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Department of Biomedical Engineering, Boston University, Boston, Massachusetts 02215
ABSTRACT
INTRODUCTION
STOCHASTIC BRANCHING PROCESS IN A DUCTAL SYSTEM
DESIGNING THE DIAMETERS OF THE BRANCHES
STOCHASTIC MODELS OF MAMMALIAN BRONCHIAL TREES
DISCUSSION
APPENDIX
ACKNOWLEDGEMENTS
FOOTNOTES
REFERENCES
ABSTRACT 
Kitaoka, Hiroko, and Béla Suki. Branching design
of the bronchial tree based on a diameter-flow relationship.
J. Appl. Physiol. 82(3): 968-976, 1997.
We propose a method for designing the bronchial tree where the
branching process is stochastic and the diameter
(d) of a branch is determined by its
flow rate (Q). We use two principles: the continuum
equation for flow division and a power-law relationship between
d and Q, given by Q ~ dn,
where n is the diameter exponent. The value of
n has been suggested to be ~3. We
assume that flow is divided iteratively with a random variable for the
flow-division ratio, defined as the ratio of flow in the branch to that
in its parent branch. We show that the cumulative probability
distribution function of Q, P(>Q) is proportional to Q
1. We
analyzed prior morphometric airway data (O. G. Raabe, H. C. Yeh, H. M. Schum, and R. F. Phalen, Report No.
LF-53, 1976) and found that the cumulative probability
distribution function of diameters, P(>d), is
proportional to dn, which supports
the validity of Q ~ dn since
P(>Q) ~ Q1. This allowed us to
assign diameters to the segments of the flow-branching pattern. We
modeled the bronchial trees of four mammals and found that their
statistical features were in good accordance with the morphometric
data. We conclude that our design method is appropriate for robust
generation of bronchial tree models.
airway; diameter exponent; flow distribution; fractals; power law
distributions
INTRODUCTION THE FUNCTION of a ductal structure is to transport
fluid to designated areas of an organ. In addition to efficient fluid
transport, for large branching ductal systems such as the airway tree,
one should also consider the optimal flow distribution at each
terminal. What kind of a branching system is required from the point of view of flow distribution at the terminals where a given amount of
fluid has to be delivered? The flow rate (Q) distribution is supposed
to have a small deviation and be stable against perturbations. Statistical descriptions (19, 28) are necessary to evaluate this aspect
of flow distribution. Previous morphometric airway models such as those
proposed by Weibel (34) and Horsfield et al. (11, 12) are not adequate
for studying the heterogeneity at the terminal ends.
In Weibel's model (34), each terminal is completely identical. This
simplification makes Weibel's model attractive and easy to handle from
the point of view of fluid mechanical computations along the airways.
However, its usage is limited because there are no such ideal branching
trees and under certain conditions (e.g., brochoconstriction) the
assumptions of symmetry and homogeneity cannot be maintained. The
airway models proposed by Horsfield et al. (11, 12) are more realistic,
because here each terminal has a different pathway, giving rise to a
natural asymmetry of the tree. This feature has been exploited to
predict the acoustic properties of the airways by Fredberg and Hoenig
(3) and more recently has been further developed by Lutchen et al. (21)
to investigate lung function (e.g., lung and airway resistances) during
heterogeneous constrictions. Nevertheless, Horsfield's models are
self-similar in the sense that the branching patterns are completely
determined by a rule of branching order, which, in turn, leads to
identical diameters (d) of the
terminals. In reality, according to the nomenclature of Boyden (2), we
can name nearly a hundred proximal branches that may be
genetically determined in the human airways. However, these airways
constitute <0.5% of the total number of branches down to terminal
bronchioles (TB), where no deterministic branching patterns have been
found. We therefore realize that a stochastic description of the
branching process is necessary, which leads to some heterogeneity of
d and flow distribution at the
terminal ends.
An optimal relationship between Q and
d was proposed decades ago (7, 17, 23,
30) as follows
where
n is the so-called diameter exponent
(22) and C is a constant that depends
on the organ in question and the fluid. Based on theoretical
considerations, the value of n has
been suggested to be 3 (17, 23). Because the viscosity of the fluid is
included in the constant C, the power
law in Eq. 1 should be applicable for
both vessels and airways containing blood and air, respectively. Thus
the relationship in Eq. 1 appears to
be quite useful for studying flow distribution in a given branching
structure. Conversely, because flow division in a branching structure
is easily solved, Eq. 1 may be used to
design large branching structures.
