Aerosol Science 38 (2007) 701 – 718
www.elsevier.com/locate/jaerosci
Asymmetric human lung morphology induce particle
deposition variation
David M. Broday∗ , Yehuda Agnon
Environmental, Water and Agricultural Engineering Department, Faculty of Civil and Environmental Engineering, Technion I.I.T.,
Haifa 32000, Israel
Received 13 February 2007; received in revised form 6 May 2007; accepted 11 June 2007
Abstract
The effect of lung morphology on the heterogeneity of regional ventilation and particle deposition in the bronchial airways is
studied using Horsfield’s regular-asymmetric lung model. Flow distribution among the airways is calculated by solving the whole
tree network, assuming laminar flow hydrodynamic resistances without accounting for gravitationally enhanced preferential airflow
distribution. The variation of morphological properties, such as the lung volume and surface area distal to any airway generation,
and physiological properties, such as ventilation and particle deposition, is calculated, and fractal dimensions that characterize
these properties and processes are computed. The close agreement between the model fractal dimension characterizing ventilation
and those found from clinical data assess the validity of the model. It is shown that the fractal dimensions that characterize the
morphological properties and the physiological processes are similar, suggesting that all are related and stem from a common
underlying attribute—the lung morphology. The variation of particle deposition in the lung, as well as the variation of ventilation
and morphological attributes, increases moderately with the lung tree asymmetry. The deposition density, regarded as a key exposure
metric or therapeutic index, does not follow a spatial scale-free distribution.
䉷 2007 Elsevier Ltd. All rights reserved.
Keywords: Deposition variation; Inhalation dosimetry; Fractal analysis; Lung ventilation; Asymmetric morphology
1. Introduction
Knowledge of particle deposition patterns in the lungs is important both for risk assessment resulted from exposure
to respirable ambient particulate matter and for accurate dosimetry of therapeutics administered as aerosolized formulations. Due to ethical and clinical limitations, particle deposition, either regional or local, is oftentimes predicted
by means of a set of validated computational algorithms. Estimation of particle deposition along the respiratory tract
(RT) by means of a mechanistic modeling requires data on the anatomy of the RT and on the physiology of breathing.
Normally, in particular when computational fluid dynamics (CFD) software cannot be utilized—i.e. when dealing with
more than 3–4 consecutive airway generations, the morphology of the RT is cast into a relatively simple structure, with
the lungs represented by a set of straight tubes or a sequence of bifurcating Y-shaped units. The most common lung
model used to date for this purpose is Weibel’s model A (Weibel, 1963), which assumes a complete (hence symmetric)
dichotomously branching airway tree of straight tubes. While commonly exploited, the most prominent drawback of
∗ Corresponding author. Tel.: +972 4 829 3468; fax: +972 4 822 8898.
E-mail address: [email protected] (D.M. Broday).
0021-8502/$ - see front matter 䉷 2007 Elsevier Ltd. All rights reserved.
doi:10.1016/j.jaerosci.2007.06.001
702
D.M. Broday, Y. Agnon / Aerosol Science 38 (2007) 701 – 718
this model is its intrinsic inability to describe the variability in structure and function, which is evident in morphometric
studies and physiological observations (Bassingthwaighte & van Beek, 1988; Horsfield, 1991; Weibel, 1991a). Hence,
using correlations for particle deposition obtained using Weibel’s lung model do not show the variation in particle
deposition that is observed in vivo. This study accounts for the inherent structural asymmetry of the lung, relates it to
the ventilation variability among different regions of the lung, and show its significance for deposition of particles of
different sizes.
The structural and functional variability of the respiratory system can be modeled by sampling airway dimensions
and breathing parameters from pertinent generation-specific probability distribution functions (Koblinger & Hofmann,
1985; Wilson & Beck, 1992). Alternatively, the airways in the lung’s tree structure can be assembled by using explicit
rules that specify deterministic asymmetric morphological attributes (airway dimensions, branching angles, etc.). Such
a lung model has been promoted by Horsfield and Cumming (1968), Horsfield, Dart, Olson, Filley, and Cummingbreak
(1971) and Horsfield (1991) and will be used in this work. According to Horsfield’s orders scheme, the airways are
classified with reference to their function (i.e. the amount of terminal units supplied by each element) so that the
airway’s order indicates the distal gas exchange capacity. In contrast, in the Weibel’s generation scheme the airways
are classified according to their proximity to the trachea.
Efficient functionality among interconnected components of the whole organism during ontogeny is unlikely attained
by a repeatedly morphogenic redesigning of primal biological systems and components (unlike e.g. synapse formation
in the brain), but through scaling laws, i.e. genetic coding that permits replication of a basic form over multiple scales,
that reflect vital requirements and constraints and warrant normal functioning of the organism (Bassingthwaighte
& van Beek, 1988; Glenny, Robertson, Yamashiro, & Bassingthwaighte, 1991; Weibel, 1991b; West, Bhargava, &
Goldberger, 1986). A branching tree configuration is believed to represent the most efficient structural transition linking
the environment to individual cells. Indeed, the three-dimensional geometry of the bronchial-pulmonary airways tree
(and of the vascular tree) is a classic example of a branching pattern that repeats itself over multiple length scales for a
remarkably complex structure, with the daughter airway dimensions recursively defined by those of the parent airway.
