Vol. 281, Issue 6, H2747-H2756, December 2001
SPECIAL COMMUNICATION
Pulmonary arterial morphometry from microfocal X-ray computed
tomography
Kelly L.
Karau1,3,
Robert C.
Molthen1,3,4,
Anita
Dhyani3,
Steven T.
Haworth1,
Christopher C.
Hanger2,
David L.
Roerig2,4,
Roger H.
Johnson3, and
Christopher A.
Dawson1,3,4
Departments of 1 Physiology and 2 Anesthesiology,
Medical College of Wisconsin; 3 Department of Biomedical
Engineering, Marquette University, Milwaukee 53201; and
4 Research Service, Zablocki Veterans Affairs Medical
Center, Milwaukee, Wisconsin 53295
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ABSTRACT |
The objective of this study was to develop an X-ray
computed tomographic method for pulmonary arterial morphometry. The
lungs were removed from a rat, and the pulmonary arterial tree was
filled with perfluorooctyl bromide to enhance X-ray absorbance. At each of four pulmonary arterial pressures (30, 21, 12, and 5.4 mmHg), the
lungs were rotated within the cone of the X-ray beam that was projected
from a microfocal X-ray source onto an image intensifier, and 360 images were obtained at 1° increments. The three-dimensional image
volumes were reconstructed with isotropic resolution with the use of a
cone beam reconstruction algorithm. The luminal diameter and distance
from the inlet artery were measured for the main trunk, its immediate
branches, and several minor trunks. These data revealed a
self-consistent tree structure wherein the portion of the tree
downstream from any vessel of a given diameter has a similar structure.
Self-consistency allows the entire tree structure to be characterized
by measuring the dimensions of only the vessels comprising the main
trunk of the tree and its immediate branches. An approach for taking
advantage of this property to parameterize the morphometry and
distensibility of the pulmonary arterial tree is developed.
cone beam reconstruction; pulmonary arterial diameter; pulmonary
arterial distensibility
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INTRODUCTION |
THE EXTANT LUMINAL
MORPHOMETRY of the pulmonary arterial tree comes almost
exclusively from measurements on plastic corrosion casts. That approach
has been important in developing present concepts regarding pulmonary
structure-function relationships. However, the data collection process
is quite tedious, and, as a consequence, the data obtained to date are
from very few lungs, i.e., usually from only one or two ostensibly
normal lungs of each species that has been studied (13, 16, 18,
33, 40). The laborious casting approach virtually precludes
accumulation of data on the progression of the pulmonary arterial
architecture during development or resulting from vascular remodeling
associated with adaptation to physiological or pathophysiological stresses.
As an alternative to corrosion casting, X-ray computed tomography (CT)
has the potential for increasing the throughput enough to make such
studies practical (20, 21). The approach has been
initiated in studies both in vivo (11, 24, 25) and in
excised lungs (21, 38). Key advantages include the fact that the data set, including all the vessels with their spatial orientation and connectivity, is rapidly available in digitized form
and that the data can be collected on the same lung under different
experimental conditions (21).
However, the problems of quantifying the complex structure and then
providing a useful quantitative summary of the large data set remain
for both imaging and casting methods. One aspect of this problem is
that there is no fully automated image segmentation method that
provides the necessary accuracy and precision. Thus, as with casting
methods, some subset of the data obtained with semiautomated methods
must be used to extrapolate to the whole.
Various approaches have been developed for summarizing complex vascular
morphometric data sets (3, 5, 6, 13, 14, 18). They vary in
applicability to CT images and in the information content of the
resulting summary. In the present study, we develop an approach for
parameterizing the rat pulmonary arterial tree structure and vessel
distensibility from X-ray CT images. The objective is a parameter
vector useful for evaluating the efficacy of interventions directed at
putative mechanisms of pulmonary vascular remodeling, as a quantitative
trait vector for identifying genes involved in pulmonary vascular
remodeling (7, 27), and as input to mathematical models
relating pulmonary vascular structure and function (5, 6,
42). The approach is based on the concept of
"self-consistency," as applied by Fredberg and Hoenig (9,
10) to the bronchial tree. In this context, the definition of a
self-consistent arterial tree is one in which all portions of the tree
downstream from a vessel of a given diameter are, in a statistical
sense, similar. That is, any artery of a given diameter subtends a
similar number of terminal arterioles with a similar connecting
network. One attribute of the self-consistent tree structure is that it
can be fully characterized by measuring the dimensions of only the
vessels that make up the main trunk of the tree and its immediate
branches, where the main trunk is the route followed from the inlet
pulmonary artery by taking the largest diameter branch at each
successive bifurcation. In what follows we develop the rationale for
parameterizing the rat arterial tree, referred to as a "principle
pathway" analysis, based on measurements of the diameters and
distances from the inlet of the main trunk and its immediate branches.
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METHODS |
Animal preparation.
