Transport in Asymmetrically Branched Structures: A Statistical Mechanical Approach to Structure-Function Relations in the Lung

BOSTON UNIVERSITY
GRADUATE SCHOOL OF ARTS AND SCIENCES
Dissertation
TRANSPORT IN ASYMMETRICALLY BRANCHED STRUCTURES:
A STATISTICAL MECHANICAL APPROACH TO
STRUCTURE-FUNCTION RELATIONS IN THE LUNG
by
ARNAB MAJUMDAR
Integ. M.Sc, Physics, Indian Institute of Technology, Kanpur, 1997
Submitted in partial fulfillment of the
requirements for the degree of
Doctor of Philosophy
2010

UMI Number: 3399562
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2010

Approved by
First Reader ' )7-^f-^ JL1
^
Second Reader
H. Eugene Stanley, Ph.D.
Professor of Physics
r - """• * ^ v—^/ j_—j_
Bela Suki, Ph.D.
Professor of Biomedical Engineering

DEDICATION
To my sister, Atreyi.

T R A N S P O R T I N A S Y M M E T R I C A L L Y B R A N C H E D S T R U C T U R E S :
A S T A T I S T I C A L M E C H A N I C A L A P P R O A C H T O
S T R U C T U R E - F U N C T I O N R E L A T I O N S I N T H E L U N G
(Order No. )
A R N A B M A J U M D A R
Boston University, Graduate School of Arts and Sciences, 2010
Major Advisor: H. Eugene Stanley, Professor of Physics
A B S T R A C T
The mammalian lung consists of an asymmetric binary tree through which air is trans-
ported to the gas exchange units, called alveoli. In this thesis, we study the asymmetric
bifurcation of the airway tree and develop models of the transport processes and relations
describing the connection between lung inflation and branching structure.
We examine the branching pattern of the airway tree of four mammalian species and
demonstrate that the bifurcations are self-similar. We derive closed form expressions for the
distribution of airway diameters as a function of generation number utilizing an asymmetric
flow-division model. Our findings suggest that the empirically observed structural hetero-
geneity of the airway tree can be explained by simple deterministic rules of the branching
pattern.
In lung diseases, airways can close during expiration. During inspiration, these closures
reopen in avalanches, leading to a series of discrete increments in lung volume with increasing
pressure. To reproduce the experimental pressure-volume (P-V) relations of the lungs, we
develop a model consisting of an asymmetric bifurcating structure with random blockages
that can be removed by the pressure of the fluid. We show that the P-V relations can
be decomposed into a linear superposition of paths connecting the root of the tree to the
alveoli. We solve the inverse problem to estimate the underlying path length distribution of
the tree by analyzing experimental P-V curves. This distribution agrees well with available
morphometric data on airway structure. When some airways collapse during expiration,

the downstream sub-tree becomes isolated from the root trapping air behind the closure.
Consequently, the P-V curve develops hysteresis. We obtain analytical solutions for this
hysteretic behavior and the amount of trapped gas during deflation.
The opening of collapsed airways is an explosive process generating an audible sound,
called crackle. We derive a relationship between the amplitude of a crackle sound and the
diameter of the airway where the crackle is generated and use this relation to estimate the
diameters of collapsed airways from measured crackle amplitudes. The results indicate that
airways with diameters within a narrow range are vulnerable to collapse and subsequent
injury if repeated opening and closure occurs.
VI

Contents
Contents vii
List of Tables ix
List of Figures x
List of Abbreviations x v
1 Introduction 1
2 Regular Branching Asymmetry of the Airway Tree 5
2.1 Airway Diameter Distributions 6
2.2 Terminal Generation Number Distribution 10
2.2.1 Exact solution 12
2.2.2 Continuum approximation 15
3 Fluid transport in branched structures with temporary closures 18
3.1 Analysis of experimental pressure-volume data from dog lungs 21
3.2 Model of lung inflation during avalanches of airway openings 26
3.2.1 Partitioning the effect of asymmetric structure from the inflation process 32
3.2.2 Models of airway opening: Correlated inflation processes 35
3.3 Characterization of the branching structure of the lung from "macroscopic"
pressure-volume measurements 41
3.4 Airway collapse, hysteresis and trapped gas 46
4 Estimating the diameter of airways susceptible for collapse using crackle
sound 56
4.1 Methods 57

4.1.1 Experimental procechire 57
4.1.2 Estimating the generation number of collapsed airways 59
4.1.3 Analysis of crackle statistics 60
4.1.4 Estimating the effect of generation dependence of threshold pressures 61
4.1.5 Estimating the diameter of collapsed airways 63
4.2 Results 64
4.3 Discussion 71
A Volume of airways 77
Bibliography 79
Curriculum Vitae 88
Vlll

List of Tables
2.1 Parameters for four species obtained from data [1, 2] and used in the model.
The mean i standard deviation of the diameter ratios Kmaj find tm jn are
obtained from the data shown in Fig. 2.2. Also shown are the values of r and
7] obtained by solving Eq. (2.4)

List of Figures
2.1 Illustration of the notation and model. The major (minor) daughter cor-
respond the daughter airways with larger (smaller) diameter. The diameter
ratios Kmaj and Kmjn defined in Eq. (2.1) are constant at every bifurcation in
the tree. The flow Qzj of the parent airway (i,j) is partitioned according to
Eq. (2.2) 6
2.2 Mean diameter ratios for the major (Kmaj, •) and minor («min, o) daughters as
a function of generation number N for one animal from each of four different
species: (a) Dog, (b) Rat, (c) Rabbit and (d) Human. The error bars show
the standard deviation of the corresponding distributions at each N. . . . 7
2.3 Linear-log plot of a typical distribution HN(D) of airway diameters D. Filled
boxes show riv(-D) for a dog lung at generation N = 7. The solid line shows
the model HN(D) as given by Eq. (2.7) using the parameters in Table 2.1. . 9
2.4 Semi-log plot of (a) mean diameters (DN) and (b) standard deviation of
the diameters a(Dj^) plotted as a function of generation N for four species:
Dog (o), Rat (A), Rabbit (v) and Human (•). Solid lines show the model
predictions using the parameters in Table 2.1, and with Dc = 0.084, 0.070,
0.053 and 0.165 respectively for the four species. The curves are vertically
shifted for clarity 11

2.5 Schematic diagram of the development of the branching pattern in an asym-
metric tree with r = 1/3 and Qc = 1/8. The numbers besides the branches
show the values of Q for the branch, with Q < Qc indicated in red. Terminal
branches are shown in red while missing branches are shown in cyan 12
2.6 Distribution LTJV of terminal generation numbers N for asymmetry parame-
ters r = 3/16, 1/4, 5/16 and 3/8 from numerical summation of Eqs. (2.16),
(2.17) and (2.18). the distributions are characterized by sharp peaks and
valleys overlaying a smoother hump 14
2.7 Total number of terminal airways Z with r, relative to the number of ter-
minals for the corresponding symmetric tree with the same Qc = 0.0008 but
with r = 1/2 15
3.1 Section of an airway showing (a) the film of liquid when the airway is open,
and (b) the liquid bridge blocking the flow of air when the airway is closed. 19
3.2 Schematic diagram of the experimental setup. The lobe is placed in a sealed
chamber (pressure Pc) with the main bronchus open to the atmospheric pres-
sure Pa. The air pressure in the chamber is slowly decreased using a vacuum
pump which creates a pressure difference P = Pa — Pc 21
3.3 Experimentally determined P-V curves of two isolated dog lung lobes A
(•) and B (o), obtained during inflation from collapsed state to total lobe
capacity. The dashed lines show asymptotic fits to VE(P) given by Eq. (3.4)
with V0 = 327.3 ml, a = 0.907, b = 0.094 /cm H20 for Lobe A and VQ =
377.1ml, a = 0.908, b = 0.075 /cm H20 for Lobe B 23
3.4 Volume fraction fv(P) of the open region of lobes (a) A and (b) B, as defined
by (3.6) 24
3.5 Distribution ijj(<f>) of opening pressures cf> of the air sacs in lobe (a) A and (b)
B, obtained by differentiating Fig. 3.4(a) and 3.4(b) respectively, according
to Eq. (3.9) 25
3.6 Convention for labeling branches of the tree according to Eq. (3.10) 26
XI

3.7 An example of an asymmetric tree T consisting of all labeled branches. Cir-
cles represent the air sacs connected by the terminal branches (shown with
underlined labels) belonging to C The double line ( = ) shows the path ^3,5
connecting the terminal branch (3, 5) to the root 27
3.8 The process of airway opening in a tree, (a)-(e) show the states of the tree
with increasing P. Branches are labeled as shown in Fig. 3.7. Open branches
are shown as outlines, newly opened branches are shown in gray and closed
branches are shown in black. The active surface is shown as a dashed line.
Inflation begins at P = 0 with all branches other than the root closed (a)
and proceeds by airway openings, either individually (b) or in an avalanche
(c), as P is increased, (d)-(e) show the pressure differences AP and states £
of three segments (1,1), (2,2) and (3,5) respectively, belonging to the path
7->
3>5, shown in Fig. 3.7. Different behavior is observed for branches on the
active surface and those embedded in an avalanche 29
3.9 Pressure along the axis of a liquid bridge in branch (i,j) when APjj is just
above pij, the liquid bridge breaks, a pair of sound waves ("crackles") are
generated and the pressure front propagates downstream 30
3.10 (a) The distribution ip{<f) of opening pressures <j) and (b) the opening proba-
bility T(P) of an airway as obtained from the three models of airway opening
(A, B and C) for a chain of six branches and with a = 0.75 for models B and
C 40
3.11 (a) The volume fraction fv of the open air sacs obtained using models A and C
obtained by fitting the experimental data for lobe A and (b) the distribution
n(n) of the generation numbers n of the terminal branches obtained from
the fit, compared to the distribution for the Horsfield model of the dog lung. 43
3.12 (a) The distribution ip{4>) of opening pressures (p using model C compared to
the experimental data from lobe A and (b) the full P-V curve reconstructed
using Eq. (3.3) and model C, compared to the experimentally obtained data. 45
xii

3.13 (a) Volume and (b) average volume as a function of pressure 47
3.14 P-V loops for (a) iV = 1 (b) TV = 2 (c) N = 4 (d) N = 8 51
3.15 Residual volume versus (a) generation number n and (b) A 54
4.1 Schematic diagram showing the attenuation of a crackle at an airway bifur-
cation as described in Eq. (4.1) 59
4.2 A representative example of pressure-volume (P-V) curves of an excised rab-
bit lung from the collapsed state, and subsequent inflations from EEP levels
of 5, 2, 1 and 0 c m f ^ O . The inflection point is prominent along the 1st and
5th inflations 64
4.3 Examples of crackle amplitude time series during inflations from: (a) the
collapsed state, (b) 5 c m H 2 0 EEP, (c) 2 c m H 2 0 EEP, (d) 1 c m H 2 0 EEP,
(e) 0 cmH20 EEP. The time series correspond to the same inflations as
in Fig. 4.1. The amplitudes are normalized to maximum value of crackle
amplitudes and displayed on a logarithmic scale 66
4.4 Histograms of the generation numbers at different EEP levels estimated from
Eq. (4.2) using all crackle data 67
4.5 A) Probability of closure p(n) for airways at generation n in the rabbit lung
for inflations from different EEP. Inset shows the peak of p(n), the generation
number of airways that are most likely to collapse at different EEP levels. B)
Fraction of collapsed airways in the rabbit lung at different values of EEP.
The symbols show the median values from six different rabbits, the error-bars
show the inter-quartile range 68
4.6 Error in estimated generation numbers of crackle origin in the airway opening
model as a function of generation dependence parameter g 69
4.7 Probability of closure in the airway tree as a function of generation number n
for several values of the generation dependence parameters g, corresponding
to distributions of closing threshold pressures 70
xiii

