Vol. 88, Issue 6, 2260-2268, June 2000
A three-dimensional model of the human pulmonary
acinus
Hiroko
Kitaoka1,
Shinichi
Tamura1, and
Ryuji
Takaki2
1 Division of Functional Diagnostic Imaging,
Osaka University Medical School, Suita City, Osaka 565-0871; and
2 Department of Engineering, Tokyo University
of Agriculture and Technology, Tokyo 184-8588, Japan
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ABSTRACT |
A three-dimensional (3-D) model of the human pulmonary
acinus, a gas exchange unit, is constructed with a labyrinthine
algorithm generating branching ducts that fill a given space
completely. Branching down to the third respiratory bronchioles is
generated with the proposed algorithm. A subacinus, a region supplied
by the last respiratory bronchiole, is approximated to be a set of cubic cells with a side dimension of 0.5 mm. The labyrinthine algorithm
is used to determine a pathway through all cells only once, except at
branching points with the smallest path lengths. In choosing each step
of a pathway, random variables are used. Resulting labyrinths have
equal mean path lengths and equal surface areas of inner walls. An
alveolus can be generated by attaching alveolar septa, 0.25 mm long and
0.1 mm wide, to the inner walls. Total alveolar surface area and
numbers of alveolar ducts, alveolar sacs, and alveoli in our 3-D acinar
model are in good accordance with those reported in the literature.
pulmonary gas exchange; lung model; three-dimensional computer
modeling; pulmonary labyrinth
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INTRODUCTION |
THE STRUCTURE OF THE PULMONARY acinus, defined by
Fletcher et al. (3) as a region of the lung distal to the terminal
bronchiole (TB), is one of the most complicated in the human body. The
TB branches into several respiratory bronchioles (RBs) in the acinus. The RB then branches into several alveolar ducts (ADs), which branch
into several alveolar sacs (ASs, blind ends of the air pathway). The
acinar structure is regarded as a branching ductal system filling the
three-dimensional (3-D) space completely. The alveoli are open surfaces
attached to the inner walls of RBs, ADs, or ASs. Therefore, the
intra-acinar surface is topologically homeomorphic to one sheet,
despite its complicated 3-D architecture.
Several morphometric studies of the pulmonary acinar
structures have been carried out by making serial histological
sections (6, 9, 11-13, 15) and by observing the casts (2, 4, 14).
However, 3-D models of acinar structure have not been
proposed. We propose a method to simulate the acinar structure
by use of an algorithm generating an intra-acinar pathway.
The basic idea of the algorithm is to construct a 3-D
labyrinth, which starts from an entrance, passes every point only once,
and includes branching. After construction of the labyrinth,
alveoli are attached on its inner walls. From a requirement that the
air transport should be effective, the algorithm should be
designed so that the path lengths to each alveolus are the
smallest. We call this algorithm a labyrinthine algorithm.
We previously proposed an algorithm to generate the airway tree down to
TBs (10). The airway tree algorithm, i.e., the tree algorithm, divides
the lung into numerous acini. According to Weibel (16), the TBs branch
three times dichotomously within the acinus; hence, the acinus is
divided into several subacini supplied by the last RBs. The
intra-acinar pathway down to the last RBs can be given by the tree
algorithm. ADs and ASs, which fill the subacinus, are given by the
labyrinthine algorithm.
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ALGORITHM GENERATING A 3-D LABYRINTH |
We start with an assumption that the whole acinus can be approximated
by a set of numerous cubic cells and that the intra-acinar pathway can
be regarded as a sequence of cubic cells (Fig. 1). Therefore, construction of the pathway is locally limited only to
extend straight or to turn at a right angle. An oblique pathway is
approximated by a sequence of zigzag paths. We fix the cell size and
the width of alveolar septa to 0.5 and 0.1 mm, respectively, and assign
alveolar septa to be attached to the inner walls at 0.25-mm intervals
(Fig. 1). These values are reasonable, since the outer and inner
diameters of ADs and ASs are then 0.5 and 0.3 mm, respectively, which
are consistent with the data previously reported (4, 17).