(1)
Here we propose a method for designing branching ductal structures where the branching pattern is stochastic and the d of each branch is determined by its Q through Eq. 1. First, we will describe a stochastic process whereby Q is iteratively divided at each bifurcation. We then examine some general properties of this branching process that can be used to generate the branching pattern or the topology of the tree (see STOCHASTIC BRANCHING PROCESS IN A DUCTAL SYSTEM). Next, we will present evidence of the applicability of the above diameter-flow relationship based on our analysis of Raabe's morphometric data (25) (see DESIGNING THE DIAMETERS OF THE BRANCHES). Finally, combining the branching pattern of flow division with Eq. 1 will allow us to design the d of the bronchial tree, and we will present simulation trees for four mammals and compare the results with actual morphometric data.
STOCHASTIC BRANCHING PROCESS IN A DUCTAL SYSTEM
Flow-Dividing Process Under Continuum Equation
First, we describe how fluid is divided at a bifurcation. We assume that mass is conserved at each bifurcation that is equivalent to the continuum equation for incompressible fluid. When the volume change of the duct during flow is negligible, the Q before branching (Q0) is equal to the sum of the Q of the two daughter branches (Q1 and Q2; Q1
Q2).
|
(2) |
0.5). Then,
Q1/Q0
is always equal to 1 
r. We
will regard r as a random variable.
The value of r determines the degree
of asymmetry of the branching pattern. If, for example, r is deterministic and fixed
to 0.5, the branching pattern is completely symmetric. Because the Q of
all branches in a tree have not yet been measured, we do not know the
actual distribution of r.
Nevertheless, we can predict it from available morphometric data, as
will be shown in STOCHASTIC MODELS OF MAMMALIAN
BRONCHIAL TREES.
In a finite structure like a living organ, there are terminal branches where the flow-dividing process stops and the fluid is delivered into the terminal units of the organ. In the lung, the TB is defined as a terminal branch of the conductive airway tree (11, 34). Although there is further branching within an acinus, the acinus is defined as the functional unit for gas exchange because the respiratory bronchioles are no longer pure conductive ducts. Accordingly, it is reasonable to assume that there is a threshold flow (Qc) below which there is no more conductive flow division. By definition, Qc provides the maximum Q at the terminal branches. In the following, we will assume that the Q is normalized to unity at the root. Thus, Qc represents the maximum fraction of the total flow that can be delivered to a TB.
The above three rules (continuum equation, flow-dividing ratio as a random variable, and the existence of a Qc as a parameter providing the maximum Q at a terminal branch) enable us to create large branching systems simulating the actual branching pattern of the bronchial tree. Specification of r and Qc that are appropriate for various mammals will be described in STOCHASTIC MODELS OF MAMMALIAN BRONCHIAL TREES.
Statistical Characteristics of Flow Distributions
Plotted on a log-log graph, Fig. 1 shows the cumulative distribution of Q, including every branch of a tree. N(Q) is the number of branches whose
Q is larger than a given value Q. The cumulative probability
distribution function P(Q) is then
given by
N(Q)/NT,
where
NT
is the total number of branches in the tree. As can be seen from Fig.
1, for Q Qc, N(Q) is proportional to Q1 so that
|
(3) |
Q), and hence it is proportional
to Q2.
Q), in a model tree plotted on a
log-log graph. In this simulation, deterministic parameter (Qc) is
0.00006 and random variable (r) is
uniformly distributed between 0.2 and 0.5. The total no. of branches,
NT, is 52,837. Note that N(Q) follows a
power law with an exponent of 1.002.
An interesting implication of Eq. 3 is that when a physical quantity is related to Q according to a power law (e.g., the d of the branch through Eq. 1), the probability-distribution function of that quantity will also be a power law, as will be discussed in the next section. Since Mandelbrot (22) proposed the concept of fractals, branching structures like the bronchial tree have often been categorized as fractal objects (1, 5, 6, 16, 18, 20, 33). Fractals are self-similar objects characterized by power-law distributions. Thus, since the results in Fig. 1 are quite independent of the particular distribution of r, the power-law distribution of flow demonstrates the general statistical self-similar property of branching structures.
DESIGNING THE DIAMETERS OF THE BRANCHES
Having generated branching patterns, we next propose a method to assign d to the individual branches by using the diameter-exponent rule of Eq. 1. Combining Eqs. 1 and 2, we obtain the following relationship between the d of a bifurcation
|
(4) |
In this section, we first reanalyze the morphometric data of four mammalian airways published by Raabe et al. (25) and then examine the validity of Eq. 1 in two ways: by directly using Eq. 4 and by examining the probability-distribution function of the diameter exponent. The significance of this is that by establishing the validity of the d-Q rule (Eq. 1), we will be able to connect the d of a branch to its Q. Thus, we can then simply transform the flow-branching pattern obtained in the previous section to diameter-branching pattern.