4.8 Simulation results showing a linear relation between normalized crackle am-
plitude and normalized airway diameter when plotted on a log-log scale. . . 71
4.9 Probability of closure as a function of airway diameter D in the rabbit lung
for inflations from different EEP 72
xiv

List of Abbreviations
P-V Pressure-Volume
EEP End-Expiratory Pressure

Chapter 1
Introduction
The primary function of the respiratory system is to deliver air to the air sacs, called alveoli,
for gas exchange. Morphological data show that the mammalian lung consists of airways ar-
ranged hierarchically in an asymmetric binary tree, the airway tree, with air sacs connected
to the terminals. Here we examine the nature of asymmetry in the bifurcation of the air-
way tree and develop models of the transport processes in the lung, obtaining quantitative
relations describing the connection between lung inflation and branching structure. Our
results have the potential for practical applications in the characterization of lung diseases.
Traditional models of the airway tree either do not address the observed heterogene-
ity in the diameters and path lengths or simply consider this structural heterogeneity to
result from random fluctuations. We examine the branching pattern of the airway tree in
the lungs of four mammalian species, and demonstrate that the simplifying assumption of
deterministic asymmetry in these branching patterns is sufficient to account for the empiri-
cally observed heterogeneity in the airway tree as represented by the distribution of airway
diameters and path lengths.
We study the distribution Un(D) of airway diameters D as a function of generation
n in the asymmetrically bifurcating airway trees of mammalian lungs. We find that the
airway bifurcations are self-similar and derive closed form expressions for Un(D) utilizing
an asymmetric flow-division model.

2
We employ the asymmetric flow-division model to obtain the distribution II(JV) of gener-
ation numbers N of the terminal airways in the tree. Our findings suggest that the observed
distributions are consistent with an underlying deterministic branching asymmetry.
Pressure-volume (P-V) relations are used to measure lung function in clinical environ-
ments. To understand lung function, it is important to determine how transport properties
of the system depend on the asymmetry of its underlying airway tree structure. We ad-
dress the problem of fluid transport in the lung by developing a novel model system that
reproduces the experimental P-V relations in mammalian lungs. The model consists of an
asymmetric bifurcating structure containing random blockages that can be removed by the
pressure of the fluid itself.
Many peripheral airways of a diseased lung collapse during expiration completely block-
ing the flow of air, and thus excluding a large number of alveoli from participating in gas
exchange. During inspiration, these closures reopen in avalanches, leading to a series of
discrete increments in lung volume as a function of pressure.
We obtain a comprehensive quantitative description of the fluid flow in terms of the
airway tree topology and the opening mechanisms of collapsed airways. We show that
the P-V relationship of the fluid can be decomposed into a linear superposition of paths
connecting the root of the structure to the air sacs.
Avalanches of airway openings during lung inflation are complicated by the presence
of audible pressure waves called crackles, which in turn can assist the opening of other
airways downstream. We develop several models to account for the influence of crackles on
avalanches and derive expressions for the opening probability of an airway at generation n.
We invert the relation between airway tree asymmetry and the P-V curve to obtain
a statistical description of the underlying distribution II(iV) of the terminal generation
numbers by analyzing experimental P-V data from dog lungs. The II(iV) obtained from
the P-V curve agrees well with available morphometric data on lung branching structure
in dogs.
During deflation, the airways tend to collapse when their internal pressure reaches a

3
closing threshold. Each collapsed airway closes off the downstream sub-tree from commu-
nication with the root and hence traps air in it. Thus, the P-V curve develops a deflation
limb which is distinct from the inflation limb, characterized by a significant hysteresis. We
obtain analytical solutions for this hysteresis behavior and the amount of trapped gas in
the lung during deflation.
In lung diseases, airways often collapse impairing gas exchange. Airways that collapse
during deflation generate a crackle sound when they reopen during subsequent re-inflation.
Since each crackle is associated with the reopening of a collapsed airway, the likelihood of
an airway to be a crackle source is identical to its vulnerability to collapse.
Using the asymmetrically branched airway tree model, we derive a relationship between
the amplitude of a crackle sound measured at the trachea and the diameter of the source
airway where the crackle was generated. This relation is then used to estimate the diameters
of collapsed airways from crackle amplitudes measured in rabbits lungs. These results indi-
cate that airways with diameters within a narrow range become unstable during deflation,
and thus are vulnerable to collapse and subsequent injury.
In addition to the amplitude of crackle sounds, we find that their time ordering contains
key information about the arrangement of contiguous airways which open sequentially dur-
ing an inflation avalanche. We develop a probabilistic approach to map these contiguous
regions of collapsed airways.
In some lung diseases, airways do not collapse completely but constrict to a smaller
diameter. Although these constrictions hinder airflow by increasing the resistance of the
lung, they cannot be identified using crackles. We thus use changes in the resistance and
compliance of the airway tree to estimate the location of the constrictions.
We examine the Poiseuille flow resistance R of unconstricted airway trees as a function
of the degree of tree asymmetry and diameter ratios. We show that for the four mam-
malian species studied, the largest contribution to the airway tree resistance comes from
the peripheral branches, regardless of the degree of asymmetry.
A rheological model of lung tissue, called the constant-phase model has been successful

4
in describing the dynamic pressure-flow relations in the airway tree. We show that for a
model with constant-phase elements attached to the terminals of an asymmetric tree, specific
relations exist between airway resistance and alveolar compliance at every bifurcation in the
tree. Diseases which change airway resistance thus lead to detectable deviations from the
constant-phase model. These deviations can be used to estimate the extent and location of
airway constrictions.
In conclusion, we show that the airway tree in the lung has an asymmetric self-similar
branching structure. We develop a general theory for fluid transport through such branch-
ing structures in the presence of randomly distributed removable closures. We show that
the pressure-volume relation as well as other analogous relations describing fluid transport,
can be expressed as a linear superposition of transport processes along one-dimensional
chains. The developed methodology provides an estimate of the distribution of the genera-
tion number of the terminal branches in the airway tree and may be used to detect several
different lung diseases.

Chapter 2
Regular Branching Asymmetry of
the Airway Tree
Leonardo da Vinci observed five centuries ago that "all the branches of a tree at every
stage of its height when put together are equal in thickness to the trunk" [3]. Similar
regularities are seen in distribution networks of plants, and the respiratory and vascular
systems in mammals [4]. The ubiquity of regular branching structures has led to the study
of underlying optimization principles [5], as well as the development of growth models
governed by local rules of branching [6]. The variation of size with generation is related to
allometric scaling of metabolic rate with body mass [7].
Most models of the lung airway tree either do not address the observed diameter het-
erogeneity or simply consider it a result of random fluctuations. Here, we examine the
branching pattern of the airway tree in mammalian lungs, and demonstrate that the sim-
plifying assumption of deterministic branching asymmetry is sufficient to account for the
observed distribution of airway diameters at any level of branching. In addition, we find
that the same form of asymmetry can determine the resistance to air flow in the lung.

6
Di.j
minor daughter
parent airway
V^.najD.,,
major daughter
( i + l , 2 j + l)
(l-r)Qv
Figure 2.1: Illustration of the notation and model. The major (minor) daughter correspond
the daughter airways with larger (smaller) diameter. The diameter ratios «maj and Km[n
defined in Eq. (2.1) are constant at every bifurcation in the tree. The flow Qij of the parent
airway (i,j) is partitioned according to Eq. (2.2).
2.1 Airway Diameter Distributions
We first introduce some notation [8]. We label each airway in the tree by a pair of indices
(i,j), where the index i is the generation number of the airway and the index j (0 < j < 2l
)
is used to distinguish between airways of the same generation. The root of the tree, the
trachea, is labeled (0,0). The daughters of a bifurcating airway (i,j) are (i + 1,2j) and
(z + l,2j + l) (Fig. 2.1).
The diameter of airway (i,j) relative to the diameter of the root is defined as Dij, with
.Do,o = 1. The daughter with the smaller (larger) diameter is termed the minor (major)
daughter and labeled by an even (odd) value of j . We define the diameter ratios Kmn
and Kmaj, respectively as the ratios of diameters of the minor and 'major daughter to their
parent (Fig. 2.1),
ftmin — L>i+,2j / J-'i,]
^maj = Di+i}2j + l/Dij .
(2.1a)
(2.1b)
We analyze the airway diameters from published data on four species: dog, rat, human
and rabbit [1, 2]. Figure 2.2 shows the mean and standard deviations of Kmn and Kmaj as

7
a
S
o
+^
I
s
1.2
1.0
0.8
0.6
0.4
0.2
0.0
(a) Dog
&
<£
Q
1.2
1.0
0.8
0.6
0.4
0.2
0.0
1.2
1.0
0.8
0.6
0.4
0.2
0.0
(d) Human
.a*liftllUlt^i
* (b)
•ii1
Rat
]I iii
'!
fit
IK
li
4
IF!
^min
10 15 20
Generation, N
25 30
Figure 2.2: Mean diameter ratios for the major («m a j, •) and minor (ftmm, o) daughters as
a function of generation number N for one animal from each of four different species: (a)
Dog, (b) Rat, (c) Rabbit and (d) Human. The error bars show the standard deviation of
the corresponding distributions at each N.
functions of generation number N for one animal from each species. For all four species,
we find that values of Kmjn and Kmaj are significantly different from each other and are
independent of N. Table 2.1 shows the average values of Kmin
and Kmaj- Figure 2.3 shows
an example of the diameter distribution HN(D) and Fig. 2.4 shows mean airway diameters
(DN) and their standard deviations a(D^) as functions of N.
The flow Qij is defined for each airway (i,j), with Qo,o = 1- At each bifurcation,
the flow Qij of a parent airway is partitioned between its daughters Qi+i,2j and Qi+i^j+i
according to
Qi+l,2j — r
Qi,j
i+l,2j + l ~ (1 — r
) Qi,j i
(2.2a)
(2.2b)
where the parameter r < determines the asymmetry of flow partitioning (Fig. 2.1).

