69
299
1,348
5,629
21,258
85,125
284,823
673,348
2,255,954
5,101,643
14,057,057
51,205,662
2
7
43
127
450
1,724
6,225
22,004
86,020
285,772
674,169
2,256,846
5,101,903
14,057,197
51,205,812
Veins
21
22
23
24
25
26
27
28
29
210
211
212
213
214
215
124
122
125
195
147
83
50
33
62
77
41
25
9
4
2
121
151
283
732
543
501
337
197
170
21
10
1
2
0
0
39,823,308
8,493,972
2,165,925
452,124
72,335
14,983
3,334
837
203
45
12
5
0
0
0
39,823,553
8,494,245
2,166,333
453,051
73,025
15,567
3,721
1,067
435
143
63
31
11
4
2
No. of extrapolations is computed by Eq. 7. Corrected total is sum
in previous 3 columns of the given order. Left lung was from 44-yr-old
man; right lung was from 24-yr-old man.
sectional areas of small arteries in our data are greater
than total cross-sectional areas of small veins, because
the total number of branches is greater in small
arteries than in small veins. However, because our data
Order
n
Diameter,
mm
115
114
113
112
111
110
19
18
17
16
15
14
13
12
11
5
19
83
181
481
608
681
846
941
636
254
73
65
112
121
15.12 6 1.81
7.31 6 1.38
4.33 6 0.68
2.81 6 0.46
1.78 6 0.25
1.17 6 0.14
0.77 6 0.10
0.51 6 0.06
0.35 6 0.09
0.22 6 0.03
0.15 6 0.02
0.096 6 0.015
0.056 6 0.005
0.036 6 0.006
0.020 6 0.003
21
22
23
24
25
26
27
28
29
210
211
212
213
214
215
37
50
79
475
431
291
192
131
323
446
255
145
19
14
2
0.018 6 0.002
0.032 6 0.006
0.066 6 0.012
0.13 6 0.03
0.23 6 0.04
0.39 6 0.06
0.63 6 0.08
0.90 6 0.08
1.39 6 0.19
2.01 6 0.26
2.90 6 0.26
4.08 6 0.52
5.95 6 0.48
8.75 6 0.99
12.97 6 1.70
Length,
mm
NS /NE
10.12 6 9.64
9.39 6 5.37
5.01 6 3.18
3.59 6 2.67
2.16 6 1.38
1.54 6 1.19
1.00 6 0.66
0.76 6 0.51
0.64 6 0.33
0.54 6 0.32
0.48 6 0.24
0.31 6 0.15
0.26 6 0.16
0.19 6 0.10
0.20 6 0.08
2.50 6 0.71
3.80 6 0.45
5.19 6 3.04
5.03 6 3.38
5.73 6 3.41
3.34 6 2.99
3.74 6 2.63
3.73 6 2.17
2.99 6 1.65
2.02 6 1.25
1.41 6 0.66
1.46 6 0.79
1.35 6 0.60
1.38 6 0.56
1.07 6 0.26
0.12 6 0.01
0.13 6 0.10
0.18 6 0.15
0.43 6 0.24
0.52 6 0.26
0.81 6 0.43
1.25 6 0.69
1.71 6 0.99
2.23 6 1.38
2.62 6 1.88
2.95 6 2.20
4.57 6 3.52
9.23 6 3.71
10.00 6 6.23
35.68 6 7.36
1.03 6 0.17
1.61 6 0.67
2.08 6 1.19
2.44 6 1.26
2.87 6 1.57
3.59 6 2.46
3.84 6 2.83
3.97 6 3.79
5.05 6 2.86
5.65 6 3.30
6.07 6 4.31
5.80 6 3.84
2.11 6 1.05
3.50 6 1.29
1.00 6 0.00
Arteries
Veins
Values are means 6 SD; n, no. of segments measured. NS /NE ,
segment-to-element ratio.
are collected from two single lungs, they could not
represent the human population.
The average blood volume of vessel elements in order
n, vn, is calculated as
vn 5
p
4
D2nLn
(10)
where Dn is the average diameter of elements in order n
and Ln is the average length of elements in order n. The
total blood volume of vessel elements in order n, Vn, is
calculated from
Vn 5
Fig. 4. Relation between order number and corrected total number of
elements in each order. If y represents logarithm of corrected total
number of elements and x represents order number, regression line is
y 5 8.40 2 0.53x for arterial tree and y 5 7.66 2 0.52x for venous tree.
p
4
D2nNnLn
(11)
where Nn is the number of elements of order n. Table 7
presents the total blood volume as given by Eq. 11 for
all orders in the arterial and venous trees. The total
blood volume in all orders of the arteries is 150.06 ml in
the left lung of a 44-yr-old man, and that of the veins is
86.59 ml in the right lung of a 24-yr-old man. These
morphologically computed data may be compared with
the experimental data given by Horsfield and Gorden
(11).