Resampling Raabe's Morphometric Data
The morphometric data of Raabe et al. (25) are based on two human lungs, two dog lungs, one rat lung, and one hamster lung. To every data set, they assigned a minimum d beyond which the measurement was complete. For branches in which d was smaller than the minimum d, the tree was arbitrary. The precision of measurement was 0.1 mm; therefore, we did not use d <0.5 mm because of the large relative measurement errors. We resampled Raabe's data so as to include only those branches that formed a complete tree with their diameters larger than the minimum d. The trunk of a tree was not limited to the trachea. As long as all branches beyond the minimum d belonging to one trunk were measured, the tree arising from this trunk was included in the statistical analysis. However, to obtain statistically meaningful results, we required that a tree contained at least 400 branches.We generated nine trees from Raabe's data (Table 1). Six of the trees were from bilateral lungs of six individuals, one tree was from a subsegmental bronchus of a human right upper lobe (HM-272), and two trees were obtained from the right apical lobe and the right intermediate lobe of a dog (DM-272).
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Calculation of the Diameter Exponent
Calculation of the d exponent was performed by solving Eq. 4 numerically at each bifurcation. For the human and dog airway trees, all bifurcations were selected where d0 was larger than the minimum d. For rat and hamster airway trees, only those bifurcations were selected where d0, d1, and d2 were all >0.5 mm. Bifurcations with d1 (or d2) d0
were excluded from the analysis, since in this case there is no finite
positive solution to Eq. 4.
The calculated values of n are
summarized in Table 1. In all cases, the SD were large. Figure
2 shows a histogram of
n obtained from the data set HM-272.
The distribution of n could be
approximated with a log-normal distribution not only for HM-272 but
also in all other cases, similar to previous reports (13, 30). As explained in APPENDIX Ab, the wide
distribution of n could primarily be
due to the high sensitivity of the calculation of n from the measured values of
d using Eq. 4. Although airways are not ideally cylindrical,
Eq. 4 is based on the assumption of
cylindrical configurations; therefore, slight changes in the measured
values of d can cause a wide
distribution of n.
There was a significant correlation between geometric mean (GM) of n and the minimum d (correlation coefficient = 0.86, P < 0.01), which is in accord with Horsfield's report (13). This correlation, however, is a result of the influence of measurement error in d (see APPENDIX Ab). Also, there was a significant correlation between GM of n and body mass (correlation coefficient = 0.83, P < 0.01). However, one should not conclude that there are species-related differences in n, because this correlation is also influenced by the correlation between GM of n and the minimum d. This analysis thus indicates that the diameter exponent rule (Eq. 1) may be acceptable in large trees.
Probability-Distribution Function of Diameters
The cumulative distributions of diameters, N(d ),
were examined by counting the number of branches whose
d was larger than a given diameter
d. When the total number of branches
in the tree is
NT, the
corresponding probability-distribution function,
P(d) is given by
N(d )/NT.
Before the calculation of
P(d ),
the measured d were first corrected
for measurement error by subtracting the absolute error, 0.05 mm, from
the values of the d. The reason is
that the number of branches with d
larger than the measured value D
should also include the number of branches whose diameters are between
D 0.05 and
D. The cumulative distribution of
d showed an apparent inverse power law
for all species, i.e., N(d ) ~ dm. Figure
3 shows examples of
N(d )
in the two human bronchial trees. The values of
m were obtained from the slope of the
regression line (Table 1). However, since the total number of branches, NT, was not
large enough in all trees to establish reliable statistics of the slope
m, m
was further corrected according to the procedure detailed in
APPENDIX Ac. This correction procedure
resulted in values of m that were
quite close to the GM of n (Table 1). Indeed, it is easy to show that m and
n should be the same. From Eqs. 1 and 3, the probability distribution of
d,
P(d ),
is derived as
|
(5) |
d),
for 2 human lungs (HM-272 and HM-373) on log-log plots. m,
Slope. Note that N(d)
follows an inverse power law with exponents of 3.4 and 3.3 for HM-272
and HM-373, respectively. Correlation coefficients are 0.995 and 0.993 for HM-272 and HM-373, respectively.
Besides Raabe's actual morphometric data, Fig.