8
Species
Dog
Rat
Rabbit
Human
Data
K
maj
0.927 ±0.085
0.865 ±0.165
0.887 ±0.263
0.876 ± 0.097
fi'min
0.574 ±0.117
0.583 ±0.182
0.529 ± 0.203
0.686 ±0.118
Model
r
0.198
0.286
0.237
0.326
V
2.92
2.32
2.26
2.97
Table 2.1: Parameters for four species obtained from data [1, 2] and used in the model.
The mean ± standard deviation of the diameter ratios Kraaj and Kmin are obtained from the
data shown in Fig. 2.2. Also shown are the values of r and 77 obtained by solving Eq. (2.4).
For simplicity, we assume the same partitioning of air flow at each bifurcation, so r is a
constant [6].
We assume that the dimensionless diameter Dij of an airway (i,j) is related to the
dimensionless flow Qij [9] as
Qij = {Dh3f (2.3)
where the exponent 77 is the same for all generations within the airway tree. Equation (2.3)
arises from the optimization of diameters of a single tube in order to minimize dissipation
while maintaining biological viability. For laminar flow the optimum value of rj = 3 [9], while
for turbulent flow 77 = 2.33 [10]. The former result has also been extended for symmetric
fractal trees [7].
We can express the diameter ratios by combining Eqs. (2.1)-(2.3),
Kmin = r1
'" (2.4a)
«maj = ( l - r ) 1 /
" . (2.4b)
Table 2.1 shows the values of r and 77 as obtained from experimental values of Kmin a n
d
«maj by solving Eq. (2.4).
We assume that when the flow through an airway falls below a critical threshold value
Qc, the gas transport transitions to diffusion and the airway is terminated by an air sac [6].
The cut-off diameter Dc at which airways terminate is given by Dc = (Qc) ' v
.
From Eq. (2.1) we see that a daughter with an even (odd) index j inherits the diameter

9
0.5
Q
£
C
>>
'55
na;
-0
>>
X5
03
X
oS-i
PH
0.4
0.3
0.2
0.1
r
Dog, r
,
^=7 ! J
-
0.025 0.05 0.1 0.2
Diameter, D
0.4
Figure 2.3: Linear-log plot of a typical distribution UN(D) of airway diameters I?. Filled
boxes show U^(D) for a dog lung at generation N = 7. The solid line shows the model
IIJV(Z)) as given by Eq. (2.7) using the parameters in Table 2.1.
of the parent airway multiplied by a factor Kmin («;maj). The diameter Dij can thus be
expressed in terms of nma and «maj using the number of even and odd steps required to
reach airway (i,j) from the root (0, 0). At generation N, the diameters D^j can take values
N-m
''maj min where m = 0 . . . N is the number of odd steps necessary to reach (N,j) from
(0, 0). Hence, we can express m as a function of the diameter I?,
m(D)
log D — N log Kn
log Kmaj logKn
(2.5)
The number flpf(m) of airways at generation JV corresponding to a particular m can be
found by enumerating the number of ways one can select the m odd steps among the N
steps, QAr(m) fN
Using the Stirling formula,
nN(m)^2 l-T7 exp
(m - N/2)'
W/2
(2.6)
Substituting m from Eq. (2.5) in Eq. (2.6), making a continuum approximation for m,
and taking into account the cut-off at Dc, we obtain the distribution UN(D) of diameters
at generation N. Thus, ILJV(-D) dD ex Q,^{rn) dm Q{D — Dc), where Q(x) is the Heaviside

10
step function. After normalization, we obtain
rijv(-D) = exp
AN,l
log2
{D/K
2Ns2
N*)
&{D - Dc (2.7)
The width and peak of the distribution IIJV(-D) are determined by s = ^ log (Kmaj/^min)
and K = y^^maj ^min respectively. The log-normal distribution is normalized by
AN = Ke
s2
/2
TTNS2
and A/v 1 reflects the effect of truncation at Dc, with
-N
AN,q erfc
(2.8)
(2.9)
where Nc = ^ log2
Dc and Nq = 2s2
/(qs2
+log2
K). Figure 2.3 shows that HN{D) predicted
by Eq. (2.7) agrees well with empirical data on the dog lung.
The mean diameter (DN) at generation N is given by
3s*/2N A
N,2
D N ne A N,l
(2.10)
For small N the effect of Dc is negligible, so the term AJV^/AJV.I ~ 1 and the mean di-
ameter decreases exponentially with N, (DJV) s=s (Ke3s
'2
)N
. However, for large N, (DN)
approaches Dc. Figure 2.4(a) shows the calculated (Djy) using the parameters in Table 2.1
in comparison to the measured data for four species.
In order to calculate the standard deviation a(Dj^) = [{Djf) — (D/v)2
] , we must
calculate the second moment
v? x _ (.„2 As2
N A
JV,3
AJV.I
(D2
N) = (K
2
e (2.11)
Figure 2.4(b) shows that a{D^) calculated using our model compares well with the observed
heterogeneity in measured data.
2.2 Terminal Generation Number Distribution
In this section we calculate the distribution of terminal generation numbers and the total
number of branches in an asymmetric tree model as a function of the asymmetry of the
airwav tree.

11
CD
• i-H
Q
CD
Dog
Rat
Rabbit
Human
0.1*
0.01
10 15 20
Generation, N
Figure 2.4: Semi-log plot of (a) mean diameters (D/v) and (b) standard deviation of the
diameters a(Dj^) plotted as a function of generation N for four species: Dog (o), Rat (A),
Rabbit (v) and Human (•). Solid lines show the model predictions using the parameters
in Table 2.1, and with Dc = 0.084, 0.070, 0.053 and 0.165 respectively for the four species.
The curves are vertically shifted for clarity.

12
Figure 2.5: Schematic diagram of the development of the branching pattern in an asym-
metric tree with r = 1/3 and Qc — 1/8. The numbers besides the branches show the values
of Q for the branch, with Q < Qc indicated in red. Terminal branches are shown in red
while missing branches are shown in cyan.
2.2.1 Exact solution
We begin with the flow partition equations as described by Eq. (2.2b). Figure 2.5 shows the
flow division in a tree with an asymmetry parameter r = 1/3. We note that flow decreases
quickest along the path on the extreme left where each generation multiplies a term r.
Similarly flow decreases slowest along the path on the extreme right where each generation
multiplies a factor (1 — r) to the flow. Thus the path on the left is going to reach Qc in the
fewest generations while the path on the right will reach Qc last. Thus the minimum and
maximum terminal generations in the tree can be written as
Nn
"log(Qc)
log(r)
iog(Qc
log(l -
(2.12a)
(2.12b)
where x is the ceiling function which provides the smallest integer k, such that k > x.

13
We note that Q can be expressed as
Q = rm
(1 - r)N
-m
(2.13)
where m is the number of "left turns" along the path from the root to the branch and N is
the generation number of the branch. Since r < 1/2, the greater the value of m for a fixed
N, the smaller the value of Q. For a given TV, we can find the minimum value of m to reach
Qc,
Q = r
m
(1 - rf-m
< Qc (2.14a)
(2.14b)
(2.14c)
r
1-r)
m
<
>
Qc
( l - r )
log Q c - i V log ( 1 - r )
logr — log (1 — r)
Thus, we define
m A^
l o g ( Q c ) - A T l o g ( l - r )
(2.15)
log(r) - log(l - r)
as the lowest value of m to reach the terminal condition Q < Qc. Next we note that
the number of airways at generation iV with a given rn is ( ). Thus the total number of
branches (including non-existent ones) at generation iV which meet the terminal criteria is
y^m ™* (')• However, these include daughters of branches which terminated at generation
iV — 1. Thus, we can write H^, the total number of airways which terminate at generation
TV as
m=m*N ' m=.m*N_Y
and the total number of terminals in the tree is given by
Y
max
Z= J^ HN. (2.17)
N=Nmm
We define the distribution 11^ as the fraction of terminal airways at generation TV, which
is given by
UN = ^ (2.18)

14
10 15 2(1 25
Terminal generation number, N
C
3
XI
Ci
S - l
10 15 20 25 3fl
Terminal generation number, N
Terminal generation number, N Terminal generation number, N
Figure 2.6: Distribution fl/v of terminal generation numbers N for asymmetry parameters
r = 3/16, 1/4, 5/16 and 3/8 from numerical summation of Eqs. (2.16), (2.17) and (2.18).
the distributions are characterized by sharp peaks and valleys overlaying a smoother hump.

15
0.2 0.3
Asymmetry parameter, r
0.5
Figure 2.7: Total number of terminal airways Z with r, relative to the number of terminals
for the corresponding symmetric tree with the same Qc = 0.0008 but with r = 1/2
Figure 2.6 shows the distributions terminal IIjv of terminal generation numbers N for
different values of the asymmetry parameter r from numerical summation of Eqs. (2.16),
(2.17) and (2.18). Figure 2.7 shows the change in the total number of terminal airways Z
with r, relative to the number of terminals for the corresponding symmetric tree with the
same Qc but with r = 1/2.
2.2.2 C o n t i n u u m a p p r o x i m a t i o n
To arrive at a closed form relations for the distribution IIjv of terminal generation numbers
iV and the total number of terminal airways Z, we approximate Eq (2.18) by assuming N
to be a continuous variable. First, we calculate the distribution P^{Q) of Q at generation
N. Similar to the calculation of the diameter distribution UN(D), the distribution PN(Q)
can be expressed as
1 - - ^ l o g 2
( Q / Q w )
PN(Q) = ^ e ~'N
7WV71
"
(2.19)

16
where
>N
IN -
iV, f l - r
2l
°g
(2.20a)
(2.20b)
We define g(N) as the fraction of airways at generation N for which Q < Qc,
g(N) dQ PN(Q)
= — erf
2
where
Nn
2 1ogQc
log (1 — r) + logr
1
> g Q c
Thus HN can be approximated as
(log(l -r) - l o g r ) '
log (1 — r) + log r
HN = 2N
g(N)~2 [2N
~'g(N-l)
= 2N
[g(N)-g(N-l)}
i A f
dg(N)
dN
and Z could be written as
where
Jo dN
i/ = ^1-ONQ log 2
(2.21a)
(2.21b)
(2.22a)
(2.22b)
(2.23a)
(2.23b)
(2.23c)
(2.24a)
(2.24b)
(2.25)

17
p
"3
10 15 20 25
Terminal Generation Number. N

Chapter 3
Fluid transport in branched
structures with temporary closures
The complex structure of biological systems [4, 11-15] and transport processes that occur in
them [16-22] are topics of much current interest, attracting researchers from engineering [23-
25], physics [26-29] and physiology [30-32]. The primary function of the respiratory systems
is to deliver air to the air sacs, called alveoli, for gas exchange. Morphological data show that
the mammalian lung consists of airways arranged hierarchically in an asymmetric binary
tree, the airway tree, with air sacs connected to the terminals [33, 34].
In this chapter, we address the problem of forcing fluid through an asymmetrically
branched structure with random closures that can be removed by the pressure of the fluid.
Such problems are often encountered during fluid flow in organ systems where the pathways
can be blocked, e.g. circulation of blood [30] and flow of air in the lung [35, 36]. Unrestricted
flow in these pathways is essential for proper physiological function, and blockages lead to
potentially lethal situations. In spite of its critical application, the problem of fluid flow
through collapsible bifurcating structures has only been marginally studied [37-40]. We
introduce a general tree model to characterize the asymmetry of the lung airway tree using
pressure-volume curves during inflation [41] and propose a method to obtain analytical

19
(a) Open Airway (b) Closed Airway
Figure 3.1: Section of an airway showing (a) the film of liquid when the airway is open, and
(b) the liquid bridge blocking the flow of air when the airway is closed.
results for tree structures and apply it to the process of lung inflation [8].
Many peripheral airways of a diseased lung collapse during expiration as the internal air
pressure and the tension of the elastic walls are insufficient to counter the surface tension of
the liquid lining [42-44]. The liquid forms a bridge or closure (Fig. 3.1) which completely
blocks the flow of air, excluding a large number of alveoli from gas exchange [36]. During
inspiration, the difference between the atmospheric pressure and the pressure surrounding
the lung, the transpulmonary pressure P, is slowly increased. As a result, a pressure
difference builds across the closures which are exposed to the atmospheric pressure through
the root of the tree. Each closure reopens when the pressure difference across it reaches
its critical opening threshold [45, 46]. Since the airways are arranged in a tree structure,
opening of one branch is not possible until all branches connecting it to the root of the
tree are open. If the threshold pressure of a daughter branch is smaller than that of its
parent, the daughter opens simultaneously with the parent. This mechanism also applies
to subsequent generations, leading to avalanches of airway openings [47].
The process of airway opening via avalanches has been studied for symmetric binary
tree models. The volume of inhaled air V during inspiration, for a fully collapsed lung, was
found to follow a simple power law in P,
V(P)ocPN
, (3.1)
where N is the generation number of the terminal branches [37-39]. Such P-V relations
are used to measure lung function in clinical environments. However, the real lung is asym-