A self-similar structure, i.e. having a hierarchical geometry for which the form is self-similar across scales, suggests
that the organ’s morphology may have fractal properties. Since fractal patterns are more error tolerant than other
structures (Shlesinger & West, 1991; West et al., 1986), they have an evolutionary advantage in providing persistent
physiological function. In biology, though, idealized (i.e. unbounded) fractal-like patterns are oftentimes modulated
by anatomic constrains (Bassingthwaighte & van Beek, 1988; Mandelbrot, 1983). Normally, these constrains affect
the larger (proximal) elements (airways, vessels), altering regular scaling at the coarse scale (i.e. which is observed
at the larger “inspection” units via a coarser “ruler”). This, however, has little effect on the overall fractal dimension
of the system, since it is the treetop that mostly matters when fractal morphology is considered (Glenny et al., 1991;
Mandelbrot, 1983). Power-law relationships, and hence fractal scaling, are discerned in the morphological-driven
Weibel lung model as well as in the functional-driven Horsfield’s bronchial airway tree (Horsfield, 1990, 1991).
A fractal branching pattern greatly amplifies the surface area and facilitates physiological processes that are related to
it, such as adsorption, gas exchange, particle deposition, etc.
Fractal morphology is encountered when successive elements of a hierarchical system show a power law distribution.
If data points obtained by means of a fractal analysis process (a log–log plot of the dependent variable vs. the independent
variable) align, the functional relationship between the dependent and the independent variables follows a power law.
The slope, S, of the regressed line is related to the fractal dimension DF by
DF = 1 − S.
(1)
DF provides a simple quantitative way to describe phenomena that are characterized by regional (spatial) heterogeneity
in structure (morphology), and are manifested in function (ventilation, gas exchange, transfer of signals, deposition of
respirable particulate matter). Nonetheless, while ideally the fractal dimension is independent of the inspection scale, it
is oftentimes found to vary with the “measuring stick” due to repeating divisions that do not obey the intrinsic scaling
rules (Glenny et al., 1991). Moreover, a system that is fractal in one feature (e.g. ventilation) need not yield power laws
for every aspect of its overall behavior (Bassingthwaighte & van Beek, 1988). Therefore, it is of great interest to see
if particle deposition in the lungs is fractally distributed, whether its nature reflects to some extent the nature of the
underlying morphology, and whether the answers to these questions depend on the particle size.
Describing the lung as a space-filling branching tree of line (one-dimensional) elements and using the airway length
as the measuring stick, Nelson and Manchester (1988) found that the fractal dimension of the lung ranges between
D.M. Broday, Y. Agnon / Aerosol Science 38 (2007) 701 – 718
703
2.64 and 2.76. Values obtained from experimental data (Nelson, West, & Goldenberg, 1990) are somewhat lower, with
DF,struct = 2.4 based on power scaling of the airways’ length and DF,struct = 2.26 when the basis was the airways’
diameter. Likewise, Weibel (1991b) reported a DF,struct = 2.35 based on scaling of the average airways’ diameter.
Alternatively, using a (two-dimensional) box counting method the fractal dimension of the lung, as calculated from a
bronchograph of an adult female, was DF,struct = 1.57 (Canals, Olivares, Labra, & Novoa, 2000). Accounting for the
different dimensionality of the methods used to calculate the fractal dimension, the main result from these studies is
the non-Euclidean nature of the lung’s morphology, in contrast with an idealistic DF,struct = 3 (e.g. West, Brown, &
Enquist, 1999). It is noteworthy that when analyzing the relative dispersion (coefficient of variation) of phenomena that
are distributed in a three-dimensional space, such as ventilation and particle deposition, the heterogeneity calculated
using box-like counting methods (as implemented in this work) has a dimension < 2 (Bassingthwaighte & van Beek,
1988; Glenny et al., 1991) that does not match the Euclidian dimension of the problem.
Measurements of both pulmonary perfusion and regional ventilation reveal substantial spatial heterogeneity that increases with the measurement resolution and is fractally distributed (Altemeier, McKinney, & Glenny, 2000; Robertson,
Altemeier, & Glenny, 2000). It is commonly suggested that the fractal dimension obtained from ventilation data
(a functional attribute) is a measure of the complexity of the lung’s structure, i.e. is induced by the regional heterogeneity of the lung structure, which itself has a fractal nature (Weibel, 1991b). Although the regional variability in
ventilation is attributed to differences in lumped airway impedance (Petak, Hantos, Adamicza, & Daroczy, 1993), only
few studies (cf. Glenny et al., 1991; Bassingthwaighte & van Beek, 1988) made an attempt to examine this conjecture
using an asymmetric morphological model of the lung. These works, however, accounted for a complete lung model,
i.e. they did not address early termination of branches as done here.
The fractal hierarchical nature of the airway tree imposes constrains on many processes that occur on top of it, such
as signal propagation (Suki, 2002), elastodynamic response (Bates, Maksym, Navajas, & Suki, 1994), and ventilation
synchrony (Altemeier et al., 2000; Robertson et al., 2000). In general, dynamic processes in the lung exhibit fluctuations
in time and spatial heterogeneity, and reveal no prominent characteristic scale, i.e. resembling a power law distribution.
Examples of such a behavior include respiratory rate (Szeto et al., 1992), lung volume (Zhang & Bruce, 2000), ventilation
and perfusion (Robertson et al., 2000; Weibel, 1991b), tidal volume, flow distribution (Altemeier et al., 2000; Kreck
et al., 2001), oxygen and carbon dioxide concentrations in the alveoli (Cernelc, Suki, Reinmann, Hall, & Frey, 2002),
and lung mechanics (Bates et al., 1994; Suki, Barabasi, & Lutchen, 1994). We propose here that the fractal nature of
processes related to normal lung functioning is important also in relation to particle deposition in the lung, and can
therefore be manifested in the spatial distribution of lesions and/or inflammations resulted from exposure to respirable
pollutants as well as in administration of aerosolized medications.