4 demonstrates that the cumulative
distribution of d both in Weibel's
(34) and Horsfield's human model (11) complies with
Eq. 5. Because Weibel used
Eq. 4 with
n = 3 for generations
between 2 and 10, the slope on a log-log plot in this range is 3.0, with a correlation coefficient of 0.995. Although Horsfield did not use
this relationship, the cumulative distribution of
d in his model also showed a good accordance with an inverse power function. The regression lines from
the trachea to preterminal bronchioles showed a slope of 3.2 with a
correlation coefficient of 0.981 in Weibel's model, and a slope of 3.1 with a correlation coefficient of 0.988 in Horsfield's model. In
summary, these together provide evidence that the slope
m of the distribution of d
is statistically equivalent to the slope
n obtained from Eq. 3, further supporting the applicability of the
diameter-flow relationship given by Eq. 1.
The Value of Diameter Exponent
What is the appropriate value for n? Arguments have been offered by several groups (29, 32, 35, 36) to explain the discrepancy between empirical values of n and the theoretical value 3 for laminar flow. Considering the results of other studies (13, 30) and our analysis of measurement errors in this study (see APPENDIX Ab), we adopted a single value of n = 2.8 for modeling the mammalian airways. The case of HM-272 in Raabe's data, a 60-yr-old human man with unknown smoking history was described as "Tissue section showed emphysematous change, typical of aging." The GM of n in this case was slightly higher (3.0 vs. 2.8) than that of the other human, HM-283, whose tissue section showed no apparent change. This slight increase of n appears to be consistent with aging (31).STOCHASTIC MODELS OF MAMMALIAN BRONCHIAL TREES
Selection of Model Parameters in Designing the Bronchial Tree
In the previous section, we established a method for determining the d of the branches where the Q is known. Combining this with the stochastic method of flow division introduced in STOCHASTIC BRANCHING PROCESS IN A DUCTAL SYSTEM now allows us to design the d of the bronchial tree. There is one deterministic parameter, Qc, and one random variable, r, in our model. We will first discuss how Qc and the probability-density-distribution function of r should be assigned, and then we will present simulation trees for four mammals based on the actual morphometric data obtained by Raabe et al. (25).The value of Qc and the distribution of
r should be assigned based on actual
morphometric data. Qc can be approximately determined from the number
of TB. If, for example, r = 0.5, the
branching is symmetric and all TB have the same flow. Recall that the
flow at the trunk was assumed to be unity. The total flow at the
terminals is the sum of the individual flows at the TB, and hence the
product of the number of terminals and the mean Q at TB is equal to 1. Accordingly, Qc is the reciprocal of the number of terminals. If now
r is a random variable with a given
mean and SD, Qc will be the maximum flow at TB, and the number of
terminal branches will be >1/Qc. Nevertheless, as a first
approximation, this argument can still be used to determine the value
of Qc. The expected value of r is
related to the distributions of the generation numbers and Q of TB. As
the expected value of r is closer to
0.5, these distributions become narrower. In the present work, we only
used a uniform distribution of r. For
example, when we assign Qc = 0.00006 and
r is distributed uniformly between 0.2 and 0.5, the distribution of generation numbers of terminal branches is
approximately normal, with a mean of 15.9 and SD of 2.0, as shown in
Fig. 5. This result agrees well with those
derived from morphometric data of human airways reported in the
literature (10, 34).
We generated 10 trees with the same Qc and the same range of r. There were no identical trees; however, the statistical features of these trees were identical. The mean ± SD of the number of terminal branches was 26,431 ± 19. The means and the SD of the generation numbers at the terminal branches were distributed between 15.8 and 16.3 and between 1.9 and 2.2, respectively. The mean Q at the terminal branches were identical, and the coefficient of variation (CV) of the Q ranged from 31 to 34%.
Comparison with Morphometric Data
In Raabe's data, TBs were reported in one of the humans (HM-272), one of the dogs (DM-272), and in the rat and the hamster lungs. The terminated branches exactly recognized as TB were assigned a T, and other terminated branches were assigned an F. We only analyzed those which were exactly recognized, and the results are given in Table 2. Because the measurement of d in rat and hamster was completed down to TB as closely as possible, we estimated the total number of TB to be between (T + F) and (T + 1.5 × F), where T and F denote the number of branches marked with a T and an F, respectively. On the other hand, in the human and dog lungs, the total number of TB were not from Raabe's data but from other reports (8, 11, 14, 35), because the population of TB measured in Raabe's data was too small. In Table 2, d are normalized with the d of the trachea except for the hamster, where it was normalized with the largest airway d available.