The lung from a 320-g fawn-hooded rat, acclimated to the normal
atmosphere at 193 m above sea level, was prepared for imaging as
previously described (21). The rat was anesthetized with pentobarbital sodium (40 µg/g body wt ip), the trachea was clamped, and the chest was opened. Heparin sodium (200 IU in 0.2 ml) was administered via injection into the right ventricle. The pulmonary artery was cannulated with a saline-filled catheter of polyethylene tubing (1.67 mm inner diameter and 2.42 mm outer diameter) via the
conus arteriosis, and the heart was dissected away. The lung was
removed from the chest and suspended from the trachea and pulmonary
arterial cannula. The lung was ventilated with 15% O2-6% CO2, balance nitrogen, at 40 breaths/min with
end-inspiratory and end-expiratory tracheal pressures of 8 and 3 mmHg,
respectively. This served to eliminate any atelectasis that may have
occurred during the dissection. The pulmonary artery cannula was
connected to a perfusion system primed with a physiological salt
solution containing 5% bovine serum albumin, and the lung was perfused for ~5 min at a flow rate in the range of 5-40 ml/min to remove residual blood from the lung vessels. The perfusate exited via the
severed pulmonary vein. Once cleared of blood, the lung, still suspended from the cannulas, was placed in a 41-mm inner diameter plastic cylinder. The cylinder axis was located at the center of a
horizontal turntable so that the lung could be rotated 360° around a
vertical axis between the X-ray source and detector with no other
significant X-ray absorbing objects passing through the beam. The
airway pressure was set at 6 mmHg, and the solution in the reservoir
connected to the arterial catheter was replaced by perfluorooctyl
bromide (PFOB) that was allowed to fill the arterial tree at a pressure
of ~20 mmHg. The PFOB provided high X-ray contrast for the vessel
lumen, and the surface tension at the PFOB-aqueous interface prevented
its entry into the capillary bed. Thus only the arterial vessels were
filled. Then the arterial pressure was set at 30 mmHg relative to the
horizontal plane through the center of the X-ray image of the lung. The
lung was rotated and stopped at each 1° increment so that 360 X-ray
images were acquired in ~10 min. The same sequence was repeated with
the arterial pressure set at 21, 12, and 5.4 mmHg, respectively, and
again at 30 mmHg. The actual intravascular pressure within each vessel relative to atmospheric pressure at the level of the vessel was obtained from the vertical distance of the vessel from the pressure reference level at the central horizontal plane of the image volume, and the PFOB density of 1.94 g/ml.
Imaging.
The X-ray system included an X-ray tube (model FXE-100.50; Fein-Focus)
with a 3 µm focal spot, an image train consisting of an image
intensifier set at the 17.8-cm aperture (model AI-5830-HP; North
American Imaging), and a charge-coupled device camera (model SMD 1M15,
Silicon Mountain Design; Colorado Springs, CO). The cylinder containing
the rat lung was placed in the scanner so that its central axis was
26.7 cm from the source. The source-to-image intensifier distance was
91.3 cm, yielding a geometric magnification of ~×3.4 and a half-cone
beam angle for the lung image of ~4°. Projection-image collection
frequency was 30 frames/s with 30 consecutive frames averaged to create
each stored image. After the lung imaging session, two additional
images were obtained. One was a flood field image with the lung removed
from the beam. The flood field image was used to correct for spatial
variations in the X-ray beam intensity and/or image intensifier gain.
The other image was of a phantom consisting of a uniform grid of
stainless steel spheres used as previously described (21)
to correct for spatial distortion (warping) inherent in the image intensifier.
Each of the 360, 8-bit planar images, consisting of a 512 × 512 array of pixels in the range of 0-255 (minimum to maximum) X-ray
intensity scale, was preprocessed as previously described (21) to correct the image intensifier spatial distortion,
to locate the axis of rotation, and to correct for nonuniform
illumination intensity. After preprocessing, cone-beam reconstruction
was performed on the projection data to yield an isotropic
reconstruction matrix of 497 × 497 × 497 voxels
representing a volume of ~3.8 × 3.8 × 3.8 cm3. The Feldkamp (8, 19) cone-beam algorithm
was utilized to account for the variation in the angle of the incident
X-ray beam at each pixel in the projection images. A filtered
[Shepp-Logan filter kernel (32)] backprojection
reconstruction was then performed.
Measurements.
The three-dimensional (3D) surface-shaded renderings of the arterial
tree in Figs. 1 and 2 help to demonstrate
the concept behind the subsequent measurements made on the
reconstructed image volume. Figure 1A illustrates a rat lung
image volume with the threshold set to eliminate the tissue and
smallest vessels, which tend to obscure the pulmonary arterial tree
structure viewed at this magnification. Figure 1B was
obtained by electronically pruning the tree so that the main pathway,
composed of the main trunk and the stumps of its immediate branches, is
separated from the rest of the tree. Figure
2 presents two views obtained after
ligating the right pulmonary artery and physically removing the right
lung before imaging. This was done to provide a less cluttered view of
the smaller branches than in the whole lung image in Fig. 1. When this
left lung image is rotated, so that a given small portion is close to
being in the plane of the page, one can appreciate the self-consistent
appearance by comparing the two circled portions. That is, one could
imagine that swapping the positions of the two circled portions would
hardly affect the overall tree structure.