Downloaded from http://jap.physiology.org/ by 10.220.32.247 on June 18, 2017
115
114
113
112
111
110
19
18
17
16
15
14
13
12
11
Table 6. Diameter and length of segments of
pulmonary arteries and veins of two human lungs
2130
MORPHOMETRY OF PULMONARY VASCULATURE
Table 7. Blood volumes of arteries in left lung and
veins in right lung
Artery
Vein
Order
Vn ,
ml
Cumulative
volume, ml
Vn ,
ml
Cumulative
volume, ml
615
614
613
612
611
610
69
68
67
66
65
64
63
62
61
8.71
10.56
15.20
13.19
13.35
11.92
10.83
12.70
15.00
11.55
7.84
7.56
4.50
3.68
3.47
8.71
19.27
34.47
47.66
61.01
72.93
83.76
96.46
111.46
123.01
130.85
138.41
142.91
146.59
150.06
9.43
8.22
5.78
10.31
7.33
6.58
7.73
4.63
5.38
5.20
4.39
6.07
2.89
1.36
1.29
9.43
17.65
23.43
33.74
41.07
47.65
55.38
60.01
65.39
70.59
74.98
81.05
83.94
85.30
86.59
arterial tree was published by Singhal et al. (22).
Horsfield (7) and Horsfield and Gorden (11) amplified
the data and used them to analyze pulmonary circulation.
The diameter-defined Strahler system used here
modifies Horsfield’s Strahler system by adding a diameter judgment at the junction where two vessels meet to
become one confluent vessel. Horsfield’s rule is to
increase the order number of the confluent vessel by 1
indiscriminately. Our rule is to increase the order
number of the confluent vessel by 1 if and only if the
Fig. 5. Distribution of total cross-sectional area of all elements of
each order with order number. A: left pulmonary arterial tree of a
44-yr-old man and right pulmonary venous tree of a 24-yr-old man
(our data). B: pulmonary arterial and venous trees in whole human
lung; values for arteries are from Horsfield (7); values for veins are
from Horsfield and Gorden (11).
Table 8. Total order number, mean diameter
of order 1 vessels, and diameter, length, and
branching ratios in pulmonary vascular trees
of cats, dogs, rats, and humans
Cats
Dogs
Rats
Humans*
Humans†
Arteries
DISCUSSION
The morphometric data of pulmonary vascular trees
of the cat, dog, rat, and human are now available in
various degrees of completeness. Table 8 summarizes
the total order number, the mean diameter of order 1
vessels, the diameter ratio, the length ratio, and the
branching ratio of the pulmonary vascular trees of
these animals. Fung (2) and Zhuang et al. (31) demonstrated the application of morphometric data in pulmonary hemodynamics. The data of the cat (29, 30) and
human (7, 11, 22) by Horsfield, Singhal, and their
colleagues were obtained by Strahler’s ordering system, whereas the other data were obtained by the
diameter-defined Strahler system. The connectivity
matrix is defined only in the latter system, for reasons
to be explained.
Cumming and Horsfield (1) pioneered the use of
Strahler’s ordering system in the morphometry of the
lung. The first set of data on the human pulmonary
Total order no. 12
Order 1 diameter, µm
21
Diameter ratio 1.72
Length ratio
1.81
Branch ratio
3.58
12
11
17
28
13
13
1.67 1.58 1.60
1.52 1.60 1.49
3.69 2.76 3.03 (6–17)
3.37 (1–5)
15
20
1.56
1.49
3.36
Veins
Total order no. 11
Order 1 diameter, µm
22
Diameter ratio 1.73
Length ratio
Branch ratio
1.53 (4–10)
2.40 (1–3)
3.52
11
x
15
15
29
1.70
x
x
18
1.58
1.56
x
3.76
x
13
1.68 (7–14)
1.49 (1–6)
1.68 (7–14)
1.48 (1–6)
3.30
1.50
3.33
Data for cats are from right pulmonary arterial and venous trees
(29, 30), data for dogs from right pulmonary arterial and venous trees
(Y. Tien, R. Z. Gan, and R. T. Yen, unpublished observations; 5), data
for rats from left pulmonary arterial tree (12). * Data from whole lung
(7, 11, 22); † data from left pulmonary arterial tree of a 44-yr-old man
and right pulmonary venous tree of a 24-yr-old man (our data).
Downloaded from http://jap.physiology.org/ by 10.220.32.247 on June 18, 2017
Vn , total blood volume of order n computed by Eq. 10. Left lung was
from 44-yr-old man; right lung was from 24-yr-old man.
MORPHOMETRY OF PULMONARY VASCULATURE
diameter of the confluence is greater than either of the
two converging vessels by a certain amount specified by
Eqs. 2 and 3.