20
metric, with many branches missing, which significantly distorts the P-V curve from the
ideal power-law behavior [33, 34, 41]. It is thus important to determine how the properties
of the system depend on the asymmetry of its underlying tree structure.
Avalanches are further complicated since the opening of an airway is accompanied by an
.audible pressure wave called crackle [48-50], which in turn can assist the opening of airways
downstream. Moreover, the air sacs are elastic and the effect of their elasticity on the P-V
curve becomes significant near the end of the inspiratory cycle, when the majority of air
sacs have been opened [38]. Although asymmetry, crackles and elasticity are important
contributors to the shape of the P-V curve, their effects are isolated to different regions and
thus it is possible to extract information about them by analyzing the same P-V curve.
We obtain experimental pressure-volume {P-V) curves of isolated dog lung lobes (Sec-
tion 3.1) and develop a model of the lung during avalanche-like airway openings (Sec-
tion 3.2). We show that when calculating the P-V relationship, it is possible to partition
the complex bifurcating structure into a set of paths connecting the root of the structure
to the air sacs (Section 3.2.1). Consequently
V(P) = VE(P)J2 n(n)Tn(P), (3.2)
n
where VE(P) is the elastic P-V relationship of the lung (Section 3.1), H(n) is the distribution
of terminals with generation number n. and Tn(P) is the opening probability of an airway
of generation n under the influence of avalanches and crackles (Section 3.2.2).
Using the analytic results of our models, we are able to fit the experimental P-V data
(Section 3.3), and obtain the distribution II(n), which is a key morphologic property of
the airway tree. Since experiments measuring P-V curves of an inflating lung are non-
invasive, this method provides a way to study "microscopic" branching structures from
"macroscopic" P-V data without the use of invasive techniques [41]. We compared these
results with known morphological data on the lung structure. The agreement of our model
with experimental data provides a better understanding of both, the general problem of
fluid flow through blocked pathways and the particular manifestation of this system in the

21
Q
J fr
Atmosphere
Pa
Pr-
Sealed Chamber
Figure 3.2: Schematic diagram of the experimental setup. The lobe is placed in a sealed
chamber (pressure Pc) with the main bronchus open to the atmospheric pressure Pa. The air
pressure in the chamber is slowly decreased using a vacuum pump which creates a pressure
difference P = Pa — Pc.
case of the lung.
3.1 Analysis of experimental pressure-volume data from dog
lungs
We experimentally determine the P-V curves of two isolated dog lung lobes, labeled A and
B. A cannula is inserted into the main bronchus and the lobe is degassed in a vacuum
chamber as described by Smith and Stamenovic [51], collapsing almost all the airways. The
degassed lobes are placed in an airtight chamber with the cannula attached to a metal tube
which is led through the lid of the chamber as shown in Fig. 3.2. We inflate the lobes, from
the collapsed state to total lobe capacity by steadily decreasing the chamber pressure Pc

22
using a suction pump. We measure the transpulmonary pressure
P = Pa - Pc ,
by recording the chamber pressure Pc with respect to atmospheric pressure Pa using a
Valydine MP-45 transducer (50 cm H2O). The airflow Q is measured at the main bronchus
using a screen pneumotachometer (resistance 5 c m # 2 0 / l / s ) attached to another Validyne
MP-45 transducer (2 c m # 2 0 ) . Pressure and airflow are both sampled at a rate of 80 Hz.
The pressure P is increased to 30 cm # 2 0 in 120 s. At this inflation rate, the time to regain
equilibrium after an airway opens is negligible compared to the total inflation time. The
volume V of inhaled air is calculated by integrating Q with respect to time,
V{t) = [ Q(t') dt! . (3.3)
Jo
The measured P-V curves are shown in Fig. 3.3. Although the two lobes have slightly
different V at maximum P, both curves show certain common features:
Region A (P < 10 cm H2O) : As P increases, V increases only slightly. At these pressures
almost all air sacs are collapsed and the slight increase in V is due to the opening of a small
number of airways and their subsequent elastic expansion.
Region B (10 cm # 2 0 < P < 20 cm # 2 0 ) : Over this range of P , V increases dramat-
ically from near 0 to near saturation. In this region, air sacs are recruited in avalanches
giving rise to the steep increase in V.
Region C (P > 20 cm # 2 0 ) : In this region, almost all air sacs are open and V increases
as a result of the elastic expansion of the opened air sacs. We fit this region using a single
exponential model for the P-V relation for the elastic expansion of the air sacs [52-54],
where V E ( P ) , the elastic volume of the lung, is given by
VE(P) = V0(l-ae-bP
), (3.4)

23
0)
£
350
300
250
200
150
100
50
^—A^K
VE(P),S
V(P)
0 w o < » » > ^ a Q l f l a
0 5 10
•oo-o
Lobe A
Lobe B
Elastic Fits
15 20
Pressure (cm H2O)
25
0
30
Figure 3.3: Experimentally determined P-V curves of two isolated dog lung lobes A (•)
and B (o), obtained during inflation from collapsed state to total lobe capacity. The dashed
lines show asymptotic fits to VE{P) given by Eq. (3.4) with VQ = 327.3 ml, a = 0.907,
b = 0.094 /cm H20 for Lobe A and V0 = 377.1ml, a = 0.908, b = 0.075 /cm H20 for Lobe
B.

24
Lobe A LobeB
^ 1.0
10 20 30 0 10
Pressure, P (cm H2O)
20 30
Figure 3.4: Volume fraction fv(P) of the open region of lobes (a) A and (b) B, as defined
by (3.6).
where the parameters VQ, a and b were determined by fitting experimental data for P >
2 0 c m i ^ O and are consistent with those previously obtained [55].
When all airways and air sacs in the lung are open, V increases only due to elastic
expansion and Eq. (3.4) describes the P-V curve. If only a fraction fv of the total volume
is open, the P-V curve can be written as
V(P) = fv(P)VE(P). (3.5)
Thus, the volume fraction fv of the open region of the lung can be calculated as
up)=^a, ( 3.6 )
and is shown in Figs. 3.4a and 3.4b for the lobes A and B.
The total volume V is the sum of the volume contained in the open air sacs, Va, and
the volume contained in the opened airways (branches), V&,
v = vb + va
(3.7)
In region A, Va « 0 as nearly all air sacs are closed and the observed volume V ~ V^. In
the fully open lung, region C, when all air sacs are open, Va is much greater than Vf>. This

25
Lobe A Lobe B
0 10 20 30 0 10 20 30
Air sac opening pressure, </> (cm H2O)
Figure 3.5: Distribution ijj{4>) of opening pressures <f> of the air sacs in lobe (a) A and (b)
B, obtained by differentiating Fig. 3.4(a) and 3.4(b) respectively, according to Eq. (3.9).
approximation is also valid for most of region B, once the first few avalanches occur. We
assume that V& <C Va and thus V ~ Va over the entire range of P; the approximation is
more accurate for higher P . If all air sacs are identical and each open air sac contributes
an equal volume, the increase in fv is due to the increase in the fraction of open air sacs fa,
Jv ~ Ja (3.8)
As P increases, more air sacs open and contribute to V. The increase in /„ is not
continuous, but occurs in steps of different sizes, corresponding to avalanches which recruit
varying numbers of contributing air sacs. The opening pressure (ft of an air sac is defined
as the pressure at which the air sac reopens. The distribution r
tp{4>) of opening pressures (f>
is an important measure of lung condition, often used to determine the applied pressures
during recruitment maneuvers [56, 57] artificial ventilation [58, 59]. When the pressure is
increased from P by an amount dP, the increase in the fraction of open air sacs d/a is the
fraction of air sacs with opening pressures <p 6 [P, P + dP). Thus the distribution ip{4>) can
be estimated as
dfa _ dfv
~ dP
p=4> a r
p=4>
W) dP
(3.9)
using the approximation of Eq. (3.8). The obtained distributions are shown in Figs. 3.5a

26
Parent branch
i + l,2j// i + l,2i + l
Left daughter Right daughter
Figure 3.6: Convention for labeling branches of the tree according to Eq. (3.10).
and 3.5b for lobes A and B respectively. Similar distributions of opening pressures have
been obtained using computed tomography [60].
3.2 Model of lung inflation during avalanches of airway open-
ings
We now develop a model of the P-V curve of an asymmetrically branched tree during
inflation. A tree is a minimally connected graph with one and only one path between any
two points [61, 62]. The lack of redundant paths makes tree structures vulnerable to edge
disruptions, since the removal of any one edge affects a large number of paths, significantly
affecting the connectivity of the structure. Although this property is the primary cause of
many obstructive lung diseases, we can exploit the strong signature of a collapsed airway
on macroscopic measurables such as the P-V curve to estimate the connectivity of the tree.
Using a simple thresholding model, we first obtain the fraction fa of air sacs open at any
pressure P and subsequently an expression of the P-V curve in terms of the tree structure.
Binary tree model
To study the inflation through the asymmetric lung we construct an incomplete binary tree
X, defined as a set of branches (airways). Each branch in T is labeled by a pair of indices
(i,j), where the index i is the generation number of the branch and the index j is used to

27
Figure 3.7: An example of an asymmetric tree T consisting of all labeled branches. Circles
represent the air sacs connected by the terminal branches (shown with underlined labels)
belonging to C. The double line ( = ) shows the path Vz.t connecting the terminal branch
(3, 5) to the root.
distinguish between branches of the same generation (0 < j < 2%
). The root of the tree is
labeled (0,0).
A branch either bifurcates into two daughters or subtends an air sac. The daughters
(i',j') of a bifurcating branch (i,j) are given by,
{ (1 + 1, 2 7) left daughter ,
J)
(3.10)
(i + 1, 2j + 1) right daughter ,
as shown in Fig. 3.6. Branches which subtend an air sac are the terminal branches or
"leaves" of the airway tree (branches with underlined labels in Fig. 3.7). The set of all
leaves of T is defined as £, where C C T.
We define a path Vij for a branch (i, j) as the set of branches connecting (i, j) to the root
of the tree (double line in Fig. 3.7). We note that according to the definition in Eq. (3.10),
the parent of (i, j) is given by ^ — 1, [j/2]), where [x] represents integer part of x. Thus,
pitj = {(i-k, [j/2k
]) : V k = 0...i}.
Each branch is either open or closed. The state (open or closed) of a branch (i, j) is