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For each mammal, we assigned values for Qc and the ranges of r (Table 3) based on the number of TB and distribution of the generation numbers of TB, and subsequently we used them to create model trees of the four mammalian bronchial trees. The diameter distributions in these model trees showed good accordance with the morphometric data summarized in Table 2. Additionally, when we changed the value of the d exponent from 2.8 to 3.0 in humans, which corresponded to the case of HM-272 in Table 2, the mean d of TB increased by 27%. This is consistent with aging actually observed in that case, as pointed out in DESIGNING THE DIAMETERS OF THE BRANCHES.
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DISCUSSION
To design the bronchial tree in this study, we introduced a stochastic
flow-dividing process instead of using deterministic branching patterns
as proposed earlier (11, 12, 34). Although we used a uniform
distribution of the flow-dividing ratio
r, there are no direct data to support
this assumption. Nevertheless, we can estimate the range of
r in Horsfield's human and dog models (11, 12) as follows. Horsfield assigned identical Q to all terminal
branches in his human model (11); therefore, the value of
r can be calculated at each
bifurcation by using Eq. 1. Although Horsfield did not use Q in his dog model (12), we can calculate the
value of r in the same way as in his
human model. The histogram of r for
Horsfield's human and dog models is shown in Fig.
6. In the human model, the branching
pattern between the lobular bronchi and TB is completely symmetric.
However, there have been several reports demonstrating that the
branching pattern within the secondary lobule is not symmetric (4, 26).
When we exclude this part of his model, the mean value of
r was 0.36, almost equal to the mean
value of r in our simulation, 0.35. In
his dog model, r has a wider
distribution than in his human model with a mean of 0.25. We need to
point out that this value is exactly the same that we used in our
simulation, although we determined r
from Raabe's data. The distributions of
r in Fig. 6 are not smooth, most
likely due to the rigid branching pattern of the Horsfield model, and
hence they appear to be less realistic than the uniform distribution we
used.
It seems feasible that the distribution of r also depends on the Q in the parent branch. We examined the correlation between r and the Q of the parent branch in both Horsfield's models and found no systematic correlation except in the central airways. We also examined the influence of a nonuniform probability-density distribution function of r on the statistical features of the tree model. For example, we replaced the uniform distribution of r with a normal distribution truncated at 0 and 0.5. We found that the distributions of generation numbers and Q at the terminal branches were almost completely determined by the expected value of r, rather than the type of the probability distribution. Therefore, we suggest that, for simplicity, the uniform distribution of r is sufficient to generate large tree models. If a more realistic branching pattern is required in the proximal part of the tree, one can assign deterministic rules to r for the first several generations. This alteration, however, will not change the overall statistical features of the tree because of the small number of such proximal bifurcations.
Several studies have pointed out that the diameters in a branching structure are distributed according to a power law (15, 24). However, to our knowledge, no study has proposed how this distribution could be related to the distribution of Q. Horsfield et al. (13) also analyzed the Raabe data and obtained the value of n in two different ways. One way was to calculate it based on his own ordering method, and the other way was by using Eq. 4. One can show that his former method is approximately equal to the method of obtaining a distribution function of d. Values of n he obtained with this method were similar to our results.
Although Eq. 1 predicts that the value
of n is constant in a branching
system, the calculated values of n
using Eq. 4 showed a wide distribution
(see Fig. 2). We therefore examined the sensitivity of
P(d) to fluctuation in
n. First, we generated a tree where Q
of all branches were assigned. Then, starting from the trunk, we
iteratively calculated the diameters at each bifurcation by applying
the following equations
|
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d) still maintained an inverse
power-law form but with an exponent identical to the GM of
n and not the algebraic mean of
n. The highest correlation coefficient
of the regression on the log-log plot was obtained when the
probability-density distribution of n
had a log-normal distribution. This result appears to be complementary
to the high sensitivity of n to
variations in d as mentioned in
APPENDIX Ab. Moreover, it also supports our choice of using GM instead of algebraic means.
We also examined the distribution of airway lengths in Raabe's data. However, there were no significant power-law distributions. Although we tried to extract relationships among d, lengths, and angles, we did not find any significant correlation that would have allowed us to build deterministic relationships among these quantities. There are two type of angles in Raabe's reports. The first is the angle of the branch relative to the direction of gravity, and the other is relative to the parent branch. These two angles are not sufficient to reconstruct the three-dimensional structure of the trees from Raabe's data. If the angles had been measured so as to determine the location in the three-dimensional space, some significant correlation might have been detected. These are further problems to be investigated for modeling a three-dimensional branching structure.