The first evident difference caused by this modification is that we now have a reasonable and systematic
way to handle the most important pulmonary artery:
the long tapered main artery from the pulmonary valve
to the periphery. As we know, the most important
formula for blood flow is that of Poiseuille. Poiseuille’s
formula states that the flow rate of a Newtonian fluid,
with coefficient of viscosity µ, in a circular cylindrical
tube of lumen diameter D and length L is related to a
pressure drop DP. D is in the fourth power. A 10% error
in D leads to a 46.4% error in flow (Q̇), whereas a 10%
error in any other parameter, such as DP, µ, and L,
leads to only a 10% error in Q̇. Obviously, among all
parameters on the Poiseuille equation, D is the most
demanding for accuracy of Q̇. In Horsfield’s way, the
main pulmonary artery has to be designated with one
order number. Because in Horsfield’s morphometry one
vessel order is given one diameter and the main
pulmonary artery has a fixed diameter, despite the
tapered phenomenon of this vessel, it is difficult to
account for the taper in hemodynamics regardless of
how the diameter is chosen. In our system the order
number increases with diameter in a systematic way,
the long tapered vessel is divided into successive orders
in a manner consistent with hemodynamics, and the
application of Poiseuille’s formula (modified with proper
Reynolds number and Womersley number effects) yields
the correct hemodynamic formula automatically with
regard to the changes of vessel diameter. Jiang et al.
(12) provide a detailed discussion about handling of
tapered vessels.
The second evident difference caused by this modification is the reduction of the standard deviation of the
diameter of vessels of each order from a Horsfield value
to ours; statistically this is achieved by eliminating the
overlaps in the ranges of the diameters of vessels of
successive orders. Jiang et al. (12) compared the statistical distributions of the diameters of each order in
these two systems. The standard deviation of the
diameter characterizes the dispersion in geometric size
of the vessels, and it influences significantly the dispersion of blood flow in the lung (significant because
diameter enters Poiseuille’s flow formula in the 4th
power). The dispersion of flow is related to the heterogeneity of oxygen transport in the lung. Hence, the
standard deviation of diameter is an important physiological parameter. We believe that the large standard
deviation of diameters obtained by Horsfield’s method
is an artifact caused by a definition of order without
regard to the size of the vessels. This is obvious from
the three pairs of inset boxes in which the differences of
the two definitions are shown in Fig. 1. According to
Horsfield’s Strahler ordering method, in two cases the
vessels became larger but the order number did not
increase. In the third case, the vessel remained the
same size, but the order number increased. With Horsfield’s Strahler ordering method, it is difficult to account for the hemodynamics in these vessels when
Poiseuille’s formula is applied.
The third evident difference of the two definitions is
the way the lengths of vessels are computed. In our
system the vessels of the same order connected in series
are considered as one element, the length of which is
the sum of the lengths of the vessels in series. In
Horsfield’s method all vessels are considered parallel.
The distinction of segment and element is an important
step in the construction of analog hemodynamic circuits, because vessels connected in series and in parallel differ tremendously in resistance. The inverse of the
length of a vessel equivalent to n vessels in parallel is
equal to the sum of the inverse of the length of each
Fig. 7. Log-log plot of mean length of elements of each order vs. sum
total of lengths of all elements of each order. Each point corresponds
to a specific order.
Downloaded from http://jap.physiology.org/ by 10.220.32.247 on June 18, 2017
Fig. 6. Log-log plot of mean diameter of elements of each order vs.
sum total of diameters of all elements of each order. Each point
corresponds to a specific order.
2131
2132
MORPHOMETRY OF PULMONARY VASCULATURE
structure is fractal, because the total number of generations is only 15 and the order 1 vessels are of finite size,
not infinitesimal. Nevertheless, we can use our morphometric data to compute the fractal dimension of human
pulmonary vasculature (19) by the method of Nelson
and Manchester (21), and the results are interesting in
the interpretation of the overall structure of the lung.
The fractal dimension (DF ) is given by the formula (20)
1 2 DF 5
log(total size of object of order n)
log(avg size of object of order n)
(12)
We obtained the fractal dimension of the diameters of
the elements 2.71 for the arterial tree and 2.64 for the
venous tree. The fractal dimensions of element length
are 2.97 and 2.86 for the arterial and venous trees,
respectively. These values were derived plots of element diameters and lengths of the pulmonary arterial
and venous trees shown in Figs. 6 and 7. Using the data
of the pulmonary vessels of the cat (29, 30), dog (5) and
rat (12), we obtained the fractal dimensions of the
diameter and length also in the range between 2 and 3.
The scope of this study is limited to morphometry. By
itself, it is not sufficient for hemodynamics, for which
data are also needed on the elasticity of pulmonary
arteries and veins and the branching angles of blood
vessels. Existing data on elasticity and branching angle
are very sketchy and need systematic measurement.