28
described by a Boolean variable £jj such that
{0 if (i, i) is closed,
1 if (i,j) is open.
Every branch (i,j) is assigned a threshold pressure pij. The threshold pressure determines
the transition of the branch from closed to open state.
Airway opening
At the beginning of inflation the lung is completely degassed and we assume that all airways
except the root are closed. Thus, £o,o = 1, and £ij = 0 otherwise. The pressure in all closed
branches of the tree is 0. The external pressure P at the root of the tree is increased from
0 by infinitesimal amounts until all branches in the tree are open. After each increase in P ,
the system is allowed to reach equilibrium, until all open branches connected to the root
are at pressure P.
All closed branches whose parent is also closed do not see any pressure difference A P
across their length. However, a closed branch (i,j) whose parent is open experiences a
pressure difference A P y . These branches form an interface between the open and closed
regions of the lung (dashed line in Fig. 3.8) called active surface [50]. Since the equilibrium
pressure in the open branches is P and that inside closed branches 0, A P J J = P. However,
transients during airway openings could cause A P , j > P for some branches on the active
surface.
Figure 3.9 illustrates the opening of an airway (i,j) under an applied pressure difference
A P j j . For A P j j < pij, the liquid bridge in the airway has a finite thickness and the
surface tension 7 of the liquid is able to sustain the pressure difference (Fig. 3.9a). When
the pressure difference A P j j across the branch exceeds its threshold pressure phj, surface
tension can no longer sustain the liquid bridge. At this point the airway opens and the
energy stored in the liquid bridge is released in the form of a pair of sound waves (one
traveling upstream and the other downstream) called crackles (Fig. 3.9b). Immediately

29
P = 0 p = p 1,0 P = Pi,i
p i , i
p
(e)
A <P2,2 = 9 1 , 1
<
P2,2 P l , l
P
Pi,I P3,5
P
Figure 3.8: The process of airway opening in a tree, (a)-(e) show the states of the tree
with increasing P. Branches are labeled as shown in Fig. 3.7. Open branches are shown as
outlines, newly opened branches are shown in gray and closed branches are shown in black.
The active surface is shown as a dashed line. Inflation begins at P — 0 with all branches
other than the root closed (a) and proceeds by airway openings, either individually (b) or
in an avalanche (c), as P is increased, (d)-(e) show the pressure differences AP and states
£ of three segments (1,1), (2,2) and (3,5) respectively, belonging to the path 7^5, shown
in Fig. 3.7. Different behavior is observed for branches on the active surface and those
embedded in an avalanche.

30
S-i
m
w
,
AP;,,
P^TN
" 7 / « i
.7/-R2
KR2
(a) APZ;i < pi* . j
Front
Negative
spike
Positive
spike
(b) AP^- = pT+
Figure 3.9: Pressure along the axis of a liquid bridge in branch (i,j) when A P y is just
above ptj, the liquid bridge breaks, a pair of sound waves ("crackles") are generated and
the pressure front propagates downstream.
following opening, the air pressure on two sides of the former liquid bridge is significantly
different and the two regions are separated by a sharp pressure front.
The pressure front diffusively propagates deeper into the tree until the two daughters
of the branch (i,j) are exposed to the external pressure P (Fig. 3.9). If the threshold
pressures of the daughters are lower than P, the daughters open simultaneously with the
parent. The process of opening is continued until all closed branches connected to the root
of the tree have threshold pressures greater than P~ and a new active surface is formed. The
simultaneous opening of a subtree following a small increase in P is called an avalanche [47].
Threshold pressures
The threshold pressure of an airway strongly depends on local variables such as the rigid-
ity of the airway walls, the amount of fluid present and its surface tension [43, 44]. Since
these quantities vary from airway to airway, the threshold pressures can be effectively con-
sidered to be independent random variables distributed according to generation dependent
distribution functions pi(p). Although we allow pi to be generation dependent, we assume

31
that branches of any given generation are statistically identical and hence their threshold
pressures are drawn from the same distribution.
A branch (i,j) is open if and only if it has an open parent and the pressure difference
APij = P across it exceeds its threshold pressure pij. Thus,
&j = ®(P ~ Pij) &-i,[7/2] . ( 3
- n
)
where
1 for x > 0 ,
e(x) ;
0 for x < 0 ,
is the unit-step function.
Opening pressures
Every open branch (i,j) other than the root undergoes a transition from being closed to
being open at a pressure defined as the opening pressure (pij of the branch,
£ij = e(p-<t>i,j). (3.12)
Using this definition and Eq. (3.11) we can write 0 ( P — (pij) = Q(P-pij) 0 ( P — <&-i,y/2])>
which has a solution
(pid = max(pjj, (pi-i^/2]) • (3.13)
Thus the opening pressure (pij of a branch (i,j) is the maximum of its threshold pressure
Pij and the opening pressure of its parent (pi-ij/2-
If the threshold pressure pij of a branch (i,j) is less than the opening pressure of its
parent </>i_ih/2], the opening pressure (pij — (
Pi-i,[j/2] a n
d thus the branch (i,j) and its
parent open simultaneously as part of an avalanche. For example, the branch (2, 2) opens
simultaneously with its parent (1,1) in Figs. 3.8c and 3.8e. For a branch (i,j) on the
active surface, the threshold pressure pij is greater than the opening pressure of its parent
(pi-iu/2] > since this is precisely the condition that stops an avalanche and produces the active
surface. Thus according to Eq. (3.13), the opening pressure (pij = pij which is greater than

32
the opening pressure of its parent 4>
i-,j/2- F°r
example, the branch (3,5) does not open
simultaneously with its parent (2,2) but at a higher opening pressure (Fig. 3.8f).
Transients
The threshold pressures pij are assigned a priori and represent the quasi-static opening
pressures of the airways. However during fast dynamic openings within an avalanche, the
actual threshold pressures and the pressure difference across the segment could be different
from their static counterparts. In particular, crackles which accompany airway openings
cause an instantaneous increase in AP. We therefore replace the step-function 0 ( P — Pij)
by a more general function Fij(P) = F(P,pij, 4>i-ij/2])i which too is a step function whose
argument depends on the opening pressure of the parent (i — 1, [j/2]) in addition to the
pressure P and the threshold pressure pij. Thus we rewrite Eq. (3.11) as
ZiJ=FiAP)Zi-i,j/2]- (3-14)
The exact form of Fij(P) depends on the model of airway opening considered.
3.2.1 P a r t i t i o n i n g t h e effect of a s y m m e t r i c structure from t h e inflation
p r o c e s s
Equation (3.14) recursively expresses the state of airway (i,j) in terms of the state of its
parent. By iterating Eq. (3.14) we write the non-recursive form as
& d ~ Fi,j Fi-i,j/2] • • • -^0,0 £o,o
= n F
kAp
)> (3
-15
)
since £o,o = 1 as the root is always open. Thus a branch (i,j) is open if and only if all
branches along the path V%j connecting it to the root of the tree are open.
We can now calculate the fraction of open air sacs at a given pressure. Since each
terminal airway subtends one air sac, the total number of air sacs in the lung is equal to

33
rix, the number of terminal airways. An air sac is open if the terminal airway connected to
it is open, and the fraction of open air sacs fa is given by
/« = — E &J> (3
-16
)
TIT
(i,j)ec
where the sum ^ £jj gives the number of open leaves of the tree.
To compare our results with experimental data, it is necessary to average over all con-
figurations of threshold pressures pij. Using Eq. (3.16) the averaged quantity {fa) can be
written as
" (3.17)
(fa) = — / Vp p{p)
rix E ^
where
/
/•oo roo
Vp p(p) = / dpofi Po('Po,o) ••• &Pi,j Pi(Pi,j) • • •
J — oo J— oo
represents an integration over all possible values of the threshold pressures of every branch in
the tree. We note that since the distributions pi{pi}j) are normalized, each of the bare inte-
grals f dpij Pi{pi,j) = 1 and their product / D p p(p) = 1. Thus the expression in Eq. (3.17)
is self-normalized. Reversing the order of the commutative operations of integration and
summation, we get
(fa) = ^ E /"PPM**
(i,j')6£
- E &J> • (3
-18
)TIT
Thus, Eq. (3.18) partitions the averaged fraction of open air sacs in the tree into a
normalized sum of probabilities of the existence of open paths from the terminal branches
to the root of the tree.
Opening probabilities
The state variable ^ j is a product of terms that are functions of the threshold pressures
of all branches along the path V%,j and the external pressure P, as expressed by Eq. (3.15).

34
Since the distribution functions p only depend on the generation number, the averaged
quantity (£i.j) depends only on the external pressure P and the generation number i. We
define Ti(P) — {^ij) which is the opening probability of a branch (i,j) at pressure P , so
Eq. (3.18) can be rewritten as
TIT ^—^
Collecting all terminal branches of the same generation n, we can rewrite the above sum as
(fa) = J2 n(n) rn(P), (3.19)
n
where U(n) is the distribution of generation numbers n of the terminal branches, i.e. the
fraction of terminal branches with generation number n.
Equation (3.19) conveniently separates the effects of morphological features of the tree
structure in a lung, given by the distribution of terminal depths II(n), from the dynamic
component described by the opening probability Tn(P). This allows us to calculate Ti(n)
from models of tree structure and Tn(P) from models of different dynamical processes in a
much simpler geometry.
We note that for a symmetric tree, all terminal branches at the same generation 7V and
thus the generation distribution II5 of the terminal branches for a symmetric tree is given
by
Us(n) = 5n^N .
Using Eq. (3.19), the fraction of open air sacs (/f) for a symmetric tree can be calculated
as
(f!) = TN(P). (3.20)
Thus Eq. (3.19) allows us to use the results obtained for symmetric trees and translate them
to asymmetric trees with different II(n).

35
The P-V curve
We can now write a comprehensive expression for the volume V of the lung as a function of
pressure P. Using the expressions of Eqs. (3.5) and (3.8) and replacing fa by (fa), we get
V(P) = VE(P)(fa),
which can by expanded using the result of Eqs. (3.19) as
V(P) = VE(P)J2 n ( n ) T n ( P ) . (3.21)
n
Although the expression in Eq. (3.21) was obtained for a binary tree, it is equally
applicable to trees of different, even heterogeneous, branching. Thus in section 3.2.2 we
calculate Tn(P) for various models on linear chains of n generations and apply those results
to the asymmetric airway tree.
3.2.2 Models of airway opening: Correlated inflation processes
We consider a linear chain of N closed branches labeled j = 1... N. The internal pressure
in the pipe is 0 while an external pressure, P, is applied at one end (j = 0). The quantity
of interest in this case is Tpj(P) which is defined as the probability of fluid flow in a pipe
with N closures at pressure P. For end-to-end fluid flow, we need all the N closures to
be open at the given pressure P. At pressure P = 0, all closures are closed and hence the
probability of flow, IV (0) = 0.
We define a probability density function ipj ((f)) such that ipj (</>) d(p is the probability for
closure j to have an opening pressure between 4> and cj) + dcj). A function, Gj(<f> (f)), can
then be defined as a conditional probability that the branch (j + 1) has an opening pressure
between 4> and cj) + d<fi , given that the j-th closure opens between pressures 4> a n
d 0 + d(j).
This allows us to write
^•+i(</>') = I d 4 Gj(<f>'<f>) ^3{<t>) (3.22)
Jo
We note that there is a one-to-one correspondence between the conditional probabilities
G(4> 4>) and the opening functions Fij(P). Defining either of these two functions completely