Our model lends itself to immediate investigation of the distribution of d of TB that can then be compared with those derived from morphometric data. Although the model is based on the relationship between d and Q (Eq. 1), in its current form, it may not be used to predict Q at TB, because the model does not incorporate the effect of local compliance which determines the regional Q (27). However, from the point of view of designing the bronchial tree, this does not seem to be a serious limitation, because the predictions of the model were in excellent agreement with morphometric data. To predict more precisely the distribution of Q at TB, one would have to know the distribution of the local compliances. There is evidence that parenchymal expansion is quite heterogeneous (27), indicating a wider distribution of local compliances. Presumably, the spatial arrangement of the peripheral airway tree and the spatial distribution of local compliances are closely related. Thus, to take local compliance into account, one would also have to know the relationship between local compliance and the terminal airway structure, which is beyond the scope of the present study. Nevertheless, we note that the model presented here may be directly applicable to designing vascular trees where tissue compliance is less of an issue.
Despite the simplicity of the model, our method of designing the bronchial tree offers two advantages over previous airway tree models. Because there are no assumptions of any kinds of unity at the terminals, we are able to investigate heterogeneity at the terminals. More importantly, our model also enables us a diameter-based analysis. Most methods investigating branching ductal structures have been based on various branching orders (1, 5, 16, 22). When we use a branching order, starting either from the top or the bottom of the tree, we have to transform this quantity into a physical quantity such as the d of the branch. The anatomic structure of the airway wall is, however, more correlated with the diameter than with the branching order, and the most effective regulation of local flow is achieved by changing the diameter of the airway. We may use our model in a statistical sense to relate diameter to various quantities in lung, such as flow or airway resistance, independently from branching order.
In summary, we have presented a stochastic design of the bronchial tree based on the d exponent law. We find that the design is robust against perturbations in its parameters and provides trees that are statistically equivalent to morphometric data. The design principle is simple and will enable us to predict distributions of various quantities related to lung function.
ACKNOWLEDGEMENTS
We thank Drs. O. G. Raabe and H. C. Yeh and also the Lovelace Foundation for providing us with data, which were obtained from research performed at the Inhalation Toxicology Research Institute supported by the National Institute of Environmental Health Science under an interagency agreement with the US Energy Research and Developmental administration (now the Department of Energy), Contract No. DE-AC04-76-EV01013.
FOOTNOTES
Address for reprint requests: H. Kitaoka, Takaki Laboratory, Dept. of Mechanical Engineering, Tokyo Univ. of Agriculture and Technology, Konganei, Tokyo 184, Japan.
Received 31 May 1996; accepted in final form 21 October 1996.
APPENDIX Aa
Distribution of Flow in a Branching Tree
There is a simple relationship (which can be symmetric or asymmetric) between the total number (NT) of branches in the tree and the number of terminal branches (E) as follows
|
(A1) |
1,
Eq. A1 can be approximated as
|
(A2) |
Qa) which we
would like to express explicitly with Qa. The terminal
branches in this partial tree have Q near Qa. However,
there will be bifurcations where only one of the two daughter branches
has a flow >Qa, and hence the other branch is not
considered as part of the partial tree. When the branching pattern is
almost symmetric, the number of such bifurcations is negligible. By
using Eq. A2, the number of terminals
in the partial tree E
is given by
NT /2 where
NT is the total number of branches in the
partial tree which is exactly
N(Qa)/2. However, since the flow at the terminals of the partial tree is approximately equal to
Qa, the product of E and Qa should give
the total flow at the trunk, i.e., unity. Therefore
|
(A3) |
|
(A4) |
Q)
on a log-log graph in Fig. 1.
When the branching pattern is asymmetric, the number of bifurcations
where one of the two daughters does not belong to the partial tree
cannot be neglected. The total flow coming out of the terminals of the
partial tree, Q, is <1 because of the missing branches. The
number of terminal branches is ~Q/Qa which is also <N(Qa)/2. However, the
total flow coming out of the partial tree is also smaller than unity by
an amount Q" = 1 Q. The number of missing
daughter branches can then be approximated as the ratio of Q" to the
mean flow through the missing branches. The Q of the missing branches
are ranged from 0 to Qa with an average of ~Qa/2. Thus, the number of missing branches is estimated
to be 2Q"/Qa. If we now add the missing daughters to the
partial tree, this new tree has a complete set of bifurcations, and so
Eq. A1 would again be applicable.