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Fig. 1.
A: surface-shaded rendering of the rat
pulmonary arterial tree with the threshold chosen to obscure the
smallest vessels to provide an uncluttered view of the overall
structure. B: main trunk, with the stumps of its branches
attached, was obtained by electronically pruning the image on the left.
Measurements of diameter of the main trunk along its length and the
positions and diameters of branches off of the main trunk compose the
principal pathway data set.
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Fig. 2.
Surface-shaded renderings after ligating the right
pulmonary artery and physically removing the right lung before imaging.
This was done to provide a less cluttered view of the smaller branches
than in the whole lung image in Fig. 1. When this left lung image is
rotated so that a given small portion is close to being in the plane of
the page, one can appreciate the self-consistent appearance exemplified
by two circled portions. The graphs below the two images are of the
diameter vs. distance along the trunk within the circle as a means of
making a quantitative assessment of the similarity between the two
regions. CCW, counterclockwise.
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In what follows, the full data set, with no artificial threshold, was
used to make quantitative measurements wherein the gray scale number
(GSN), obtained from the reconstruction is the measure of X-ray
absorption within a given voxel (21).
Mapping the principle pathway.
For this analysis, a trunk is a pathway through the tree defined by
following the path of the largest diameter branch at each bifurcation.
The main trunk is the trunk that starts at the main pulmonary artery.
Because of the correlation between the distance between bifurcations
and diameter, the main trunk is either the longest, or very nearly the
longest, pathway through the tree. A minor trunk can start at any
bifurcation. To begin the principal pathway analysis, several trunks
were followed through transaxial slices of the three-dimensional volume
from their respective inlets to their most distal resolved vessel
segments. The first step in this process was to identify axial
x,y,z coordinates along a trunk that
could be used to specify locations for measuring trunk and branch
diameters and their distances from the trunk inlet. These coordinates
were identified interactively using an approach similar to that
described previously (21) but with modification required
to track a vessel regardless of its angle relative to the rotation axis
of the image volume.
In the description of the method for identifying
x,y,z coordinates, the terms trunk and
branch can be interchanged. The only distinction is that downstream
from a bifurcation the trunk has the larger of the two diameters. In
the most common case, the trunk and branch cross sections were oblique
to the transaxial slices and the branch and trunk diverged on
progression through the transaxial slices. This is shown
diagrammatically by the trunk and branch A in Fig.
3. In this case, a bifurcation was
located by observing the trunk cross-section while progressing through the sequence of transaxial slices until the single ellipsoid became bilobular and then split into two. The slice number, in which there was
visible separation between the two new ellipsoids (the crotch of the
bifurcation), was taken as the z coordinate as depicted in
Fig. 3 (gold-colored diamonds). The central x,y coordinates of each of the two ellipsoids on the slice at which the single ellipsoid completely separated into two, were recorded along with the
z coordinate.
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Fig. 3.
Identification of vessel segments in the image volume.
The numbered horizontal lines through the surface shaded rendering on
the upper left designate the numbered locations of the transaxial
slices on the right. The transaxial slices are used by the operator to
locate bifurcation points and the central axes of trunk and branch
vessel segments. This section of the tree was chosen for diagrammatic
purposes because it contains all three bifurcation types encountered on
traversal through a stack of transaxial slices. That is, branches like
branch B that converge with the trunk, or like
branch A that diverge from the trunk, or like
branch C, that are nearly in the plane of the transaxial
slices, as observed during a traversal through the stack of slices that
compose the image volume. The lower left image specifies branch types,
bifurcation points (gold diamonds), and line segments identified in
text (see METHODS). The blue circles are located on the
axes of the branches. The white circles are located on axes of the
trunk. Planes orthogonal to the axes designated by the black lines are
the planes on which the trunk diameters are measured. Red arrows are
the distance measurements contributing to the cumulative distance
x.
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The two additional cases are also illustrated in Fig. 3. When the
branch and trunk converged (branch B), so that the
ellipsoids merged on progression through the slices, z was
identified as being in the plane just before the merger. The central
x,y coordinates were then recorded in the same
manner as in the first case. The third case is evident when one of the
vessels is oriented so nearly parallel to the transaxial slices that
the lobulation indicative of a bifurcation appears elongated rather
than as a distinct ellipse (e.g., branch C in Fig. 3). The
z coordinate was identified by observing the proximal
intersection of the branch with the trunk, and by proceeding through
the volume half the number of slices encompassing the full width of the
horizontal vessel segment. The central x,y
coordinates of the oblique vessel segment were identified on this plane
as in cases A and B. The central
x,y coordinates of the horizontal segment were
placed on this same plane but on the visually interpolated edge of the
oblique vessel ellipse.