The objective of this work is to model the lungs’ fractal topology and to assess its properties (Section 2). We next
evaluate the model by comparing calculated regional ventilation with experimental data (Section 3). A study of the
resulting deposition patterns of respirable particulate matter (PM) is detailed in Section 4. Sections 5 and 6 are devoted
to discussion and conclusions. For this purpose, we implemented Horsfield’s (1990) asymmetrical lung scheme into
a robust and previously validated mechanistic inhalation dosimetry model (Broday, 2004; Broday & Georgopoulos,
2001; Broday & Robinson, 2003) and studied deposition of fine particles in the lung.
2. Lung model
Normally, the symmetric Weibel’s lung model is used for studying deposition of particles in the human lungs.
However, the symmetric lung model oversimplifies the true structure of the lung, showing significant deviations between
modeled and measured regional ventilation (Glenny et al., 1991; Horsfield et al., 1971), gas exchange and lesion
distribution. Hereafter, a modified Horsfield’s regular-asymmetry lung model (Horsfield, 1990) is used to describe the
structure of the lung. Specifically to this scheme, morphological relationships are defined among airways ordered from
the periphery towards the trachea, with the order type reflecting classification based on delivery of air to distal regions
rather than on structural hierarchy, as in standard generation-numbering schemes.
Morphological data suggest that in proximal regions, where the lung follows a complete dichotomous branching
scheme, the average diameters of successive airways correspond to each other as
di+1 = di 2−1/3 ,
(2)
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D.M. Broday, Y. Agnon / Aerosol Science 38 (2007) 701 – 718
where i is the Weibel’s generation counter (the trachea being i = 0). This expression was found valid also when
airway classification follows Horsfield’s ordering scheme (Horsfield, 1990, 1991 see Eq. (7)), and is in agreement
with theoretical considerations of minimization of the hydrodynamic dissipation in laminar flows within a multiple
dichotomously branching tree-like system. Similarly, the mean length of succeeding airways in the lungs’ tree structure
also follows the 2−1/3 rule, in agreement with the length exponent of a space-filling tree (West, Brown, & Enquist,
1997).
A symmetrical dichotomous branching structure is characterized by a branching ratio of Rb = 2. In a regularasymmetric lung model Rb depends on the inherent asymmetry of the system, and is defined as the exponent of the
(absolute value of the) slope of the regression line in the plot of the logarithm of the total number of airways of any
order vs. the order number (Horsfield, 1991)
Rb NI
> 1,
NI +1
(3)
where NI is the number of airways of order I . Horsfield’s regular-asymmetric lung model assumes no variation in
diameters and lengths within orders and the asymmetry solely results from a difference in the daughter airways’orders at
any bifurcation. Specifically, the order of the parent airway in any bifurcation is higher by one than the order of the largest
daughter airway, and a fixed order difference, max , is applied (whenever possible) between the two daughter airways
(Fig. 1). When it is impossible to apply the prescribed order difference due to proximal termination of an airway branch
(the tree periphery always ends with a symmetric bifurcation, = 0, of order 1 airways), the maximal order difference
possible is used (0 max ; Horsfield, 1990). This forces a gradual morphological variation of airways from the
conduction airways into the acini. According to this model, although airways always branch dichotomously distinct
lung pathways have different attributes (e.g. lengths, resistance to flow, etc.). It is noteworthy that an asymmetrical
tree-like geometry like the one described here results in a self-similar tree structure with a repeating branching pattern.
The relationship between the overall average branching ratio, Rb , and the anatomic variable max in such regular
asymmetrical trees is (Horsfield, 1990; Horsfield, 1976)
Rbmax (Rb − 1) = 1.
(4)
For a maximal order difference of max =3, a common bifurcation geometry in the human lungs (Horsfield & Cumming,
1968), the average branching ratio is Rb =1.38 (see Table 1) whereas for max =2 the average branching ratio is Rb =1.47,
in agreement with Nelson and Manchester’s (1988) Rb = 1.46. Note that anatomic data suggest that max varies among
I+1
I
I-
Fig. 1. Typical bifurcation in an asymmetric lung model. Airways classification follows Horsfield and Cumming’s (1968) ordering scheme.
Table 1
Relationships between the degree of asymmetry of bifurcating airways, max and different system parameters
max
Rb
Rl , R d
lI
dI
l I − , dI − 0
1
2
3
4
5
2
1.62
1.47
1.38
1.33
1.29
0.79
0.85
0.88
0.90
0.91
0.92
1
1.17
1.29
1.38
1.46
1.52
−1/3
−1/3
Rb —overall branching ratio, Rd = Rb —average diameter ratio, Rl = Rb
which goes through the larger daughter airway during inspiration.
l¯i+1 d̄i+1
, d̄
l¯i
i
0.79
0.79
0.78
0.77
0.77
0.76
0.5
0.62
0.68
0.73
0.76
0.78
—average length ratio, —the fraction of flow in the parent airway
D.M. Broday, Y. Agnon / Aerosol Science 38 (2007) 701 – 718
705
airway bifurcations in the range 0 max 5 (Horsfield, 1990, 1991), with = 0 representing a symmetrical branching
tree and larger ’s representing an increasingly asymmetric structure. A symmetric branching pattern is oftentimes
assumed in the distal respiratory airways (Asgharian, Hofmann, & Bergmann, 2001; Horsfield et al., 1971; Yeh &
Schum, 1980) with the dimensions of the terminal bronchioles, dc and lc , usually considered body-size invariants
(West et al., 1997).