Accordingly, for this new tree, the total number of branches is
N(Qa) + 2Q"/Qa. Therefore, Eq. A1 can now be written as
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APPENDIX Ab
Influence of Measurement Errors on the Calculation of Diameter Exponent
When calculating the d exponent with the use of Eq. 4, the influence of measurement errors in d is extremely important. In Raabe's data, the precision of measurement was 0.1 mm. If it were 0.01 mm, there would be 1,000 possible combinations of the measured d of the three branches at a bifurcation. We examined how n is distributed corresponding to all possible combinations. In the following, we define di (I = 0, 1, or 2, with 0 denoting the parent, and 1 and 2 the daughters) as a measured value of d and Di as its true value. For example, let us examine the combination of measured values of d0 = 0.8 mm, d1 = 0.7 mm, and d2 = 0.5 mm. The calculated value of n at this bifurcation is 2.61. If D0 = 0.75 mm, D1 = 0.74 mm, and D2 = 0.54 mm, the n is 7.90. If D0 = 0.84 mm, D1 = 0.65 mm, and D2 = 0.45 mm, the n is 1.83. We calculated n for all possible combinations, and obtained the distribution of n as shown in Fig. 7. The distribution of n is approximately log-normal, having a higher mean value than the original value. When the diameters are larger, the distribution of n is narrower and the mean value of n is closer to the original value, because the relative measurement error is smaller for larger diameters. This is one reason for the significant correlation between the minimum diameter and the mean value of n in the analysis of Raabe's data. Another reason is as follows. When D1 and/or D2 is close to D0, the value of n is high. However, such a bifurcation is often excluded when the relative measurement error is large (e.g., a combination of D0 = 0.74 and D1 = 0.73, both of which would be measured at 0.7 mm when the precision is 0.1 mm). Therefore, at a smaller diameter, there are less bifurcations with higher values of n, resulting in smaller mean of n in the analyses of Raabe's data.APPENDIX Ac
Correcting the Slope m for Smaller Trees
In practice, when the NT of branches in a tree is not large enough to neglect the1 in
Eq. A1, the estimated value of
m should be corrected for it. When
NT is smaller,
the influence of 1 is bigger and the slope obtained from the
log-log plot of flow distribution becomes slightly >1, as shown in
Fig. 1. The slope of P(Q) on a
log-log plot (a) can then be
calculated as
|
1. There is another condition under which Eq. A3 breaks down, namely, when Q is below Qc. In this
analysis, we did not include small diameters near TB, and hence it was
not necessary to correct for the effect of Q being smaller than Qc.
REFERENCES
| 1. |
Bennett, S. H.,
B. W. Goetsman,
J. M. Milstein,
and
J. S. Pannu.
Role of arterial design on pulse wave reflection in a fractal pulmonary network.
J. Appl. Physiol.
80:
1033-1056,
1996.
|
| 2. | Boyden, E. A. Segmental Anatomy of the Lungs. New York: McGraw-Hill, 1955. |
| 3. | Fredberg, J. J., and A. Hoenig. Mechanical response of the lung at high frequencies. J. Biomech. Eng. 100: 57-66, 1978. |
| 4. | Gamsu, G., W. M. Thurlbeck, P. T. Macklem, and R. G. Fraser. Peripheral bronchographic morphology in the normal human lung. Invest. Radiol. 6: 161-170, 1971. [Medline] |
| 5. |
Glenny, R. W.,
and
H. T. Robertson.
Fractal properties of pulmonary blood flow: characterization of spatial heterogeneity.
J. Appl. Physiol.
69:
532-545,
1990.
|
| 6. | Goldberger, A. L., D. R. Rigney, and B. J. West. Chaos and fractals in human physiology. Sci. Am. 262: 43-49, 1990. |
| 7. | Groat, R. A. Relationship of volumetric rate of blood flow to arterial diameter (Abstract). Federation Proc. 7: 45, 1948. [Medline] |
| 8. |
Hansen, J. E.,
and
E. P. Ampaya.
Human air space shape, sizes, areas, and volumes.
J. Appl. Physiol.
38:
990-995,
1975.
|
| 9. |
Horsfield, K.
Diameters, generations, and orders of branches in the bronchial tree.
J. Appl. Physiol.
68:
457-461,
1990.
|
| 10. |
Horsfield, K.,
and
G. Cumming.
Morphology of the bronchial tree in man.
J. Appl. Physiol.
24:
373-383,
1968.
|
| 11. | Horsfield, K., G. Dart, D. E. Olson, G. F. Filley, and G. Cumming. Models of the human bronchial tree. J. Appl. Physiol. 31: 202-217, 1971. |
| 12. |
Horsfield, K.,
W. Kemp,
and
S. Phillips.
An asymmetrical model of the airways of the dog lung.