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Fig. 1.
Three-dimensional (3-D) alveolar model. There are 8 alveoli in a cubic
cell with a side dimension of 0.5 mm. A 0.1-mm-wide alveolar septum is
attached with inner walls at 0.25-mm intervals.
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The labyrinthine algorithm is applied to the subacinus, the shape
of which can be assumed to be a rectangle or more
realistic forms given by the tree algorithm (10). The starting point of the labyrinth is chosen at a cell including the end point of the last
RB. The basic idea of the algorithm is to determine a branching pathway
that passes all cells only once, except at branching cells, without
loop formation. The mean path length (MPL) from the starting point to all cells is required to be as small as possible.
First, we consider a rectangular subacinus consisting of L × M × N cells. Figure
2 shows an example of a cubic subacinus
made of 6 × 6 × 6 cells. The starting point is assigned at
a corner of the subacinus. Figure 3 shows
some simple examples of labyrinthine walls and pathways at the lowest
cross section (Fig. 2). The MPL becomes the largest when there is no
branching (Fig. 3A). In this case, the pathway goes up to the
second level at the top left cell. The MPL is calculated as
(LMN 
1)/2. This is the least effective
pathway. For derivation of MPL see APPENDIX A.
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Fig. 2.
Cubic subacinus with 6 × 6 × 6 cells. A: outer
appearance. B: cross section at lowest level shown in Fig. 3.
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Fig. 3.
Cross sections of cubic labyrinth starting at bottom left.  ,
Starting points;  , branching points;  , end points. Paths are
indicated by thin lines. A: case without branching, where mean
path length to each cell (MPL) is largest. B: a simple example
giving smallest MPL. Branchings occur at left, and end points
are located at right. C: 1 more branching is added
without changing MPL. D: branching with a detour, which has a
larger MPL.
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On the other hand, a simple example with the smallest MPL is shown in
Fig. 3B, where branchings occur horizontally and vertically at
every cell located on the left side and every branch ends at the right
side. In this case, MPL is (L + M + N 3)/2 (see below) and the number of ends is L × N.
When the wall configuration is given as shown in Fig. 3C, the
MPL remains unchanged, although the numbers of ends and branchings increase by 1.
It is obvious that the path length to each cell becomes larger when it
includes detours, as shown in Fig. 3D. Here we define a detour
as a route containing reverse directions. The MPL and the standard
deviation (SD) are theoretically given by the following equations as long as there are no detours
In
these formulas, the summations are taken for all paths. When L = M = N, MPL = 1.5(L 1), and SD = 0.5L.
Hence, we have constructed a labyrinthine algorithm generating a
branching pathway without detours. When a cell has multiple directions,
the direction generating a path is selected by a random process. Hence,
this algorithm generates many kinds of labyrinths with the same MPL.
The detail of the algorithm is explained in APPENDIX B.
There are no detours in any labyrinth generated with the above
algorithm as long as the starting point is located at the corner. On
the other hand, if the starting point is located inside the subacini,
which is usual in an actual lung, we must modify the algorithm to avoid
detours. Thus we have introduced a concept of priority direction for
all i, j, and k axes according to the spatial
relationship between the starting point and the center of the
subacinus. Priorities are given to directions from the starting point
to the center. Figure 4 shows priority
directions for three different starting points. When several priority
directions are possible, a direction is chosen randomly from them. If
there is no priority direction, a direction is chosen randomly from possible nonpriority directions. This rule prevents activated cells
from selecting directions that are the reverse of those from the
starting point to themselves (see APPENDIX B).
Figure 5, A and B, shows
examples of directions with and without priority. The starting points
move from the corner toward the center of the subacinus by 1 for every
direction. There are no detours in Fig. 5A; however, there are
three detours in Fig. 5B, as shown in Fig. 5C,
resulting in a longer MPL than in Fig. 5A. When a starting
point is located at a corner, there is no cell in nonpriority direction
from the starting point. Therefore, the algorithms with and without
priority are the same in that case.
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Fig. 5.