From first geometric principles, the length ratio between the larger order daughter airway (I ) and the parent (I + 1)
airway (Fig. 1) scales as (Horsfield, 1990)
lI
−1/3
< 1,
≡ Rl = Rb
lI +1
(5)
in agreement with the 2−1/3 rule in the symmetric lung model. Likewise, if minimization of dissipation during transport
of oxygen (nutrients) and removal of carbon dioxide (wastes) by means of a network of branching airways (vessels) is a
hidden design criterion in biologically bifurcating systems, optimal system design requires the diameter ratio between
the larger order daughter airway (I ) and the parent airway (I + 1) to scale according to the reciprocal of the cubic root
of Rb (Horsfield, 1990),
dI
−1/3
≡ Rd = Rb
< 1.
dI +1
(6)
Values of Rl (max ) and Rd (max ) agree with those reported in the literature (Table 1). This model provides a reasonable
range of values for the average ratios of diameters and lengths of conjugate daughter airways. Namely, observations
reveal that the generation-wise average length ratio of proximally branching daughter airways (larger-to-smaller)
is 1.61 ± 0.05 (Horsfield, Relea, & Cumming, 1976; Weibel, 1963, 1991a), close to the theoretical prediction for
a large max (Table 1). Similarly, the observed mean diameter-ratio among proximally branching daughter airways
(larger-to-smaller) ranges from 1.16 (Phalen, Yeh, Schum, & Raabe, 1978; Weibel, 1963, 1991a) to 1.274 (Horsfield &
Woldenberg, 1989; Horsfield et al., 1976) and corresponds to the theoretical values appearing in Table 1 for a small
max .
Since any Weibel’s generation contains different Horsfield’s order airways (Fig. 1), the ratio of the average airway
diameters between consecutive airway generations is
dI (1 + Rd− ) Rd (Rd + 1)
d̄i+1
=
< 1.
≡ i =
−(+1)
2
d̄i
2dI Rd
(7)
Similarly,
lI (1 + Rl− ) Rl (Rl + 1)
l¯i+1
≡ i =
=
< 1.
−(+1)
2
l¯i
2lI Rl
(8)
Anatomic data suggest that i and i are system invariants and take constant values and , respectively (Weibel,
1963, 1991a; West et al., 1997), in agreement with the invariance of Rd and Rl . Moreover, Table 1 reveals that and are almost independent of the asymmetry parameter of the system, max , and in excellent agreement with the observed
values. The identical scaling of airway diameters and lengths, Eqs. (5) and (6) and Eqs. (7) and (8), gives rise to a
generation-invariant average length-to-diameter ratio. Experimental observations suggest that this ratio ranges from
3.25 (Weibel, 1963) to 2.8 (Kitaoka, Takaki, & Suki, 1999; Phalen et al., 1978) to 2.56 (Yeh & Schum, 1980). This,
however, is at odds with findings that the diameter of the daughter airways reduces on average by ∼ 0.86 between
consecutive generations whereas their length decreases by ∼ 0.62 compared to the parent airway (Weibel, 1963, 1991b).
These last scaling parameters are not captured accurately by the lung model when assigning a power of − 13 in Eqs. (5)
and (6).
As discussed above, since the seminal work of Mandelbrot (1983) the lung is described as a self-similar assembly
that reflects structural hierarchy which scales according to a power law, thus having fractal properties. The salient point
is that Horsfield’s deterministic asymmetrical lung model produces a tree with fractal properties although exponential
relationships exist among the airway orders. This happens because these relationships translate into power-law relationships when the airway generation is used as the counter. By definition, the structural fractal dimension DF,struct of
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D.M. Broday, Y. Agnon / Aerosol Science 38 (2007) 701 – 718
such an ideal tree-like system is
DF,struct = −
log Rb
= 3.
log Rl
(9)
This value agrees with the findings of Sapoval, Filoche, and Weibel (2002) on the fractal dimension of the human acini.
Yet assuming a somewhat different objective function for the lung’s airway tree optimization, namely implementing
constant dissipation for a unit surface area across airway generations (which translates into invariance of the specific
power u), Bengtsson and Eden (2003) suggested that the ratio of diameters between subsequent generations is =
−2/5
Rb . This translates into a difference of ∼ 5% in Rd . Accordingly, distinct powers are found for the recursive scaling
of airway lengths and diameters. For example, West et al. (1999) suggested that DF,struct = 3, following Eq. (9), whereas
≡ − log Rb / log Rd ranges between 2 and 3.
As per the bifurcation’s geometry, no consistent trend was found for the branching angles, although in general, they
tend to increase towards the periphery. With the increase in max the lung morphology becomes more monopodial and
observations suggest a marked difference between the branching angles of the major and the minor daughter airways
(Thurlbeck & Horsfield, 1980). The branching angles can be calculated once the partitioning of the parent flow among
the daughter airways is known (Kitaoka et al., 1999), with the latter depending only on the linear dimensions of the
interconnected system, see Eq. (14).
3. Fractal nature of ventilation
3.1. Theory
Measurements of pulmonary perfusion and of lung ventilation reveal substantial spatial heterogeneity that increases
with the spatial measuring resolution (Robertson et al., 2000). From a mechanical perspective, regional ventilation
must be determined by the morphology of the lung and by the physiology of breathing. With respect to the former,
the heterogeneity of regional ventilation is attributed to the distribution of lung impedance that results from a coherent
variation of attributes of the bronchial tree, lumping together the geometric variation in airway dimensions and the
distribution of terminal branches due to proximal termination of lung pathways. Hence, since the airway structure
that carries the air has fractal properties, ventilation should be fractally distributed as well (Glenny et al., 1991). As
for the physiology of breathing, the distribution of inter-breath intervals as a function of the subject’s age follows a
power law form (Suki, 2002) and carries information (anatomic, physiological, functional, and pathological) across
spatiotemporal scales throughout ontogenesis. These power law functions are closely related to the apparent fractal
nature of ventilation.