Once these x,y,z coordinates of the
main trunk (white circles in Fig. 3) and the upstream ends of the
branches (blue circles in Fig. 3) were located, the upstream branch
coordinates were revisited. Each branch was followed as it diverged
from the trunk until the first bifurcation was reached. The downstream
x,y,z coordinates were then identified
in the same manner as described for cases A, B,
and C. For most branches, the process ended at this first
bifurcation off the main trunk. However, nine branches were designated
minor trunks and followed to their respective terminations as was the
main trunk.
Having identified these coordinates, the midpoints
[(x1 + x2)/2,
(y1 + y2)/2,
(z1 + z2)/2] of the line
between consecutive trunk
xi,yi,zi
coordinates and between upstream and downstream branch
xi,yi,zi
coordinates were identified. Two vectors intersecting these midpoints,
orthogonal to each other and to the central axis, specified an
orthogonal plane through the midpoint of the central axis line segment.
The thick black lines in Fig. 3, bottom left, represent one
of each pair of these vectors. Cubic interpolation was performed to
obtain the GSN values of the orthogonal slice on this plane. The vessel
segment diameter was estimated from its cross-sectional image in this
slice as previously described (21). The distance along a
trunk was computed as the cumulative distances between consecutive
midpoints, beginning at the x,y,z of
the inlet segment.
Note that Fig. 3 represents the identification of the coordinates and
central axes. The diameter measurements were made on the orthogonal
slices wherein the vessel cross sections appear as circles rather than
ellipses as shown previously (21). Whereas the
identification of the x,y,z
coordinates was interactive, once the coordinates were specified, the
diameter measuring algorithm was automatic. One can appreciate that the
orthogonal slice through the midpoint between the case
A and case B bifurcations on Fig. 3 would be
circular, whereas the orthogonal slice through the midpoint between
case A and case C would be
distorted due to the fact that the two branches come off the trunk less
than a full branch diameter apart. The effect of this
distortion is averaged in the vessel diameter estimate by the
optimization procedure (21). These cases contribute to the
apparent roughness of the taper of the trunks observed in the diameter-
versus-distance plots presented in Principal pathway
parameterization. If the trunk segment was distorted to the extent
that the diameter-measuring algorithm did not converge, the next
downstream trunk location was assigned two branches rather than one
(the algorithmic definition of a trifurcation). The overall result was
robust with regard to the operator choices. For example, as case
A or case B bifurcation approaches a case C
bifurcation, the choice of method makes little difference to the
outcome. Confusion with regard to a case A or case
B does not occur.
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RESULTS |
The data obtained was a contiguous sequence of axial segments
consisting of the main trunk and nine minor trunks, and the diameter of
the branch off of the main trunk associated with each segment.
Principal pathway parameterization.
The distances through the lumen from the inlet to the ends of the
contiguous sequence of axial segments consisting of the main trunk and
nine minor trunks are plotted on Fig. 4,
A, C, and E. To further investigate
the self-consistency of the arterial tree, we begin by observing that
the diameter, D, vs. distance, x, along the main
trunk has an appearance reminiscent of the following functional form
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(1)
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where D(0) is an approximation to the trunk diameter at
x = 0; Ltot would be the total
length of the trunk if extended until D = 0, i.e.,
D(0)/Ltot is a measure of the average
taper of the entire trunk, and c is the measure of the
concavity-convexity of the taper, i.e., when c < 1, the curvature of the taper is convex to the trunk axis, and when
c > 1, the curvature of the taper is concave to the
axis. A conceptual representation of Eq. 1 is shown in Fig.
5.
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Fig. 4.
A: same surface-shaded renderings as in Fig.
1, with 10 trunks identified by colored lines. B: The 10 colored lines have been bent and placed along the main trunk to
demonstrate diagrammatically where the minor trunks would superimpose
on the main trunk. C: diameter (D) versus
distance (x) relationships for the 10 individual trunks
identified in the top images, with Eq. 2 fit through the
data. The distance, x = 0, is where the branch
leaves the main trunk. D: superposition of
the nine trunks on the main trunk by shifting the distance axis by
si to the right. E: cumulative number
of branches (NBr) vs. x along the 10 trunks
identified at top. On this graph, the branches start at
their respective locations, si, along the
main trunk. The lines are Eq. 4 fit though the data.
F: superposition of the 9 minor trunks on the main
trunk accomplished by shifting up by
ni.
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Fig. 5.
Schematic representation of the parameters of Eq. 1 describing the main trunk geometry. D(0),
approximation to the trunk diameter at x = 0;
Ltot, total length of the trunk if extended
until D = 0; c is the measure of the
concavity-convexity of the taper.