36
defines the dynamics of the system.
To calculate ipj, we need an initial state, which can be calculated by defining a hypo-
thetical closure at j = 0 and assuming that this closure is permanently open, that is 4>o =
0.
Thus,
V>o(</>) = £(</>). (3.23)
The opening probability, r V ( P ) , can thus be written as
YN{P) = f ^V;vW>) (3.24)
Jo
In the following subsections, we define three specific models of airway openings, con-
struct their respective conditional probabilities G(</)'4>) and calculate the opening proba-
bility r V ( P ) . The first, Model A, describes the simplest process of avalanching. Models B
and C add the effect of transients, especially crackles, to the opening process by modifying
the threshold pressures of the segments permanently or temporarily. Pressures are normal-
ized such that the maximum threshold pressure Po in the tree is 1. In all three models we
assume that the threshold pressure distribution p(p) is uniform between 0 and 1. These
models then allow us to fit the experimental P-V curve using Eq. (3.21).
Model A : Simple avalanching
This is the simplest model of airway opening. To construct G(4>'(f>), we look at the processes
by which a branch opens. If the opening pressure <fi' of the (j + l)-th branch is less than
that of the j - t h branch, </>, the branch (j + 1) will open simultaneously with the branch j as
a part of an avalanche. We could thus write G for this part as 4>8{4>' — <f>) where the factor
<fi is numerically the probability that <p' is less than </>, since the distribution of threshold
pressures is uniform. The (^-function reflects the fact that the (j + l)-th branch opens at the
same pressure as the j - t h one. However, if <// is greater than <f>, the (j + l)-th branch will
open independently and G will contain a term Q{<f>' — <j>), 0 being the unit step function,
reflecting the ordering of the opening pressures. The function G is thus given by
GA
{<l>'(f>) = (f> 5{<P' -$) + 6 ( 0 ' - (/>). (3.25)

37
Using Eq. (3.23) and (3.25) and by repeated application of Eq. (3.22) we find
^Jf{(j))=34P- (3.26)
Thus using Eq. (3.24), we are able to derive the opening probability as
r £ ( P ) - = PN
. (3.27)
This is identical to the expression in Eq. (3.1) that can be derived using other methods [37-
39].
Model B : Permanent effect of pressure wave
In this case we slightly alter the algorithm for the change of state of a closure. In addition
to opening only when the pressure across the closure exceeds its threshold pressure, we take
into account the added effect of a pressure wave. When closure j opens, a pressure wave is
set up in the fluid which facilitates the opening of closure (j + 1). We take this into account
by changing the opening pressure of the closure (j + 1) as
<pJ+i -> a (j)j+i (3.28)
where, a(< 1) is a constant. In this model, the reduction of the threshold pressure is
permanent, i.e. once a parent opens the threshold pressure of the child is maintained at
the reduced level for the duration of the experiment. Thus for all practical purposes, the
threshold pressures of all generations greater than 1 are distributed uniformly between 0
and a while that of the first generation is distributed between 0 and 1 (as it cannot be
opened in the wake of the pressure wave from the parent).
We can then modify Eq. (3.25) to write the function GB
((f)'(f>) for Model B as
GB
{<j,'<j>) = ^ e ( a - < £ ) < ^ ' - 0 )
a
+ - G(a - <f>') Q((f>' - </>)
a
+ e(<f> - a) 6(<l>'-</)). (3.29)

38
The first term again represents the avalanche part of the closure opening, but in this
case the renormalization of the opening ((f)) increases the probability factor by 1/a. A step
function is also included, which distinguishes the behavior of the closures for pressures less
than a from the automatic opening at pressures greater than a. The second term represents
the independent opening of a closure and the probability is again rescaled by a factor 1/a.
The two step functions in this term not only reinforces the distinction in the first part but
also restrict the possible values of <f>' to less than a. The final term is included to take into
account the automatic opening of the closures at pressures greater than a.
Using the result of Eq. (3.29) in Eq. (3.22) and the initial condition from Eq. (3.23) we
can derive
Vf«>)=j(^Y G(a-0) + e(<f>-a). (3.30)
Again the fluid flow probability can be derived using Eq. (3.24) as
/ p N-l
T%{P) = p(- Q(a-P) + P e(P-a). (3.31)
Model C : Transient effect of pressure wave
The depression of the opening pressure due to the pressure wave in Model B (Eq. (3.28))
is a permanent phenomenon. This means that once the threshold is lowered by the pres-
sure wave, it does not regain its original value. Thus all thresholds after the first one are
distributed between 0 and a and not between 0 and 1. However apart from this renor-
malization, there is very little that is different between Models A and B. We shall now
try to explore a more intricate model in which the reduction of opening pressure is only
a temporary phenomenon and the threshold regains it's original value after a short time,
unless the closure is opened instantly. We shall deal only with instantaneous reduction of
the threshold which facilitates the avalanche like opening of the closure but has no effect
on the independent change of state.

39
The conditional probability GClA'
GC(±!
for this model is given by
+ e(« - ft) e(ft - ^)
a
+ O(0 - a) 5(0' (3.32)
As mentioned earlier, the process of avalanching in this model is identical to Model B
and thus the first term of Gc
is identical to that in Eq. (3.29). However, the second term,
describing independent opening, is markedly different in this case. Not only is there no
rescaling of the opening pressures in this event, there is also the absence of the restricting
step function on ft'. Thus ft can now take values greater than a and give rise to delayed
large avalanches. The final term is again identical to that in Eq. (3.29). This is because at
pressures greater than a all closures are opened in large avalanches.
The Eq. (3.32) can now be used to solve for the probability density function, tp^((b),
which is given by
Aj(a) ft'1
0 ( a - 0 )
3-1
VfW
+
3-1
1 + ^Bkia)
k=l
where
Aj(a)
Bj(a)
n(i+fc=i
a
a k
6 ( 0 -a)
and
A,-_i(a) a-
J
for j > 1, and Ao(a) = BQ{U) = 1.
Upon integrating Eq. (3.33) with respect to 0, we get
' pN
-.c?N(P) A J V - l (a)
N
0(a - P)
+ r ^ > ( P ) 0 ( P - a ) .
(3.33)
(3.34)
(3.35)
where
N-l R , s
T%>{P) = BN{a)+YJ^fl{rk+l
a
fe+i (3.36)

40
o• r—I
• I—I
+^>
CO
Q
6
4
2
0
(a) Model B .'!
/I
Model C
Model A
0.0 0.5 OL i.o
Opening Pressure,
1.5
1.0
0.5
0.0
(b)
0.0
Model B —• //— Model A
f Model C
0.5 OL l.o
Pressure, P
1.5
Figure 3.10: (a) The distribution ip(cj>) of opening pressures <f> and (b) the opening proba-
bility T(P) of an airway as obtained from the three models of airway opening (A, B and C)
for a chain of six branches and with a = 0.75 for models B and C.

41
The opening probability T^v and the distribution ip of opening pressures <j> f°r
the three
models are compared in Fig. 3.10. The distribution ip for model C (Fig. 3.10a) is visually
similar to the distributions obtained from experimental data (Fig. 3.5). We note that ]?jv is
identical to the open fraction in a symmetric tree (Eq. 3.20). Thus for the same maximum
threshold pressure and number of generations, models B and C recruit more air sacs than
the simple avalanching model A (Fig. 3.10b).
We can construct more sophisticated models of airway opening by extending these basic
models. The pressure wave could have a partly instantaneous and partly permanent effect
on 4> by combining models B and C. The parameter a could be distributed instead of being
a fixed number. The threshold pressure distributions could be made non-uniform as well as
generation dependent. In each case, the technique described in this section could be used
to obtain an analytical solution for ip{<f>) and T^(P). These results can then be combined
with a distribution of generation numbers of terminal branches H(n) and the elastic P-V
curve VE(P) to obtain the final pressure-volume relationship of the lung.
3.3 Characterization of the branching structure of the lung
from "macroscopic" pressure-volume measurements
We fit the fv(P) curves obtained from experimental data (Fig. 3.4) with polynomial func-
tions Y2n
a
n{P/Po)n U
P to the inflection point P x in the curves. The maximum threshold
pressures PQ is given by the pressure above which all branches are open and thus fv = 1.
The inflection points in the curves, determined by numerically differentiating the curves for
fv and finding the first maxima. For model A, we determine Po by fitting the curve up
to Px and extrapolating it to fv = 1. For models C, P x represents the point of crossover
from avalanche-like behavior to pressure-wave mediated behavior and thus the parameter
a = Px/P0.
We use polynomials of order 48, since this is the known maximum depth in a dog
lung [34]. The large number of coefficients makes simple regression unstable, and we use

42
an additive diagonal term in the coefficient matrix to regularize the results. The raw fit
thus obtained is then fine tuned by randomly updating each coefficient by a small amount
and recalculating the fitting errors simultaneously in the normal and logarithmic scales, to
ensure the accuracy of the coefficients for small n.
For model A, fv is given by using Eqs. (3.19) and (3.27) as
</,?)= £n(n) fyY (3.37)
n ^
and the coefficients of the fitted polynomial an — n(n), the distribution of terminal gener-
ations. For model B, the expression for fv for pressures less than a is given by
and the distribution can be calculated from the polynomial fit as H(n) = anan
~l
. Similarly,
for model C, the fv is given by
for pressures up to a. Thus, the distribution of generation numbers of the terminal segments
can be estimated by U.(n) = nan/An(a).
For models B and C, we fit the region P > a using the expressions for Tn(P) in this
region as given by Eqs. (3.31) and (3.35) and the same II(n) as obtained by fitting the
region P < a. The fitted curves for fv using models A and C for the lobe A are displayed
in Fig. 3.11a. The distribution H(n) obtained using model C is shown in Fig. 3.11b.
The distribution H(n) in Fig. 3.11b has two distinct regions, a narrow peak for n < 5
(shown as open rectangles) and a broad distribution for 15 < n < 40 (shown as filled
rectangles). The second part of the distribution has two main peaks in the region 22 < n <
30.
We compare n(n) to a known model for the airway tree structure, the Horsfield model [34]
which is an asymmetric self-similar description of averaged experimental data obtained by
physical measurements on a polymer cast of the airway tree. The Horsfield distribution

43
O-1—i
• i—i
a
.S-i
P
1.0
0.8 h
0.6
0.4
0.2 h
(a)
Lobe A
Q.Qcbo o o o o o
OGXDCQCQ3Q0Q
Data
Model A
Model C
o
0 10 20
Pressure, P (CU1H2O)
30
o. 1—1
5-1
-M
CO
.1—1
Q
0.15
0.10
0.05
0.00
(b)
Lobe A
Horsfield
0 10 20 30
Generation number, n
40
Figure 3.11: (a) The volume fraction fv of the open air sacs obtained using models A and
C obtained by fitting the experimental data for lobe A and (b) the distribution I7(n) of
the generation numbers n of the terminal branches obtained from the fit, compared to the
distribution for the Horsfield model of the dog lung.