Paths of cubic labyrinth, where starting cell is shifted from corner
toward center of labyrinth by 1 unit in all directions. Cross section
is at 2nd level. A: case with priority direction, where no
detours are recognized. B: case without priority direction
allowing 3 detours. C: pickup of 3 detours in B.
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When the location of the starting point is shifted from the corner to
the center by l, m, and n in the three directions,
respectively, the MPL is calculated theoretically as follows
There is a very simple but important relationship between the number of
inner walls and the number of cells. A cell along the pathway without
branching has four walls. On the other hand, an end cell has five
walls, and branching decreases the number of walls. For example,
dichotomous and trichotomous cells have three and two walls,
respectively. Because an increase in branching is associated with an
increase in end cells, the total number of walls does not change by
generating branching. Hence, for a sufficient number of the cells
(Nc), we have the following relation with the
number of inner walls (Nw) independent of the
structure of the labyrinth
This
means that the surface area of inner walls per unit volume is constant
for any labyrinth structure. Because alveoli are attached along inner
walls of the pathway, the number of alveoli and the total surface area
of alveoli per unit volume should be constant.
There is another simple relationship between the total number of
branches (B), the number of branching nodes (T), and the number of end
points (E) in a branching tree structure as follows (8)
where the starting point is not
regarded as an end point. This relationship is independent of whether
branching is dichotomous, trichotomous, or quadrichotomous. In the
human pulmonary subacinus, the trunk is the last RB, and the number of
ASs is equal to E. On the other hand, the sum of the numbers of ADs and
ASs is equal to B 1. Therefore, the number of ADs is calculated
as B E 1, which is equal to T 1.
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CHARACTERIZATION OF 3-D LABYRINTHS |
We examined the characteristics of the 3-D labyrinth generated by our
algorithm, first for a cubic subacinus and then for more realistic
acinar contours generated by our airway tree algorithm (10).
Here, the acinar size is discussed. Acinar volumes have been
investigated by several researchers. Hansen and Ampaya (5) and Kitaoka
and Itoh (9) reported the average acinar volume at
three-fourths total lung volume (TLV) as 183 and 173 mm3, respectively. This means that the acinar volume is
~230 mm3 at maximum inspiration. On the other hand, the
airway tree generated by our algorithm contains 27,000 TBs within the
TLV of 6.37 liters (10). According to Weibel (16), the lung parenchyma
is 90% of TLV. Hence, the average acinar volume at the maximum
inspiration in our airway tree model is calculated as 212 mm3 (= 6.37 × 106 × 0.9/27,000).
This value is close to those obtained with the fixed lungs mentioned
above. Finally, we assigned the average acinar volume as 216 mm3, which corresponds to a cube with a side dimension of
6 mm.
When a 0.1-mm-wide alveolar septum is attached to the inner walls at
0.25-mm intervals, as mentioned above, one cell contains eight alveoli
on average. The surface area of one alveolus is calculated as 0.255 mm2 (= 0.25 × 0.25 × 2 + 0.25 × 0.1 × 4 +0.15 × 0.1 × 2), and the alveolar surface area
per cell is calculated as 2.04 mm2 (= 0.255 × 8).
Because the volume of one cell is 0.125 mm3, the number and
the total surface area of alveoli per unit volume of lung parenchyma
are calculated as 64/mm3 (= 8/0.125) and 16.3 mm2/mm3 (= 2.04/0.125), respectively.
The total number of alveoli and the total surface area with TLV of 6.37 liters are then estimated as 370 × 106 (= 6.37 × 106 × 0.9 × 64) and 93.4 m2 (= 6.37 × 0.9 × 16.3), respectively, which are in good
accordance with the data previously reported (1, 5, 7, 13, 17).
Cubic subacinar model.
As a model of subacinus, we assumed a cube with a side dimension of 3.0 mm, which contains 63 = 216 cells. First, we assigned the
starting cell at a vertex. The MPL is always 3.75 ± 1.5 (SD) mm,
equal to the theoretical values as mentioned above. Under the
assumption that lengths of all RBs in the acinus are 1.5 mm, the MPL
from the first RB is calculated to be 8.25 ± 1.5 mm. These values are
in good accordance with the morphometric data of Haefeli-Bleuer and
Weibel (4), who reported 8.8 ± 1.4 mm.