J. Appl. Physiol.
52:
21-26,
1982.
|
| 13. | Horsfield, K., and A. Thurlbeck. Relation between diameter and flow in branches of the bronchial tree. Bull. Math. Biol. 43: 681-691, 1981. [Medline] |
| 14. | Horsfield, K., and M. J. Woldenberg. Diameters and cross-sectional areas of branches in the human pulmonary arterial tree. Anat. Rec. 223: 245-251, 1989. [Medline] |
| 15. |
Howorth, S. T.,
J. H. Linehan,
T. A. Bronikowski,
and
C. A. Dawson.
A hemodynamic model representation of the dog lung.
J. Appl. Physiol.
70:
15-26,
1991.
|
| 16. |
Jiang, Z. L.,
G. S. Kassab,
and
Y. C. Fung.
Diameter-defined Strahler system and connectivity matrix of the pulmonary arterial tree.
J. Appl. Physiol.
76:
882-892,
1994.
|
| 17. | Kamiya, A., T. Togawa, and A. Yamamoto. Theoretical relationship between the optimal models of the vascular tree. Bull. Math. Biol. 36: 311-323, 1974. [Medline] |
| 18. | Kitaoka, H., and H. Itoh. Spatial distribution of the peripheral airways. Application of fractal geometry. Forma 6: 181-191, 1991. |
| 19. | Koblinger, L., and W. Hofmann. Analysis of human lung morphometric data for stochastic aerosol deposition calculations. Phys. Med. Biol. 30: 541-556, 1985. [Medline] |
| 20. |
Krenz, G. S.,
J. H. Linehan,
and
C. A. Dawson.
A fractal continuum model of the pulmonary arterial tree.
J. Appl. Physiol.
72:
2225-2237,
1992.
|
| 21. |
Lutchen, K. R.,
J. L. Greenstein,
and
B. Suki.
How inhomogeneities and airway walls affect frequency dependence and separation of airway and tissue properties.
J. Appl. Physiol.
80:
1696-1707,
1996.
|
| 22. | Mandelbrot, B. B. The Fractal Geometry of Nature. San Francisco, CA: Freeman, 1982. |
| 23. |
Murray, C. D.
The physiological principle of minimum work. I.
Proc. Natl. Acad. Sci. USA
12:
207-214,
1926.
|
| 24. | Phillips, C. G., S. R. Kaye, and R. C. Schroter. A diameter-based reconstruction of the branching pattern of the human bronchial tree. Respir. Physiol. 98: 193-217, 1994. [Medline] |
| 25. |
Raabe, O. G.,
H. C. Yeh,
H. M. Schum,
and
R. F. Phalen.
Tracheobronchial geometryhuman, dog, rat, hamster. Albuquerque, NM: Lovelace Foundation for Medical Education and Research, 1976. (Rept. no. LF-53)
|
| 26. | Reid, L. The secondary lobule in the adult human lung, with special reference to its appearance in bronchograms. Thorax 13: 110-115, 1958. |
| 27. |
Roddarte, J. R.,
M. Chaniotakis,
and
T. A. Wilson.
Variability of parenchymal expamsion measured by computed tomography.
J. Appl. Physiol.
67:
226-231,
1989.
|
| 28. | Soong, T. T., P. Nicolaides, C. P. Yu, and S. C. Soong. A statistical description of the human tracheobronchial tree geometry. Respir. Physiol. 37: 161-172, 1979. [Medline] |
| 29. | Suwa, N. Supracellular structural principle and geometry of blood vessels. Virchows Archiv. A390: 161-179, 1981. |
| 30. | Suwa, N., T. Niwa, H. Fukasawa, and Y. Sasaki. Estimation of intravascular blood pressure gradient by mathematical analysis of arterial casts. Tohoku J. Exp. Med. 79: 168-198, 1963. |
| 31. | Thurlbeck, W. M. Morphology of the aging lung. In: The Lung: Scientific Foundations, edited by R. G. Crystal, and J. B. West. New York: Raven, 1991, vol. 2. |
| 32. | Uylings, H. B. M. Optimization of diameters and bifurcation angles in lung and vascular tree structures. Bull. Math. Biol. 39: 509-519, 1977. [Medline] |
| 33. |
Weibel, E. R.
Fractal geometry: a design principle for living organisms.
Am. J. Physiol.
260 (Lung Cell. Mol. Physiol. 4):
L361-L369,
1991.
|
| 34. | Weibel, E. R. Morphometry of the Human Lung. New York: Academic, 1963. |
| 35. | Woldenberg, M. J. Finding the optimal lengths for three branches at a junction. J. Theor. Biol. 104: 301-318, 1983. [Medline] |
| 36. |
Zamir, M.
The role of shear-forces in arterial branching.
J. Gen. Physiol.
67:
213-222,
1975.
|
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