44
corresponds in shape and position with the l~I(n) obtained by fitting the P-V data. We are
able to recover the two main peaks at approximately their correct positions. The small-n
part of the distribution (n < 5), that we obtain from our data, does not correspond to
the branching structure of the tree since the dog lung is not known to have terminals with
depths n < 13.
We attribute the existence of the small-n part of H(n) to the airway wall elasticity and
the volume of air contained in the airways before any air sacs open (Appendix A). The
first few branches of the airway tree are held open by cartilaginous rings, and the expansion
of these branches at low P also contributes to the small-n part of II(n). We ignore this
region when focusing on the branching structure and normalize the Horsfleld model to only
the area under the second part of the distribution. The Horsfield model is an idealized
description of the dog lung and does not account for the differences between individual
dogs. In contrast, with our approach we can also identify the variation in structure among
specific samples.
Finally, using Eq. (3.5), we combine the effect of elasticity to obtain the full P-V curves
of our models using the expression for VE(P) from Eq. (3.4) along with the parameters
obtained from the fits shown in Fig. 3.3. Figure 3.12a compares the distribution of opening-
pressures using model C with that obtained using the experimental data. The resulting
P-V curves are compared in Fig. 3.12b.
The P-V curve of model C has a small deviation from the experimental curve near the
maximum threshold pressure (Fig. 3.12b) due, we believe, to an underestimation of the
maximum threshold pressure, i.e. the pressure at which all airways are opened. Our as-
sumption, that the maximum threshold pressure of the branches correspond to the pressure
at the point of inflection is only true when the distribution of threshold pressures is uniform
and generation independent. However, if the threshold pressures are generation dependent,
our method underestimates the maximum threshold pressure [38, 57]. To estimate the ef-
fect of generation dependence, we simulated inflation of randomly branched trees using a
simple generation dependent threshold pressure distribution with overlapping domains. We

45
3
ef
o• r—I
• i-H
CO
• I—i
Q
0.3
0.2
0.1
0.0
(a)
Lobe A
Model C
V +< Data
0 10 20 30
Opening Pressure, (j> (CU1H2O)
300
200
100
(b)
Lobe A
Q cpo 0 0 , 0 0 O-Q-
0
Data o
Model A - - -
Model C
10 20
Pressure, P (cmi^O)
30
Figure 3.12: (a) The distribution ip((f)) of opening pressures <fi using model C compared to
the experimental data from lobe A and (b) the full P-V curve reconstructed using Eq. (3.3)
and model C, compared to the experimentally obtained data.

46
found that the inflection point shifts to a pressure smaller than the maximum threshold
pressure, independent of the exact distribution or the degree of randomness in branching.
The high pressure in this region would allow a more significant contribution from the open-
ing of the deeper air sacs (Eq. (3.21)), which we are unable to probe accurately. However,
in real lungs, these air sacs (n > 30) are few in number (Fig. 3.11b) and do not contribute
significantly to the shape of the P-V curve.
3.4 Airway collapse, hysteresis and trapped gas
The collapse of airways during expiration has important consequences. Airway collapse
causes air to be trapped in the sub-tree behind the closure. This trapped gas keeps part
of the lung inflated and consequently the lung volume does not decrease to zero at end
expiration. The volume of the lung at end expiration is called "residual volume" and is
typically 10% of the maximum lung volume in normal humans. Another consequence of
airway collapse is the hysteresis observed in the P-V curve where the inflation limb is
characterized by lower volume and the deflation limb by a higher volume. Although factors
such as viscoelasticity of the lung tissue, resistance of the airways and the hysteresis of the
lung surfactant that lines the lung contribute significantly to the hysteresis, we show that
airway closures is a major cause of hysteresis, especially during slow breathing.
We model airway closure as a threshold phenomena similar to our model of airway
opening. During deflation, the pressure P in the lung continuously decreases. When the
pressure in an airway falls below its closing threshold pressure C, the airway collapses. We
assume that, for each airway, the threshold pressure for airway closure is a fixed fraction A
of its opening threshold pressure. Thus,
C = AT, (3.40)
where the constant A < 1 to ensure that an open airway does not close immediately during
inflation.

47
Deflation
First Inflation
_J L
c T C + T
Pressure. P
Deflation -
Second
Inflation
First
Inflation
1 + A
Pressure. P
Figure 3.13: (a) Volume and (b) average volume as a function of pressure.

48
We illustrate the phenomena of airway opening and collapse during cyclic breathing
using a simple system consisting of a single airway connected to an alveolus (Fig. 3.13a).
Initially, the lung is degassed, the airway is closed and thus the alveolus has zero volume V
and is at zero pressure. During first inflation, the pressure P at the open end of the airway is
increased quasi-statically, and the airway experiences a pressure difference of P (black line).
However as long as P < T, the opening threshold pressure of the airway, the airway remains
collapsed and V remains zero. When P = T, the airway suddenly opens and the pressure
in the alveolus equilibrates with the applied pressure P and the V = VE{P) as described
by Eq. (3.4). As pressure continues to increase, V increases along the elastic curve. During
the deflation phase (red line), P is quasi-statically decreased at the airway opening. The
volume V decreases along the elastic curve as long as P > C, the closing threshold of the
airway. When P = C, the airway collapses and traps air in the alveolus at pressure C and
and the trapped residual volume Vmin = VE(C) remains in the alveolus even as P decreases
to zero. During the second inflation (green line), the airway has an internal pressure C.
Thus when a pressure P is applied at the airway opening, the pressure difference across
the airway is P — C. The condition to open the airway is given by a pressure difference
greater than the opening threshold T, and thus the airway opens only when the pressure
P > T + C. The opening pressure is higher during the second inflation than during the
first inflation due to the pressure of the air trapped int he alveolus. For P < T + C, the
volume V = Vmm. At P = T + C, the volume jumps to V = VE(T + C) and subsequently
the volume follows the elastic curve VE(P)-
Now we consider a population of alveoli each having an opening threshold pressures Tt
drawn from an uniform distribution in the range (0, f) but with a same ratio A of closing
to opening threshold Cj/Tj. Such a situation would represent a lung in which only the
smallest airways collapse each trapping air in the alveolus it subtends. We attempt to find
the relation between the applied pressure P and the mean volume (V) of the population
(Fig 3.13b). During the first inflation, the probability of an airway to be open is identical

49
to the probability that P > Ti, which is given by
rP P for P < 1,
Probability(P > T) = / d P p(T) = I (3.41)
0
l l f o r P > l ,
where p(T) is the probability density function of T, which is uniform between 0 and 1.
Thus, for P < 1, the volume Vt of an alveolus is
0 with probability 1 — P,
Vi(P) = { (3.42)
VE{P) with probability P.
Since all alveoli are statistically identical, the population mean would be equal to the
configuration mean and thus,
,PVE(P) f o r P < l .
(1/(1
>) = { (3.43)
VE{P) for P > 1
where the superscript 1 refers to the first inflation (black line). During the deflation phase,
the airway connecting a alveolus closes when P < A Tj. The volume of an open airway is
given by VE(P). However if the airway is closed at pressure P, the volume contained in the
alveolus depends on the pressure at which the airway closed. Thus, for P < A,
rP/X f l
(V{
-]
)= dTp(T)VE(P)+ dT p(T)VE(XT). (3.44)
Jo Jp/x
where the superscript refers to the deflation curve. For P > A, the airway is always open
and the volume follows the elastic curve. Using the expression in Eq. (3.4) for V#(P), we
can write,
{^o fl +(IJ
f^- ~ "^r^- (1 + bP)] for P < A
°i bX bX [
> ~ (3.45)
V0 (l~ae-bP
) for P> A
This curve is shown as a red line in Fig. 3.13b. For the second inflation, each airway
remains collapsed until the pressure difference reaches the opening threshold T^. Since the
air trapped in the airway is at pressure AT, the airway only reopens when P — XT > T.

50
Thus for P < (1 + A)r the volume of the alveolus is VE(XT) while for P > (1 + X)T the
volume is given by VE{P). Thus, for P < (1 + X)T
(VM)= /1 + A
dTp(T)VE(P) + [ dT p(T)VE(XT). (3.46)
where the superscript refers to the second inflation curve. For P > A, the airway is always
open and the volume follows the elastic curve. Using the expression in Eq. (3.4) for VE{P),
we can write,
(!/(+))
Vr ! + & (ebX
~e~^ )-ffxe~bP
for P < 1 + A
(3.47)
V0(l~ae-bP
) for P > 1 + A
This curve is shown as a green line in Fig. 3.13b. The residual volume, VRV is given by
^RV - (^(_)
>(0) = <V«)(0) = Vb [l ~ ~ ( l - e~bX
)] (3.48)
To extend the results to a tree, we first consider an alveolus connected by a chain of n
airways with opening threshold pressures 7, T2, .. .Tn. In this case, the entire path con-
necting the alveolus is open only if the pressure P exceeds the maximum opening threshold
pressure Tmax = max(Ti,T2,... Tn). We assume that each T{ is an independent random
variable distributed uniformly in the range (0,1). Thus the distribution pmax(n
,Tmax) of
Tmax among n threshold pressures Tj is given by
Pmaxn
i J-max) =
"•'max (0.4yJ
The averaged volumes can now be obtained by replacing the threshold pressure distribution
p with the maximum threshold pressure distribution pmax in Eqs. (3.41), (3.44) and (3.46).
The P-V during first inflation is described by
(VW)= f dT Pmax(n,T)VE(P) (3.50)
Jo
for P < 1, where pmax(^,?1
) is the probability of existence of an open pathway from the
airway opening to the alveolus and VE(P) is the volume of the open alveolus. If an open

51
i i 1 1 1—i i
0 0.5 1 1.5 II 0.5 1 1 5
Pressure, P Pressure, P
i , 1 1 1 1 i
Figure 3.14: P-V loops for (a) N = 1 (b) N = 2 (c) N = 4 (d) N = 8.

52
pathway does not exist, the volume of the alveolus is zero. Using the expression in Eq. (3.49),
Eq. (3.50) evaluates to
{V0 (l-ae-bP
) Pn
f o r P < l ,
(3.51)
V0 (l-ae-bP
) for P> 1.
The black lines in Fig. 3.14 show the P-V curves during first inflation for different n.
During deflation, the V = VE{P) for P < X, the maximum closing threshold. However,
for P < A, the volume of the alveolus is given by VE{P) if the path to the airway opening is
open, or VE(XT) if the airway path is closed, where AT is the maximum closing threshold
in the airway path. Thus, the volume V can be written as
(!/(-))= / dT pmax(n,T)VE(P) + dT Pmax(n,T) VE(XT). (3.52)
Jo JP/X
Using the expression in Eq. (3.49), Eq. (3.52) evaluates to
(
<y(-)> =
V0 l + a n £ i _ „ ( f t A ) - a (f)™ {e-bP
+ nE^n(bP)} for P < X ,
(3.53)
V0 l - a e - &
n for P> A,
where
oo p—xt
En(x) = dt— (3.54)
tn
is the J5n-function, related to the exponential integral. The red lines in Fig. 3.14 show the
P-V curves during deflation for different n.
During the second inflation, the alveolus initially contains trapped air at pressure AT,
where AT is the maximum closing threshold in the airway path. The maximum opening
pressure along the airway path is T. Thus the airway path reopens when P — XT > T, or
P > (1 + A) T. Since the maximum value of T is 1, the airway path is always open when
P > 1 + A and the volume of the alveolus is given by V = VE(P). For P < 1 + A, the
volume of the alveolus can be wriiten as
,. p
,-i
(V<-+)
) = / 1+A
dT pm a x (n,T) VE(P) + / dT pmax(n,T) VE{XT). (3.55)