Next, we assigned the starting point inside the subacinus (Fig. 4). We
performed 200 trials with and without priorities. When priorities were
given, MPL was always equal to the theoretical value, 2.75 mm. On the
other hand, MPL without priorities became >2.75 mm and ranged from
2.79 to 3.14 mm.
The numbers of ADs and ASs are equal to E and T 1, respectively, as mentioned above. These numbers are different at every trial because of random processes. We performed 200 trials and examined
distributions of these numbers. The number (mean ± SD) of ADs was
37.2 ± 2.3 (range 33-41), whereas the number of ASs was 49.4 ± 1.3 (range 46-51). These values are also consistent with those
reported previously (4, 12).
Figure 6 shows the 3-D structure of the
pathway and alveolar walls within the subacinus for the case shown in
Fig. 5A. Figure 6A is the whole pathway, and Fig.
6B is a sketch indicating connectivity of the branching
structure of the pathway, a connectivity diagram. The resulting lengths
of ADs and ASs are, respectively, shorter and longer than morphometric
data of actual pulmonary acini (4, 12) because of frequent branchings
at the proximal parts. However, the branching pattern and the total
path length are consistent with the values previously reported (4, 12).
Figure 6, C-E, shows alveolar walls on three different
cross sections passing the starting point. In Fig. 6C, the
cross section is parallel to the alveolar walls, as in Fig. 5A.
When the cross sections are oblique, as in Fig. 6, C and D, the
alveolar shapes give, for example, a hexagon (Fig. 6D) and
tetragon (Fig. 6E). Hexagonal ductal walls with alveolar septa
in Fig. 6D are very similar to those in real histological
sections of the lung.
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Fig. 6.
3-D presentations of pathway and alveolar walls of cubic subacinus in
Fig. 5A. A: 3-D structure of whole pathway. Alveolar
ducts (ADs) and alveolar sacs (ASs) are drawn with black and gray
lines, respectively. Starting point, branching points, and end points
are blue, green, and red, respectively. B: connectivity diagram
of pathway. C-E: 3 cross sections of alveolar walls. Cross
section in C is the same as Fig. 4A. All alveolar walls
are parallel or perpendicular to cross section. D: cross
section perpendicular to line connecting 2 opposite vertices of cube.
ADs and ASs are presented as hexagons. E: cross section with
direction between those in D and C. Alveolar walls are
presented as parallelograms.
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From these results, we can conclude that our labyrinthine algorithm
with priority of directions is suitable for modeling the effective air pathway.
Acinar models with realistic boundaries.
We generated an airway tree model down to the last RBs supplying
subacini with the tree algorithm. Each subacinus was approximated by a
set of cubic cells with side dimension of 0.5 mm, and the labyrinthine
algorithm with priorities was applied to it.
We show three acinar models in Figs. 7 and
8 and Table
1. Figure 7 shows model A,
consisting of eight subacini. The total volume is 256 mm3,
which is somewhat larger than the average in actual cases. Figure 7,
A and B, shows the outer appearance and the cross
section, respectively. Figure 7, C-E, shows the
intra-acinar pathways. The view in Fig. 7C is the same as in
Fig. 7A; the view angle in Fig. 7D is slightly
different. Figure 7E shows the pathway of the subacinus. The
intra-acinar pathway is radially extended from the end of the RB.
Although the shape of the subacinus is irregular, there is no detour or
loop in the pathway. Its connectivity diagram is shown in Fig.
7F. The volume of the subacinus is 40 mm3, and the
MPL from the starting point is 2.7 mm. The numbers of ADs and ASs are
68 and 83, respectively.
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Fig. 7.
Acinar model A consisting of 8 subacini. Bronchioles are black,
and alveolar walls are different colors according to subacini.