The fractal character of lung ventilation has been studied by a variety of techniques with only small variation in
the calculated fractal dimensions (Table 2). All the techniques suggest that the nature of the airway structure is the
major determinant of the fractal properties of the lung. Yet, the size and concentration of tracer aerosol particles used
in such studies may affect to some extent the observed fractal dimension due to inherent particle deposition properties,
irrespective of the lung structure. Normally, the fractal nature of ventilation is revealed by calculating the local coefficient
of variation of dispersive processes at successive finer scales. Similar to dispersion of a passive tracer through porous
media, fractal topology may be revealed with time evolution that depends on the Peclet number, Pe = ud/D, where
u and d are typical air velocity and airway size, respectively, and D is the tracer’s diffusion coefficient. In the limit
of Pe → ∞ (very fast convective process), the fractal dimension of the front is close to the fractal dimension of the
structure underlying the dispersive process, much like a percolation pattern formed by fluid flowing through a porous
medium (Martys, 1994). In relation to particle deposition, this condition may be attained by relatively large particles,
for which the diffusion coefficient is small. On the other hand, for small Pe (e.g. for very small particulate or gaseous
tracers) the front may show self-affine properties independent of the underlying morphology. Transport of particles
along the tracheobronchial tree (TB) is characterized by a varying Pe as a result of gradual changes in the airways
diameters (about two orders of magnitude) and function (changes of eight orders of magnitude in u). Hence, from
a purely theoretical standpoint it is unclear whether the fractal dimension that characterizes the ventilation may also
characterize deposition of particles with a wide spectrum of sizes, from diffusive 0.01 m particles up to inertial 10 m
particles, as they advance along the varying geometry of the TB tree.
D.M. Broday, Y. Agnon / Aerosol Science 38 (2007) 701 – 718
707
Table 2
Experimentally observed fractal dimensions in lungs
Specie
Posture
Fractal dimension
Morphology
Remarks
Perfusion
Ventilation
Pig
Prone
1.192 ± 0.061 (Altemeier
et al., 2000)
1.157 ± 0.041 (Altemeier
et al., 2000)
Pig
Supine
1.148 ± 0.043 (Altemeier
et al., 2000)
1.15 ± 0.03 (smallest
scale) 1.29 ± 0.04
1.12–1.16 (Conhaim
et al., 2003)
1.08 ± 0.02 (Glenny et al.,
1991; Glenny and Robertson, 1990, 1995)
1.132 ± 0.006 (Parker,
Ardell, Hamm, Barman,
& Coker, 1995) 1.12 ±
0.02 (Glenny & Robertson, 1995)
1.092 ± 0.038 (Altemeier
et al., 2000)
1.587 (Canals et al., 2000)
Rat
Dog
Supine
Dog
Prone
Rabbit
Supine
Ventilation assessed by
aerosolized 1 m fluorescent micro-spheres
Ventilation assessed by
broncho-graphy
1.3–1.46 (Gilliard,
Pappert, & Spragg, 1995)
Sheep
1.07–1.17 (Caruthers &
Harris, 1994)
Human being
2.4–2.9 (Horsfield &
Thurlbeck, 1981) 2.8
(Kitaoka & Suki, 1997)
1.57 (Canals et al., 2000)
Ventilation assessed by
broncho-graphy
Experimentally, the heterogeneity of ventilation can be characterized by the airflow relative dispersion. The underlying spatial correlation of the ventilation heterogeneity is a measure of the airflow distribution among neighboring
regions of the lung, with the latter normally characterized by the coefficient of variation, CV = / (also called the
relative dispersion—RD). Alternatively, it can be measured by means of the dispersion index, DI = 2 /. Such an
analysis involves plotting in a log–log chart the coefficient of variation of the dispersion metric against a decreasing
sample size. A negative correlation coefficient between these variables implies increasing heterogeneity with the decrease in sample volume. If the data points align along a linear regression line with a negative slope S, lung ventilation
follows a power law. Thus, it is assigned fractal attributes that characterize the measurement protocol, the underlying
morphology and physiology, or both. Mathematically, this is expressed as
CV(v)
=
CV(v0 )
v
v0
S
,
(10)
where v0 is an arbitrarily chosen sample size, usually taken to be the specimen size at the finest resolution. The fractal
dimension, DF,vent , is defined using Eq. (1). Note that the ones digit of the obtained fractal dimension represents the
analysis (a two-dimensional counting) rather than the physical characteristics of the underlying process.
A complete homogeneity of the airflow distribution in the lung is characterized by CV = 0 at all scales, leading
to DF,vent = 1. However, DF,vent = 1 can also reflect a process characterized by a non-vanishing constant (i.e. scale
independent) CV. Such a scale-independent coefficient of variation represents an ordered spatial pattern (e.g. a Sierpinski carpet) for which the flows in neighboring regions are totally correlated. For a complete random distribution
at all scales (e.g. uncorrelated distributed flow) DF,vent = 1.5. Situations characterized by DF,vent > 1.5 indicate inversely (negatively) correlated flows whereas situations that appear to have 1 < DF,vent < 1.5 indicate scale-independent
heterogeneous distributions of partially correlated flows.
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D.M. Broday, Y. Agnon / Aerosol Science 38 (2007) 701 – 718
The spatial correlation, rs , among airflows can be specified in terms of DF,vent as (Van Beek, Roger, &
Bassingthwaighte, 1989),
rs ≡
Cov(Q1 , Q2 )
= 23−2DF,vent − 1.
Var(Q)
(11)
Thus, as expected, for a uniform flow distribution (DF,vent = 1) rs = 1 whereas for a random flow distribution (DF,vent =
1.5) rs = 0. We conclude that if lung ventilation is fractally distributed than the flow to neighboring regions is correlated
and, therefore, deposition of respirable particles in the lung is also expected to be spatially correlated. It should be noted
that the correlation between regions falls off with the increase in separation between them at a rate that asymptotically
has a slope of 2(1 − DF,vent ) on a log–log plot (Glenny et al., 1991).