We expect the branching angle to be important where
inertial force is significant, i.e., in large arteries and
veins where the Reynolds and Womersley numbers are
large. For smaller vessels, where the Reynolds and
Womersley numbers are ,1 and the viscous forces
dominate, the branching angle is not expected to be a
significant morphometric parameter. On the basis of
the data of morphometry, including the connectivity
matrix and the branching angle, and elasticity, the
hemodynamic analyses will be done and compared with
the theoretical models of Krenz et al. (16, 17) and
explain the theory of Hakim et al. (6) on the regional
distribution of pulmonary blood flow in humans.
Present address of W. Huang: Dept. of Bioengineering, University
of California, San Diego, La Jolla, CA 92093-0412.
Address for reprint requests: M. R. T. Yen, Dept. of Biomedical
Engineering, The University of Memphis, Memphis, TN 38152.
Received 17 January 1996; accepted in final form 15 July 1996.
REFERENCES
1. Cumming, G., L. K. Harding, K. Horsfield, K. Prowse, S. S.
Singhal, and M. J. Woldenberg. Morphological aspects of the
pulmonary circulation and of the airway. In: Fluid Dynamics of
Blood Circulation and Respiratory Flow. Neuilly-sur-Seine: North
Atlantic Treaty Organization, 1970, p. 230–236. (AGARD Conf.
Proc. No. 65)
2. Fung, Y. C. Biodynamics: Circulation. New York: SpringerVerlag, 1984.
3. Fung, Y. C., and S. S. Sobin. Theory of sheet flow in lung
alveoli. J. Appl. Physiol. 26: 472–488, 1969.
4. Fung, Y. C., S. S. Sobin, H. Tremer, R. T. Yen, and H. H. Ho.
Patency and compliance of the pulmonary veins when the airway
pressure exceeds blood pressure. J. Appl. Physiol. 54: 1538–
1549, 1983.
5. Gan, R. Z., Y. Tian, and R. T. Yen. Morphometry of the dog
pulmonary venous tree. J. Appl. Physiol. 75: 432–440, 1993.
Downloaded from http://jap.physiology.org/ by 10.220.32.247 on June 18, 2017
vessel. This is evident in an analog electric circuit. Each
segment is a resistor. If two resistors of resistance R
each are connected in series, the total resistance is 2R.
If they are connected in parallel, the total resistance is
R/2. Hence, if several segments are connected in series
into elements, they must not be mistaken as in parallel.
In hemodynamics it is important to determine whether
the n vessels are connected in series or in parallel.
Hence, Horsfield’s method seems to emphasize the
connectivity to the degree of ignoring diameter change
at the points of connection, yet it ignores the connectivity by treating all vessels of the same order as if they
are all parallel. The range overlap of the diameter and
the large standard deviation are consequences of Horsfield’s basic definition of vessel order number. We
believe that it is logical to introduce a modification of
his definition. In 1991, Horsfield (9) recognized that all
vessel segments of the same order are not all in parallel
and suggested a ‘‘Strahler method stage 2,’’ which
considers the segments of the same order in series if a
tapering vessel is intersected only by the smaller
branches; however, no data were reported.
How the connectivity feature among vessels of different order is described is very important for hemodynamic modeling. The importance has been extensively
discussed by Jiang et al. (12). In 1971, Horsfield (8–10)
developed a method of ‘‘delta’’ to describe the connectivity and asymmetry in a bronchial tree, where delta is a
number designed to show the difference in the size of
two vessels. A map and statistics of delta are needed.
However, no data have been reported. In this study, we
use the connectivity matrix to describe the connectivity
of blood vessels from one order to another and to
calculate the total element number for each order.
Our definition of order 1 vessels is different from
Horsfield’s. We define order 1 vessels as the smallest
noncapillary vessels (28). Yen and Sobin (28) showed
the relationship between the capillary bed and the first
several orders of arterioles and venules in detail. In
Horsfield’s data (7), order 1 vessels are defined as those
between 10 and 15 µm diameter that are first encountered going down any pathway. Horsfield’s order 2
vessels are equivalent to our order 1 vessels (28).
According to our data, the total element number of
order 1 arteries in the left lung is ,5 3 107; if a similar
number is assumed for the right lung, then the total
number of elements of order 1 arteries for the whole
lung is ,1 3 108.
Because there is no way to obtain a complete set of
human morphometric data on living persons, the use of
postmortem material is the logical answer. In this
study, the left lung was from a 44-yr-old man and the
right lung from a 24-yr-old man. Unfortunately, we
could not obtain a pair of lungs from either man. A
study in a pair of lungs from one person would provide
more information, but human lung specimens are
extremely difficult to obtain. We hope for the opportunity to obtain more data to represent the human
population.
A number of authors prefer to look at the lung
structure as fractal. We do not think that the lung
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