A: outer appearance of acinus approximated by a set of cubes
with side dimension of 0.5 mm. B: a thin layer with thickness
of 0.02 mm cut from acinus. C and D: all pathways
within acinus. Alveolar walls are indicated with tiny dots. View angles
of C and A are the same, but view angle of D is
slightly different. E: pathway within 1 subacinus (yellow in
A). F: connection diagram of pathway within yellow
subacinus.
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Fig. 8.
Acinar models B (blue) and C (pink). Terminal
bronchioles (TBs) of models B and C arise from the same
pre-TB. A: outer appearance of models B and C.
Shapes are more elongated than shape of model A. B:
thin layer with thickness of 0.02 mm cut from acini.
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Figure 8 shows a combination of models B and C; TBs
arise from the same pre-TB. They consist of seven and five
subacini, respectively. The morphometric data of models A, B,
and C are summarized in Table 1. Models B and C
have larger MPL than model A, because the acinar shapes of
models B and C are more anisotropic.
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DISCUSSION |
Results of our model analysis have shown that our labyrinthine
algorithm leads to reasonable predictions of some characteristic quantities in the human pulmonary acinus. Here we compare these predictions more precisely with past studies.
Detailed morphometric studies of intra-acinar structures were performed
by Parker et al. (12), Hansen et al. (6), and Haefeli-Bleuer and Weibel
(4). According to these studies, the TB branches three times with
nearly symmetric dichotomy, and the last RB branches 2-12 times
with asymmetric dichotomy. Parker, Hansen, and Haefeli-Bleuer
and their co-workers reported an average of 6, 8, and 7 branchings of
the TB from the last RB to ASs, respectively, and ~800, 4,900, and
2,000 ADs and ASs per acinus, respectively. The deviations in these
values may be due to difficulties of shape definition for irregular
branching structures. Number and size of segments in a branching system
depend on the definition of branching. If one regards a slight
protrusion of an AD as a branched AS, the branching times and the
number of ASs would increase. In fact, the number of ASs reported by
Hansen et al. is much larger than that reported by Parker et al. Hansen
et al. estimated that the average number of alveoli per AS was 3.5, whereas Parker et al. estimated the number to be 13 (there was no
statement about the alveolar number in the report of Haefeli-Bleuer and
Weibel). The ratio 3.5/13 is approximately reciprocal to the ratio of
their numbers of ASs (2,500/400). Our acinar model A has an
average of 398 ADs and 468 ASs, and the average number of alveoli per AS is 18. These values are close to those reported by Parker et al.
(12).
Haefeli-Bleuer and Weibel (4) estimated the path length through the
intra-acinar airway and reported an MPL of 8.8 ± 1.4 mm from the
first RB to ASs. The values in our models A and B are
in good accordance with their values. Model C has a larger MPL
because of its long shape. In all our models, numbers of ADs and ASs
are almost proportional to the acinar volume. These values are within
the range of values previously reported (4, 5, 12).
There are limitations of our 3-D acinar model. We have assumed that the
intra-acinar pathway is a sequence of cubic cells and that construction
of the pathway is limited only to extend straight or to turn at a right
angle; hence, the geometry of the intra-acinar pathway is locally
unrealistic. However, the whole pathway can simulate an oblique or a
curved branching system well. This kind of discrete approximation is
convenient for computer simulation. Another limitation in our model is
that all alveoli are assumed to be congruent. However, it is possible
to regard the alveolar size as one of the variables representing the
state of the alveoli. Hence, we do not regard these
limitations as serious. We have preliminarily succeeded in
simulating remodelings of acinar structures in idiopathic
interstitial pneumonia and pulmonary emphysema. The acinar model
presented here will enable us to simulate respiratory diseases and give
us a better understanding of the structural-functional correlation of
the lung.
Conclusion.
Our labyrinthine algorithm reproduces well the intra-acinar pathway,
despite its simplicity. Moreover, it is expected that the 3-D acinar
model generated by this algorithm might enable us to simulate
respiratory diseases and to understand the structural-functional correlation of the lung.
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APPENDIX A |
MPL from the starting point to all cells in a rectangle space.