3.2. Airflow calculation and ventilation modeling
The airflow in the trachea is provided to the model as an input, based on the subject’s age and activity. The data are
taken in part from ICRP (1994). The mean airflow within any airway is calculated based on the total hydrodynamic
resistance to flow distal to any bifurcation. Clearly, this requires a priori knowledge of the complete lung morphology,
in order to assess the total hydrodynamic resistance distal to all the equi-hierarchical bifurcations. The hydrodynamic
resistance, R, depends on structural attributes of the inherent asymmetrical lung model. For simplicity, we assume that
all the airflows are laminar throughout the lung. Although this assumption is common for flow calculations in multigeneration hierarchical trees (Bassingthwaighte & van Beek, 1988; Kitaoka et al., 1999), it represents an idealistic
approach. This simplification can be lifted if resistance in turbulent flows is accounted for in the proximal (central)
airways, based on the local Reynolds number. Due to the excess complexity introduced into the calculation when
accounting for proximal turbulent airflows this option has been deferred at present. Also, it is well known that the
lung compliance modifies the airflow distribution and, consequently, particle deposition (Asgharian & Price, 2006;
Asgharian, Price, & Hofmann, 2006). However, as common in mechanistic and CFD studies on airflow distribution, for
simplicity we disregard the pressure drop due to the lung expansion and contraction when calculating flow partitioning.
It is noteworthy though that the asymmetric nature of the airways’ tree and the proximal termination of airway branches
is correlated with, and therefore represents, the distal volume behind each daughter airway, and reflects its ventilation
requirements. Assuming laminar flows throughout the airways, the flow is proportional to the hydrodynamic resistance
in a Poisseuille flow, Ri , which is a highly non-linear expression of the airway geometry,
Ri =
128 li
,
di4
(12)
where is the viscosity of air. Since the geometry of the lung tree is deterministically specified using Horsfield’s
order scheme, the airflows can be solved by a standard network solver that proceeds from the tree’s top (trachea) to
bottom (periphery) once all the individual airway resistances are calculated and lumped (from bottom to top). The
assumption behind this approach is that all the alveoli are exposed to a uniform pleural pressure, i.e. we disregard the
small variation in pleural pressure induced by the gravity pull on parenchymal layers.1 Hence, at each bifurcation the
airway-pleural pressure difference gives rise to distinct airflows within the two daughter branches (accounting for all
the airways distal to them). It should be noted, however, that unlike the simplified (i.e. based only on mass conservation)
allocation of airflows to each airway order (cf. Horsfield et al., 1971), the present calculation, which employs both mass
and (average) momentum balances, does not force equal airflows in airways of an identical order (but in the case of a
complete symmetrical case, = 0). Nonetheless, the variation of flows within airways of the same order is relatively
small. The variation of airflows within any (Weibel’s) generation is calculated once the airflows in the lung tree are all
known.
Following the method used when studying ventilation distribution experimentally, adjacent regions of the lung were
clustered into progressively larger sections and the CV of the airflows within the airways supplying these sectors was
calculated. Based on these calculations, a fractal dimension was determined using Eqs. (10) and (11). Similarly, fractal
dimensions that correspond to the CV of the volume and of the surface area distal of the cut (beyond the main duct
1 The parenchymal variability (Rodarte, Chaniotakis, & Wilson, 1989) and its effect on the overall impedance in response to pressure modulations
is beyond the scope of the present work.
D.M. Broday, Y. Agnon / Aerosol Science 38 (2007) 701 – 718
709
supplying each section) were also determined. The division of the lung into subsections started at the main bronchi and
stopped once the cut did not traverse a doubling number of airways relative to the previous cut, due to termination of
one of the branches. Normally (in particular for the larger max ), our analysis stopped ahead of the spatial resolution
at which homogeneous mixing within the acini might be apparent ( Butler & Tsuda, 1997; Tsuda, Otani, & Butler,
1999). Hence, we did not expect to see a leveling of the curve in the fractal analysis plots (Altemeier et al., 2000).
The method used to quantify the spatial variability of different lung attributes (anatomic and physiological) provides
fractal dimensions that reflect the physical scale that extend from the third–fourth Weibel airway generations and up to
the terminal bronchioles. Namely, our analysis is bounded both from below and from above and is therefore relevant
to certain length scales, which nonetheless cover most of the physical extent of the lungs (excluding the most distal
alveolar ducts and alveoli).
Figs. 2a–e depict the airflow variation in bisected airways for cuts through different airway generations in the
asymmetric lung models. The number n of airways bisected (x-axis) at each cut corresponds to “divisions” in a normal
box-counting method, and is related to the number of proximal airway bifurcations. Therefore, it is directly related to the
size/volume of the TB tree distal of the cut. The largest number of airways bisected at the highest (most distal) resolution
is designated by ntot . The apparent linear behavior of the coefficient of variation for smaller inspection units (i.e. larger
ntot /n) in the log–log plots suggests power-law relationships. It should be noted that when DF,vent is calculated from
experimental data, measurements that are obtained using the largest measurement scale, especially when the natural
subdivision of the underlying process is unknown or varying, should be excluded from further analyses, since the
best estimate of DF,vent is obtained using the smaller morphological segments (Glenny et al., 1991). Correspondingly,
the linear regression is performed on data obtained for the larger n’s. Fractal dimensions characterizing the simulated
variations in ventilation are summarized in Table 3. As the lung becomes more and more asymmetric (i.e. with the
increase in max ), DF,vent increases slightly (< 15%), in parallel with the fractal dimensions that represent the variation
in morphological properties, i.e. the total distal volume, DF,vol , and the total distal surface area, DF,surf . Specifically,
the total distal surface area and volume show variation within any generation similar to the variation of ventilation
but DF,vol and DF,surf are ∼ 2–3% higher (not statistically significant). This suggests that (i) the spatial variation in
ventilation results from the variation in the intra-lung morphology among neighboring regions, and that (ii) variation
in ventilation is expected even at zero-gravity conditions (cf. Glenny & Robertson, 1995). Consequently, variation in
deposition of respirable particles is also expected due to the internal variation in lung structure.