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Self-consistency implies that if Eq. 1 describes the
D versus x relationship for the main trunk, then
the D versus x graph of any trunk would be
described by Eq. 2
![D(x)=D(0)[1−(x+s<SUB>i</SUB>)/L<SUB>tot</SUB>]<SUP><IT>c</IT></SUP>](/content/vol281/issue6/fulltext/H2747/img002.gif)
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(2)
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where D(0), Ltot, and
c would be the same for all trunks, and
si would be the distance along the
x-axis of Fig. 4C that an individual trunk would
need to be shifted to match the equivalent distal portion of the main
trunk. Thus fitting Eq. 2 to all of the data from the 10 trunks on Fig. 4C simultaneously yields 12 parameters, 9 unique si values (si for
the main trunk is 0) and 1 common value each for D(0),
Ltot, and c for the 10 pathways. Because of its distortion due to the cannulation, the main pulmonary artery is not included in the main trunk data set. Instead,
x = 0 occurs at the bifurcation of the pulmonary
artery, so that D(0) is the Eq. 2 approximation
to the right pulmonary artery inlet diameter. The fitting of this and
the subsequent equations to the data was carried out using the FMINS
routine in the Matlab version 6 optimization toolbox. Because a
value of D(x) at x > Ltot is not a real number (i.e., a negative
number raised to the power c), Eqs. 1 and 2 were set to 0 when x > Ltot (or x + si > Ltot) occurred
during this and subsequent optimizations for equations having similar
form. To construct Fig. 4D, the data for the minor trunks
were horizontally translated to the right by the distance
si. The resulting superposition is a
characteristic of self-consistency.
Another characteristic of self-consistency is that the branches off any
trunk should be similarly distributed. Figure 4E again shows
the 10 trunks, but with the number of upstream branches (NBr) plotted versus x. On the left,
each trunk begins at its respective si. The
progression of the curves is again suggestive of self-consistency.
To further investigate this aspect of self-consistency of the arterial
tree, we note that NBr versus x along
the main trunk has the appearance of the following functional form
![N<SUB>Br</SUB>(<IT>x</IT>)<IT>=N</IT><SUB>tot</SUB>{1<IT>−</IT>[1<IT>−</IT>(<IT>x/L</IT><SUB>tot</SUB>)]}<SUP><IT>b</IT></SUP>](/content/vol281/issue6/fulltext/H2747/img003.gif)
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(3)
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where Ntot is an approximation to the total
number of branches off the main trunk, i.e.,
Ntot/Ltot is a measure of
the average number of branches per unit length, and b is the
measure of how the branches are distributed along the length, i.e.,
when b is >1, the number of branches per unit length
increases with x and when b is <1, the number of
branches per unit length decreases with x. Thus a change in
Ntot/Ltot would reflect a
change in branch density, whereas a change in b would
distinguish the large versus small vessel contribution to such a change.
Self-consistency implies that if Eq. 3 describes
NBr versus x relationship for the
main trunk, then the number versus x graph of any trunk
would be fit by Eq. 4
![N<SUB>Br</SUB>(<IT>x</IT>)<IT>=N</IT><SUB>tot</SUB>{1<IT>−</IT>[1<IT>−</IT>(<IT>x+s<SUB>i</SUB></IT>)<IT>/L</IT><SUB>tot</SUB>]}<SUP><IT>b</IT></SUP><IT>−n<SUB>i</SUB></IT>](/content/vol281/issue6/fulltext/H2747/img004.gif)
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(4)
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where Ntot and b would be the
same for all trunks and ni would be the number
of branches an individual trunk would need to be shifted to match the
equivalent distal portion of the main trunk. Thus, fitting Eq. 4 to all of the data from the 10 trunks shown in Fig.
4E, simultaneously yields 11 parameters, 9 unique ni values (ni for the
main trunk is 0) and 1 common value each for
Ntot, and b for the 10 pathways.
Because a value of x > Ltot is
not a real number, x > Ltot (or
x + si > Ltot) were ignored if they occurred during this
and subsequent optimizations for equations having similar form. The
resulting superposition, in Fig. 4F, along with that in Fig.
4D, is the basis of the following parameterization of the
tree structure.
Under the assumption of self-consistency, only the dimensions of the
main trunk and the diameters of its immediate branches need to be
measured to fully characterize the tree structure. With Eq. 1 as the representation of the main trunk diameter versus distance
relationship, Eq. 5 can represent the diameters,
DBr, of the vessel segments branching off the
main trunk at x
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(5)
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where DBr(0) is an approximation to the
diameter of the first branch off the main trunk (the left pulmonary
artery), and, assuming self-consistency,
DBr(0)/D(0) is an average ratio of branch-to-trunk diameters. Although represented by a continuous function, the branches are actually placed only at the number, Ntot, of discrete locations along the main
trunk, and their diameters vary with x proportionately much
more than the main trunk diameter. The 
is a variable diameter
increment reflecting this heterogeneity of the branch diameter along
the principle pathway. Figure 6
represents an entire principle pathway data set with Eqs. 1 and 5 fit simultaneously with four free parameters:
D(0), DBr(0),
Ltot, and c, and with = 0. was then represented by a coefficient of variation
(CVBr) obtained from the variance in the
measured branch diameter around the fitted Eqs. 1 and 5, namely
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(6)
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where data refers to the measured value and fit refers to the
value predicted by the fit of Eq. 5 to the data for each of the Ntot branches along the principal pathway.