The location of a cell is denoted
C(i, j, k), as shown in Fig. 2, where 0 
i L 1, 0 j K 1, and 0 k N 1. The path length
from the starting point,
C0(i0, j0, k0),
to the cell C(i, j, k) is denoted
P(i, j, k)
In
the case of Fig. 3A, path lengths increase one-by-one from 1 to
the maximum value LMN 1. Therefore, we have
In
the case of Fig. 3B, path lengths in the first line on the
first level are 0, 1, 2, 3, 4, and 5, respectively. Those in the second
line on the first level are 1, 2, 3, 4, 5, and 6, respectively, and so
on. On the second level, the path length of each cell increases by one
from that on the first level, and so on. Therefore , P(i, j, k) = (i + j + k), and we have
In
general, P(i, j, k) is always equal to
(i + j + k) as long as the starting point is at
the corner and there are no detours. In Fig. 3C, the pathway of
the cell at the right side of the newly made wall is the same as in
Fig. 2B.
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APPENDIX B |
Labyrinthine algorithm.
The algorithm generating a branching pathway is constructed with a
method of queue ordering cells. The queue is a linear array of cells
that can generate paths. At every step, only a cell located at the front of the queue can generate one path by selecting
one of the possible directions. We call this cell an
"activated cell." When an adjacent cell is already passed, the
direction toward that cell is forbidden to prevent loop formation.
If the activated cell has no possible direction, it is
deleted from the queue. If it has only one possible direction, it
generates one step of path and is then deleted from the queue. If it
has more than one possible direction, it generates one step and is then
put at the rear of the queue. The cell connected with the activated
cell via a newly produced path is put at the rear of the queue. Thus paths are generated sequentially until the queue becomes empty. The
number of paths is always equal to the cell number minus 1, because
there is no loop.
Figure 9 shows an example of
paths in a space with 3 × 3 × 1 cells. Because
the activated cells continue to be shifted outward from the starting
point, there is no chance for activated cells to select backward
directions. Therefore, there are no detours in the constructed
pathway.
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Fig. 9.
Labyrinthine algorithm in 3 × 3 × 1 rectangular subacinus.
Starting point is at bottom left. Activated cells are indicated
with squares at respective steps. Possible directions are indicated
with arrows. For cells with >1 possible direction, selected direction
is indicated by thicker arrow. Numbers within diagrams show orders
within queue for activation; step 1 is starting cell. Dark
cells have no possible directions. After step 8, a branching
labyrinth without loops or detours is constructed. Constructed pathway
is indicated by thin lines. , Branching points; , end points.
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Figure 10 shows cases without and with
priority directions with 4 × 4 × 1 cells. In Fig.
10A, the activated cell at the seventh step with number 6 has a
possible upward direction, which leads to a detour. However, in Fig.
10B, cells located in the priority directions can select only
the priority directions. Likewise, cells located opposite the priority
directions can select only opposite directions. Therefore, in the case
with priority directions, there is no chance for all activated cells to
make detours.
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Fig. 10.
Labyrinthine algorithm in 4 × 4 × 1 rectangular subacinus,
where starting point is inside space without (A) and with
(B) priority directions. In A, 2 detours are generated;
there are no detours in B. In this case, priority directions
are in positive directions in i and j axes. Priority
and nonpriority possible directions are indicated with solid and open
arrows, respectively. Nonpriority possible directions are not indicated
if there are prior possible directions, except at start. For
cells with >1 possible direction, selected direction is indicated by
thicker arrow.
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This algorithm is applicable to an acinus with any shape as long as it
is approximated by a set of cubic cells.
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FOOTNOTES |
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. §1734 solely to indicate this fact.
Address for reprint requests and other correspondence: H. Kitaoka,
Div. of Functional Diagnostic Imaging, Osaka University Medical School,
2-2 Yamadaoka, Suita City, Osaka 565-0871, Japan (E-mail:
kitaokah{at}image.med.osaka-u.ac.jp).
Received 29 July 1999; accepted in final form 25 January 2000.
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