The variation in ventilation increases with the asymmetry of the lung, as evident from Figs. 2a–e. However, the
fractal dimension that characterizes this variation changes only slightly with max (< 15%, Fig. 3), suggesting a robust
functional response of the lung to structural variations. It therefore suggests that the functional outcome of mutations
that have occurred along the evolutionary pathway probably propagated more slowly than the morphological changes
(i.e. having a larger time constant). The salient point is that while small structural variations of the recursive genetic
coding are expected due to meiosis errors or environmental mutagenic and genotoxic agents, the overall function of
the interconnected system can resist such structural variations by means of curtailing their effects and keeping vital
processes in a relative homeostatic state. This conclusion is supported by the findings of Kitaoka et al. (1999) that
a small amount of random fluctuations added to the recursive structural rule in their lung model produced almost
identical statistical properties for all the trees’ features. Finally, the change in DF,vent among the different (although
still idealistic) lung morphologies suggests that it is sensitive to a certain degree to pathologic changes (Suki, 2002).
Table 3
Fractal dimensions characterizing the lung’s volume, surface area, ventilation and deposition of particles with diameter spanning four orders of
magnitude (0.01–10 m) as a function of the morphological asymmetry of the airways’ tree
max
DF,vol
DF,surf
DF,vent
DF,dep 0.01 m
DF,dep 0.1 m
DF,dep 1 m
DF,dep 10 m
1
2
3
4
5
1.14
1.19
1.23
1.27
1.30
1.12
1.18
1.21
1.26
1.29
1.11
1.15
1.20
1.23
1.26
1.12
1.14
1.16
1.20
1.22
1.10
1.13
1.15
1.19
1.21
1.10
1.11
1.13
1.15
1.20
1.12
1.13
1.14
1.16
1.22
Breathing parameters represent sitting conditions for an adult.
710
D.M. Broday, Y. Agnon / Aerosol Science 38 (2007) 701 – 718
1.5
1.5
flow
volume
surface area
Dep. 0.01 um
Dep. 0.1 um
Dep. 1.0 um
Dep. 10 um
ln (CV)
0.5
0
flow
volume
surface area
Dep. 0.01 um
Dep. 0.1 um
Dep. 1.0 um
Dep. 10 um
1
0.5
ln (CV)
1
-0.5
-1
0
-0.5
-1
-1.5
-1.5
max=1
-2
max=2
-2
0
2
4
6
8
0
10
2
4
ln (ntot/n)
10
1.5
1.5
flow
volume
surface area
Dep. 0.01 um
Dep. 0.1 um
Dep. 1 um
Dep. 10 um
0.5
0
flow
volume
surface area
Dep. 0.01 um
Dep. 0.1 um
Dep. 1.0 um
Dep. 10 um
1
0.5
ln (CV)
1
ln (CV)
8
6
ln (ntot/n)
-0.5
0
-0.5
-1
-1
-1.5
-1.5
max=3
max=4
-2
-2
0
2
4
ln (ntot/n)
6
0
8
4
6
8
ln (ntot/n)
1.5
flow
volume
surface area
Dep. 0.01 um
Dep. 0.1 um
Dep. 1 um
Dep. 10 um
1
0.5
ln (CV)
2
0
-0.5
-1
-1.5
-2
max=5
0
2
4
ln (ntot/n)
6
8
Fig. 2. The coefficient of variation of the ventilation, distal supplied volume, distal surface area and particle deposition vs. the normalized number of
inspection units (log–log scale). The slope of the regressed lines, excluding the largest inspection units (see text), is related to the fractal dimension
of the phenomena via Eq. (1). (a) max = 1, (b) max = 2, (c) max = 3, (d) max = 4, (e) max = 5.
The consistency of the fractal dimensions that characterize the variation of different lung parameters, anatomical as
well as functional, suggests that the underlying morphology is the major factor that affects the spatial distribution of
these attributes. Indeed, disease induced anatomic changes (e.g. bronchoconstriction, emphysema) are known to affect
the normal functioning of the lung as well as its fractal dimension (Gillis & Lutchen, 1999; Nagao & Murase, 2002).
It should be noted that our results conform with findings (Horsfield, 1991; Suki, 2002) that the flow within any
airway follows a power law form, Qi = kd i , with d the airway diameter, k a constant, and ≡ − log Rb / log Rd .
D.M. Broday, Y. Agnon / Aerosol Science 38 (2007) 701 – 718
711
1.35
Volume
Surface area
Ventilation
1.3
DF
1.25
1.2
1.15
1.1
0
1
2
3
4
5
6
δ max
Fig. 3. Variation of the fractal dimension of ventilation, distal volume and distal surface area with the lung structural asymmetry (max ).
Morphological data suggest that ranges between 2.3 and 3 (see the text following Eq. (9)). Furthermore, following
a regular (recursive) asymmetrical lung model similar to the one used here but without early termination of airway
branches (i.e. a complete airway generation tree) and assuming a laminar flow within the airways, Bassingthwaighte
and van Beek (1988) showed that the distribution of flows at any generation has a binomial shape,
fi (Q) = k (1 − )i−k Q0 ,
(13)
with i the generation counter, the fraction of flow during inspiration which goes down the larger daughter airway
in each bifurcation, and Q0 the airflow at the trachea. In our notation (i.e. based on the hydrodynamic resistance
calculation portrayed above) Bassingthwaighte and van Beek’s (1988) definition of is
=
Rbmax
Rbmax + 1
,
(14)
and its values are reported in Table 1. For a daughter airways diameter ratio of 1.274 Glenny et al. (1991) reported
= 0.635, in good agreement with Eq. (14) for max = 1, 2.