The rationale behind Eq. 6 is the concept that there is an
underlying smooth taper to the main trunk that can be represented by
Eq. 1, and that the variability in measured data about the
fitted line reflects the irregular distortion of the smooth taper near
bifurcations as well as errors in the diameter estimates. Assuming that
these sources of variance also exist in the branch diameter data,
subtracting the variance about the fit to Eq. 1 from that
around Eq. 5 provides an approximation to the variance in
the branch diameters due to the heterogeneity in the branch-to-trunk
diameter ratio.
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Fig. 6.
Principal pathway data including the main trunk diameter
at x from the bifurcation of the main pulmonary artery and
the diameters and x locations of the branches off of the
main trunk. Data are from images obtained at an arterial pressure of 21 mmHg. Below the x-axis is another schematic representation
of the main trunk to emphasize the concept that the principal pathway
is a tapered tube with branches, represented in this depiction as
depressions or scars where the branches have been broken off.
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From Eqs. 1, 3, and 6, the parameter vector
summarizing the tree morphometry at a given vascular pressure would be
[D(0), DBr(0),
Ltot, Ntot, c,
b, CVBr]. However, as illustrated in
Fig. 7, the vessel diameters are also a
function of vascular pressure, which over the pressure range studied
can be approximated by Eq. 7 (1, 39, 42)
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(7)
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where 
is a distensibility coefficient, namely, the fractional
change in diameter per unit change in pressure. Assuming that is
diameter independent (1, 21), Eqs. 2 and 4 can be written as
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(8)
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and
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(9)
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Thus the [D(0,0), DBr(0,0),
Ltot, Ntot, c,
b, CVBr, ] vector can be obtained
by fitting Eqs. 8, 9, and 3 to all of
the principal pathway data obtained over the entire pressure range
studied. The approach was to first simultaneously fit Eqs. 8 and 9 to the D(x,P) and
DBr(x,P). The two surfaces
representing this fit are shown in Fig.
8. Then the Ltot
was used as an input to Eq. 3, which was then fit to the
NBr(x) data as shown in Fig.
9 to obtain the estimates of
Ntot and b for all pressures
simultaneously. Although a slight increase in
Ltot with increasing pressure can be observed in
Fig. 9 it was not considered to be of sufficient magnitude to warrant
consideration in the model expressions. We suspect that it is mainly
due to a small stretching of the vertically oriented lungs as the lung
weight increased with increasing mass of PFOB. In any case, there is no
evidence for a substantial effect of intravascular pressure on vessel
lengths. The parameter values obtained in this manner for the lung
studied are as follows: D(0,0) = 1.34 mm,
DBr(0,0) = 0.610 mm,
Ltot = 41.2 mm,
Ntot = 37, c = 0.695, b = 1.67, CVBr = 54.6%,
and = 2.25% mmHg.
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Fig. 7.
Principal pathway data similar to Fig. 6, but from images
obtained with the arterial pressure set at 30 mmHg (A) and
12 mmHg (B) to demonstrate vessel distensibility.
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Fig. 8.
The entire principal pathway diameter versus distance
data set from the images obtained at all pressures. The two surfaces
are fit simultaneously from Eqs. 8 and 9 to the
entire data set.
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Fig. 9.
Entire NBr vs. x data
set from the images obtained at all pressures. The line is Eq. 3 fit simultaneously to the entire data set.
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One objective of this study was to develop a method for morphometric
analysis with high enough throughput that comparisons can be made
between lungs and between groups of lungs in future studies. We
envision this parameter vector to be the basis of such comparisons. In
addition, elements of the parameter vector can also be used to
extrapolate the relatively small number of measurements made to an
entire tree structure. An example approach would be as follows. The
first step is to determine the distances between branches,
L(NBr), in the model tree using
Eq. 10
![L(N<SUB>Br</SUB>)<IT>=L</IT><SUB>tot</SUB>{[1<IT>−</IT>(1<IT>−</IT>(<IT>N</IT><SUB>Br</SUB><IT>/N</IT><SUB>tot</SUB>)]<SUP>1<IT>/b</IT></SUP>}](/content/vol281/issue6/fulltext/H2747/img010.gif)
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(10)
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Then the branch x positions along the trunk are
identified from the cumulative
L(NBr). In a self-consistent tree,
the main trunk diameter must decrease monotonically from the inlet.
Therefore, the next step is to calculate the model main trunk diameter
using Eq. 8 for the particular pressure chosen for the
model. The branch diameters (the average from the five pressures for
each branch normalized to the Eq. 8 trunk surface) can then
be used to determine the model branch diameters. This is done by
reassigning each branch diameter the trunk diameter closest to the mean
branch diameter, unless the branch diameter was smaller than the
smallest measured trunk diameter. In the latter case, the trunk was
extended along the Eq. 1 line to include the branch
diameter. An entire tree down to the resolution of the image data can
then be constructed using the self-consistency recursion relationship
that results from all vessels down stream from a given
NBr having the same dimensions. As with any
model specified by a smaller number of parameters than the dimensions
of the real system, the tree constructed is a stereotypical
approximation to the real tree. However, it retains representations of
asymmetry and heterogeneity missing from some morphometric synopses
(6) and in a way that can provide input to a
computationally efficient hemodynamic model (9). This
model tree is depicted in Fig. 10 in
the format of Figs. 6 and 7 except that lines representing the minor
trunks and their immediate branches have been added to give an
impression as to how the principal pathway data are extrapolated to the
entire tree. The entire tree would be represented by many more lines because the branches off of the minor trunks become minor trunks and so
on.