53
Using the expression in Eq. (3.49), Eq. (3.55) evaluates to
1 + o n £ i _ n ( 6 A ) - a f ^ l ie-^ + nEir ^ n ^ + ^i-n(m) for P < 1 + A
<y(+)> = ;
Vn (1 -ae~bP
) for P> 1 + A
(3.56)
The green lines in Fig. 3.14 show the P-V curves during second inflation for different n.
Figure 3.14 shows the effect of generation number n on the P-V loops. We note that
with increasing generation number the area of the P-V loop increases. We note that if the
pressure at end-inspiration Pma,x > 1 + A, all subsequent cycles follow the (V^-
)) and (V^+
')
curves. The residual volume V^y, defined as the volume of the deflation limb at P — 0 and
can derived from Eq. (3.53) as
F(nV "
VRV = lim (V(
-}
) = V0
P-+0
1 + an [ Ei-nibX) (3.57)
bn
n
,
where T(n) is the gamma-function. Figure 3.15a shows that the VRV increases with gen-
eration number, but saturates beyond n K, 8. Figure 3.15b shows that the parameter A,
which is the ratio of closing and opening threshold pressures, has a stronger influence on
VRV which goes from nearly 0 at small A to about 80% of the maximum volume at A —> 1.
In conclusion, we have derived a general theory for quasi-static fluid flow through col-
lapsible bifurcating structures. We show that while calculating the pressure-volume curve
or analogous average descriptions of fluid transport, the complex branching structure can
be partitioned into a linear superposition of one-climensional chains. Using this result we
constructed a comprehensive model of the lung P-V curve based on the topology of the
lung airway tree, the elasticity of the lung tissue and the mechanisms of airway openings.
We have shown that transient pressure waves during the process of airway openings sig-
nificantly affect the shape of the P-V curve. Although the full P-V curve is a result of
the combination of influences, we have been able to separate the effect of each of these
factors using a single measurement. The resulting method also provides an estimate of the
distribution of generation number of the terminal branches in the airway tree, or the depth
of the air sacs in the lung. Since the estimated distributions compare favorably to available

54
0.40
1.0
0.8
B 0.6
£
0.2
0.0
0.0
4 6 8
Generation number, n
0.2 0.4 0.6
A
10
1 1 1 •
;V = 2 — —
^"^^^"
* ^ i i i
_^*--*^*^*^*"^
i
0.8 1.0
Figure 3.15: Residual volume versus (a) generation number n and (b) A.

55
morphological data, our approach should be useful in clinical situations as well as in devel-
opmental studies. In general, our results, particularly those involving tree partitioning and
the general solution of the opening process, are equally applicable to other physical systems
involving transport in asymmetrically branched structures.

Chapter 4
Estimating the diameter of airways
susceptible for collapse using
crackle sound
The alveolar gas exchange regions deep in the lung are supplied with fresh air through the
branching structure of the airways [63]. In the normal lung, airways remain patent during
breathing [64]. In the diseased lung, airways can collapse due to excessive fluid accumula-
tion [65], increased smooth muscle contraction [66], decreased parenchymal tethering [67]
or some combination of these mechanisms. An accurate localization of closure along the
airway tree is important since the larger the size of the occluded airway, the more severely
it affects gas exchange. Using histological methods, the site of closure has been estimated
to be in the small airways having diameters of about 1 mm [68]. Computed Tomographic
imaging revealed, however, that the direct application of methacholine can result in the
closure of much larger airways [69]. While the first method is in vitro, the second can image
only a few airways and hence both are limited in their ability to assess the number and
location of closed airways.
When a closed airway reopens during inflation a crackle sound is generated [48, 49].

57
Crackles are "explosive" sound energy packages consisting of a sharp initial negative deflec-
tion in pressure followed by some low-frequency ringing [70] and can be measured either on
the chest wall [71] or at the trachea [49]. The presence of crackle sound during inspiration
signifies airway openings which in turn imply that airway closure occurred during the pre-
vious expiration. Recently, we presented crackle sound data and a theory that relates the
amplitude of crackle sound measured at the trachea to the attenuation of the crackle as it
passes through successive bifurcations [49]. This attenuation in turn can be expressed in
terms of the diameter of the airway from which the crackle sound was originated. In our
current study, we extended this theory to determine the number and location of collapsed
airways. We measured the pressure-volume () curves and crackle sound in isolated rabbit
lungs during inflation from several end-expiratory pressure () levels. From these data, we
estimated the distribution of airway generations and the corresponding airway diameters
vulnerable for closure as a function of EEP.
4.1 Methods
4.1.1 Experimental procedure
New Zealand White rabbits (N = 6, body weights: 2.53.0 kg) were anesthetized with pen-
tobarbital sodium (50mg/kg body weight) and exsanguinated by severing the abdominal
artery. The intact lungs were then cannulated using a 3.5-mm tube and excised. The ex-
perimental protocol was approved by the Institutional Animal Care and Use Committee of
Indiana University.
Crackle sound and P-V curves were simultaneously measured during inflation maneu-
vers. The excised lungs were suspended in a 2-litre glass bottle, in order to maintain the
humidity of the lung surface and minimize the contamination of the acoustic detection of
lung sound by the environmental noise. The bottle was open to atmosphere via a thick
walled silastic tube (length: 1 m, inner diameter: 3 mm) providing further acoustic insu-
lation of the apparatus. The tracheal cannula was led through the lid of the bottle and

58
attached to a Y-piece; one arm of which contained a commercial microphone (Monacor Inter-
national Ltd, Bremen, Germany; outer diameter: 5 mm), and the other arm was connected
to infusion pump (Harvard Apparatus, Boston, MA) with two 60-ml syringes in parallel via
a polyethylene tube (length: lm, inner diameter: 2mm) which served as a pneumotacho-
graph and a low-pass mechanical filter suppressing the unevenness of the flow delivered by
the pump. The pressure drop across the tube and the tracheal pressure (P) with respect
to atmospheric pressure were measured with Validyne MP-45 differential transducers (±2
and ±30 cmH^O, respectively). Flow (V) during inflation and P were low-pass filtered at
25 Hz and sampled at a rate of 128 Hz with a custom-made data acquisition system run by
a personal computer. The pre-amplified microphone signal was fed into another computer
and was recorded by using a GoldWave sound editor. The infusion pump drove two 60-ml
syringes in parallel, and the rate was adjusted to V = 40 ml/min. Each inflation started
with a 5 s baseline recording to establish V = 0 ml/min and either lasted for a maximum
of 3 minutes or was stopped when P reached 35 C111H2O. The first inflation started from
P = 0cmH2O, i.e. the collapsed state of the lung. Following the inflation the lungs were
kept at 30cmH2O for 1 min, and slowly deflated ( 1 min) to an EEP of 5 cmH20. After
another 1-min period of equilibration, the inflation-deflation maneuver was repeated with
successive EEP levels of 2, 1 and 0 cm H2O. Inflation volume (V) was obtained by numerical
integration of V.
The sound recordings were first high-pass filtered at 2 kHz with the GoldWave sound
editor in order to improve the temporal resolution of crackles by enhancing the sharp initial
transients of the crackles and suppressing the lower frequency ringing [70, 72]. After the
high-pass filtering, the maximum level of background noise was estimated in each recording
from the baseline and the end-inflation segments that contained no crackles, and a threshold
was determined for the minimum discernable crackle amplitude. The recordings were then
divided into intervals of length AT. For each interval i, the sound energy (AEi) was
computed as the sum of the squared amplitude. If the ratio AEj+/ AEi computed from two
successive intervals exceeded a preset ratio A, a crackle was registered. The AE values were

59
Ao
Ps
Figure 4.1: Schematic diagram showing the attenuation of a crackle at an airway bifurcation
as described in Eq. (4.1).
computed from subsequent intervals while AE increased monotonically (i.e. AE{ < AE{+i,
...). The amplitude s of the crackle was then defined as the maximum sound amplitude
within the last interval. Based on preliminary investigation whereby the identified crackles
were confirmed by listening to segments of the original unflltered recordings, the parameters
A T = 0.33 ms and A = 3 provided the most reliable crackle identification results with the
minimum number of missed or falsely identified acoustic events.
4.1.2 Estimating the generation number of collapsed airways
To analyze the crackle time series, we invoked a previously developed model [49]. We assume
that each crackle is generated with a constant amplitude so in an opening airway segment,
called the source. At each bifurcation between the source and the trachea, the crackle
amplitude is attenuated by a factor b which can be expressed in terms of the geometry of
the bifurcation (Fig. 4.1) as follows:
2A-j
A0 + Ai + A2
(4.1)

60
where AQ is the cross sectional area of the parent airway into which the attenuated crackle
propagates, A is the cross-sectional area of the daughter airway which the crackle arrives
from and A2 is the cross-sectional area of the other daughter airway. We further assume
that the airway tree is symmetric with A = A^ and that the factor b is the same for all
bifurcations. Using literature values, we obtain an average value of b — 0.52 for the rabbit
lung [2, 73]. After passing n bifurcations, the crackle reaches the trachea and its amplitude
is attenuated by bn
. We can thus calculate the generation number n from the amplitude s
at the trachea of each identified crackle as,
n = k g ' - k g ' o . ( 4 .2 )
logo
Since we assume that each crackle has the same amplitude so at the source, we use the
largest recorded crackle amplitude as a first estimate of so- The smallest discernable crackles
from all inflations should have identical amplitudes determined by the sensitivity of the
equipment as well as the detection algorithm. Hence, we adjusted the value of SQ such that
the maximum computed generation numbers from all inflations coincided.
4.1.3 Analysis of crackle statistics
Using Eq. (4.2), we calculated the histogram of generation numbers of crackle sources for
each EEP, combining the identified crackles from the inflation data of six rabbit lungs. Since
during inflation a crackle was generated only if the airway collapsed during the preceding
deflation, the number of crackles detected also signifies the number of airways that were
closed. We can thus convert each histogram to a probability curve by dividing the number
of crackles associated with each generation n by the total number of branches at that gen-
eration, 2n
. This probability curve represents the likelihood that a branch at generation
n collapses during deflation to a given EEP. Preliminary calculations, however, resulted in
probabilities larger than unity for certain n values. The likely reason for this is that the cen-
tral airways with significant cartilage content do not collapse during a passive deflation [68]
and thus do not produce crackles. In order to take this effect into account, we increased

61
all generation numbers in a step-wise manner until all closure probabilities became smaller
than unity. Finally, the fraction of collapsed airways in the lung was calculated by divid-
ing the total number of recorded crackles for a particular inflation by the total number of
airways after accounting for the shift in n due to the central airways.
4.1.4 Estimating the effect of generation dependence of threshold pres-
sures
We investigated the effect of generation dependence of opening and closing threshold pres-
sures, since both the generational trend of the threshold pressures as well as the amount
of overlap among threshold pressures at different generations are likely to have an impact
on the observed pattern of airway collapse. When the closing threshold pressures of adja-
cent generations have sufficient overlap in their distributions, a parent airway can collapse
earlier than its daughter airways. In such a case, the daughter airways are "protected"
from collapse as they are not subject to further pressure changes experienced by the open
region of the lung. The collapsed parent thus traps air in the subtree below it, reducing
the fraction of collapsed segments and subsequent crackle generation at higher generation
numbers. We also address our assumption that the amplitude of every crackle sound at the
source is identical, irrespective of the generation of the source airway and its opening pres-
sure. We study the case where the source amplitude of the crackle sound is proportional to
the opening pressure of the airway. In the case of generation dependent threshold pressure
distributions, this also makes the crackle sound amplitude generation dependent.
We construct a symmetric airway tree of N = 11 generations. Each airway at generation
n is assigned an opening threshold pressure Po drawn from a distribution pn dependent on
the generation number n. The distributions pn are uniform between an upper and a lower
limit given by the following interval:
~g2
-gn
gN
-gn
(4.3)
These distributions are constructed such that for n = 11, the upper bound is always 1 and

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