View larger version (29K):
[in this window]
[in a new window]
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Fig. 10.
Self-consistent model tree, for a pressure of 21 mmHg,
constructed from the parameter vector obtained from the Eq. 8 and 9 fit to the surfaces in Fig. 8 and Eq. 3
fit to the data in Fig. 9. Model trunk diameters
( ) at the x locations correspond to each
branch diameter
( ). The
solid lines are the individual trunk diameter versus x
relationships for the main trunk and those minor trunks beginning at a
branch off the main trunk. Dashed lines are the Eq. 9 for
each minor trunk.
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DISCUSSION |
The results suggest that the rat pulmonary arterial tree is
self-consistent and conducive to principal pathway analysis. This property may be useful for summarizing the pulmonary arterial morphometric data available in the three-dimensional volume
reconstruction of the lung. A practical consequence of
self-consistency is that it minimizes the number of measurements needed
to characterize the complex tree structure consisting of thousands of
vessel segments. This will be important, at least until completely
automated image segmentation methods become available. Another
practical consequence is that self-consistency lends itself to a
computationally efficient means of evaluating the hemodynamic impact of
changes in the structure (9).
The expectation is that the parameter vector obtained as described will
serve three interrelated purposes as indicated in the introduction. One
is to provide a basis for comparing the lumenal morphometry of
pulmonary arterial trees in animals exposed to different remodeling
stimuli and to interventions directed at putative mechanisms involved
in the remodeling. Also, if any element of the vector is found to
exhibit differential susceptibility to a particular remodeling stimulus
in a specific rat strain, it may be a useful phenotype for quantitative
trait localization by genetic linkage analysis (7). A
third purpose is as input to mathematical models relating pulmonary
vascular structure and function in normal and remodeled pulmonary
arterial systems (2, 6, 42). In this regard, the parameter
vector can be translated into a structure amenable to hemodynamic
modeling similar in concept to the airway modeling of Fredberg and
Hoenig (9, 10). The recursion relationships that result
from self-consistency (alluded to in Fig. 10) provide computational
efficiency for large heterogeneous asymmetrical structures. The
objective of that kind of modeling would be to determine how changes in
the morphometry and vessel mechanics, revealed by changes in the
parameter vector, contribute to the changes in function, e.g.,
pulmonary hypertension, abnormal distributions of flow, shear stress, etc.
We chose the rat lung for this developmental study because of its
history of use in studies of pulmonary vascular remodeling (28,
29, 36), because of its increasing importance in genetic studies
(7, 17, 31, 34), and because a small lung takes advantage
of the geometric magnification possible with microfocal imaging. For
the combination of image-acquisition procedures in the present study,
the smallest arteries for which diameters were confidently measured
were ~50 µm in diameter (21). The diameters of
terminal pulmonary arteries, i.e., arteries connecting directly to
capillaries, have reported to range from ~13 to 30 µm in diameter in various studies and species (5), which is just beyond
the resolution in the present study. Improvements in computational capacity to take advantage of the higher resolution capabilities of the
detector system and the use of volume of interest reconstruction (23, 30) to allow for higher geometric magnification than in the full lung reconstruction used in the present study, are examples
of feasible enhancements for future development.
The algorithms developed to quantify the 3D image volumes are partially
automated (containing some interactive steps). Whereas advances are
being made in the field of vessel segmentation in micro CT images
(11, 37) that may be applicable to lung images in the
future, the method described for locating a bifurcation seems to
produce a satisfactory result and can be easily and robustly implemented by the operator. Each branch is associated with a segment
of the trunk, but no attempt was made to establish a specific location
along the segment. Other approaches are possible. For example, Wood et
al. (38) defined the bifurcation (branch) point as the
intersection of the central axes of the parent and two daughters, and
they developed a means for identifying that point. One criterion for
judging the value of adding such a method in the future might be the
sensitivity of a hemodynamic model to how the branches are distributed
along the total length of the principle pathway. Given the relative
sensitivity of the pressure-flow relationship to the diameter versus
the length of a vessel segment, one might anticipate that the
sensitivity will not be great. In fact, the robustness with regard to
individual length variations may be one of the reasons that
evolutionary forces have not done more to narrow the wide variation in
segment length-to-diameter ratio in the pulmonary vascular bed
(35), or in other vascular beds (35, 41).
With the approach taken, cumulative distance along a pathway, which is
insensitive to the definition of the length of an individual segment,
is given primacy.
The functional forms fit to the data in the present study were chosen
because they represent major trends in the data with a minimum number
of parameters, the forms are similar to those resulting from a previous
symmetrical tree analysis (5), and the parameters have
clear interpretations with respect to the shape of the principal
pathway (Figs. 5 and 6). It remains to be seen whether this parameter
vector has the flexibility to represent all of the systematic and
important variations that might occur. However, it might be considered
a nested version of a higher dimensional model that might need to be
constructed to include such variations. For example, the representation
of the vascular distensibility by a single diameter-independent
parameter may need adjustment if the diameter-dependent remodeling of
the vessel walls, which has been reported in rats exposed to various
stimuli (22, 26, 28), also results in diameter-dependent
distensiblity. It may be surprising that the distensibility can be
represented by a single parameter even in the normal rat. This
uniformity has been observed previously for the rat (21)
and other species (1), and it can be appreciated by
observing the proportionate changes in the diameters of the trunk and
branches along the trunk length in Fig. 7 as well as in the surface fit
in Fig. 8.
Although the rat was used in the present study, it seems likely that
other mammalian species will have self-consistent pulmonary arterial
trees. The morphometric analysis of Liu et al. (24) on the
portions of the dog pulmonary arterial tree having dimensions accessible in whole animal CT scans would appear to support that contention. Likewise, insofar as the arterial tree follows the airway
structure, airway self-consistency (9, 15) would predict arterial self-consistency. When the entire pulmonary arterial tree is
in the field of view, species differences in gross structure are
generally discernible (6). One of the distinguishing
features at the whole tree level of observation is the degree to which the structure is dominated by a branching pattern wherein the two
daughter branches at a bifurcation are more nearly equal in diameter
[dichotomous (4, 14)] or a pattern wherein one of the
two daughters tends to be much smaller than the parent [monopodial (4, 14)]. These are sometimes referred to as maple tree
versus pine tree-like structures, respectively. The latter has a more obvious main trunk than the former. Application of the principal pathway analysis does not imply a predominantly monopodial branching pattern. Such a pattern may make it easier to decide which pathway to
follow at a bifurcation. However, if two equal branches were ever
encountered (an uncommon event even in the most dicotomously branching
trees), according to the self-consistency concept, following either
pathway would produce the same result. As a related point, gross
differences between species that can be observed at low magnification
tend to disappear at high magnification (6). This is
probably related to the fact that the functional respiratory units also
tend to have similar species-independent structures and the fact that
the arteries eventually have to hook up to the capillaries that are
essentially all one size. Self-consistency, as we have used the term,
implies that, having traversed the longest pathway through the tree,
the structure downstream of any vessel of a given diameter is known.
Thus, using local diameter as the landmark, self-consistency requires a
continuous taper along each pathway (i.e., that the daughter branches
at a bifurcation are smaller than the parent). This self-consistency
does not extend into the capillary bed because local capillary diameter
provides no information about how much of the capillary bed lies
downstream from that location. This self-consistency does not require
the scale independence of a fractal structure, but it does imply
that, as one progresses through the tree, the local diameter specifies (in a statistical sense) the downstream structure. A self-consistent tree structure has attributes of a fractal in entreating simplified mathematical representation of the complex structure. Whereas a fractal
tree can be self-consistent, self-consistency does not require scale
independence. However, the transformation imposed by certain ordering
schemes can reveal fractal relationships in the transformation space
that are not necessarily evident in the direct appearance of the
structure (6, 18, 41).
In conclusion, the results demonstrate self-consistency in the rat
pulmonary arterial tree. An approach for exploiting this property is
presented. The approach may be one way that high spatial resolution
volumetric X-ray CT can contribute to the information required for
understanding normal pulmonary arterial structure-function relationships and how these relationships are affected by pulmonary vascular remodeling associated with pulmonary vascular development, adaptations to environmental stress, or disease.
| |
ACKNOWLEDGEMENTS |
This study was supported by National Heart, Lung, and Blood
Institute Grant HL-19298, The Whitaker Foundation, The W. M. Keck Foundation, The Falk Medical Trust, and the Department of Veterans' Affairs.
| |
FOOTNOTES |
Address for reprint requests and other correspondence: C. A. Dawson, Research Service 151, Zablocki Veterans Affairs Medical Center, 5000 W. National Ave., Milwaukee, WI 53295 (E-mail:
cdawson{at}mcw.edu).
The costs of publication of this
article were defrayed in part by the
payment of page charges. The article
must therefore be hereby marked
"advertisement"
in accordance with 18 U.S.C. Section 1734 solely to indicate this fact.
Received 25 June 2001; accepted in final form 4 September 2001.
| |
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Am J Physiol Heart Circ Physiol 281(6):H2747-H2756
TOP
ABSTRACT
INTRODUCTION
![<IT>−</IT>{1<IT>−</IT>[1<IT>−</IT>[(<IT>N</IT><SUB>Br</SUB><IT>−</IT>1)<IT>/N</IT><SUB>tot</SUB>)]<SUP>1<IT>/b</IT></SUP>]}](/content/vol281/issue6/fulltext/H2747/img011.gif)