Following Bassingthwaighte and van Beek (1988), the fraction of the ith generation airways having the kth airflow
in the tracheobronchial tree is
i!
k!(i − k)!2i
and the mean flow is Q̄i = 2−i Q0 . The discrete probability distribution function of the flow is highly right-skewed for
= 0.5 (i.e. for max = 0), with the distribution variance and skewness increasing with . The coefficient of variation
of the flow within any generation of the recursive tree is
CV =
(2[2 + (1 − )2 ])i − 1.
(15)
Fig. 4 compares the variation of ventilation at different airway generations as calculated numerically in this work and by
Eq. (15). The shape of the curves and the fractal dimension obtained from the slopes of the regressed lines are in good
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D.M. Broday, Y. Agnon / Aerosol Science 38 (2007) 701 – 718
Theoretical CV
3
2
δ=5
δ=4
δ=3
1
δ=2
δ=1
0
0
1
2
3
Numerical CV
Fig. 4. Comparison between model results (numerical) and the theoretical (Bassingthwaighte & van Beek, 1988) coefficient of variations of ventilation
in an asymmetrical lung. The higher theoretical values result from the complete morphology embedded in the theoretical expression, in contrast with
the present lung model which accounts for early branch termination.
agreement. The mismatch between the data and the theoretical expression stems from the more idealistic morphological
nature of Eq. (15), which generates larger variation than the current model as the lung’s asymmetry increases.
4. Particle transport and deposition
Particle deposition is calculated by accounting for the three major deposition mechanisms: inertial impaction, gravitational settling, and diffusion. Detailed expressions of the deposition efficiencies pertinent to each mechanism and
how to account for their simultaneous effect in a ‘vectorial’ sense were described elsewhere (Broday, 2004; Broday &
Georgopoulos, 2001; Broday & Robinson, 2003). In short, in each airway we calculate the deposition efficiency for
particles of different sizes and then add these “binomial” deposition trials to calculate the actual fraction of particles
of a given size that deposit in any airway. Clearly, the calculation of the deposition fraction in distal airways accounts
for deposition in proximal airways. The calculation stops either when all the inhaled particles deposit or when the
breathing cycle completes. While the depth of penetration equals the tidal volume, mixing within the airways due
to interaction between freshly breathed air and resident air is expressed by a dispersion coefficient that replaces the
Brownian diffusion.
Figs. 2a–e depict the variation of particle deposition distal of the cut, i.e. in different equi-hierarchical lung regions,
in different lung morphologies (max = 1, 2, . . . , 5). Deposition variation is shown for particles with sizes that span
over four orders of magnitude, covering the spectrum of respirable particles of interest in environmental and therapeutic
applications.Apparently, the relative dispersion of the deposition depends on the particle size and is somewhat smaller for
0.1 m particles than for 10 m particles (note that 0.1–1 m particles are more persistent since deposition mechanisms
are less effective in removing them, resulting in a prominent minima of the deposition efficiency). Fig. 5 depicts
the fractal dimension that is calculated from the relative dispersion (CV) of the particle deposition distribution using
Eq. (1). In general, DF,dep varies slightly with the particle size (Table 3). Our results suggest that DF,dep takes a minimum
value for particles with diameter of about 0.1–1 m, which are the particles known to be more airborne persistent.
Fig. 5 also reveals that DF,dep varies with the lung morphology and, in agreement with our previous findings, increases
with the increase in the lung asymmetry. This result is in accord with the reported increase of the fractal dimension
that characterizes the perfusion heterogeneity with the lung asymmetry (Glenny & Robertson, 1995). The apparent
linear dependence of DF,dep on the lung asymmetry max is summarized in Table 4. The differences between DF,dep
and DF,vent are small (< 6%) and tend to increase as the lung tree becomes more asymmetric, most likely reflecting
minute differences in the slopes of the regressed lines.
Particle deposition that follows a complete spatial randomness (CSR) can only originate from a homogeneous planar
Poisson process (Conhaim, Watson, Heisey, Leverson, & Harms, 2003). For CSR conditions, knowing the location
of any deposited particle does not infer on the prospect of finding other deposited particles nearby. A CSR pattern
implies that the mean number of particles per subdivision equals the variance of the particle count regardless of the
D.M. Broday, Y. Agnon / Aerosol Science 38 (2007) 701 – 718
713
1.25
DF,dep
1.2
1.15
0.01 um
0.1 um
1.1
1 um
10 um
1.05
0
1
2
3
4
5
6
max
Fig. 5. Variation of the fractal dimension of particle deposition with the lung structural asymmetry (max ).
Table 4
Variation of the fractal dimension that characterizes particle deposition in asymmetric lungs for particles of different sizes, DF,dep = a · max + b
dp (m)
a
b
R2
0.01
0.1
1
10
DF,vol
DF,surf
DF,vent
0.0242
0.0266
0.0256
0.0219
0.0413
0.0409
0.0375
1.0938
1.0741
1.0621
1.0893
1.1025
1.0864
1.0756
0.9853
0.9885
0.9435
0.8299
0.9943
0.9914
0.9906
Also included the variation of the fractal dimension that characterizes other properties of the lung, DF = a · max + b.
subdivision size. Hence, CSR conditions are characterized by a dispersion index (DI = 2 /) that equals a unity and
by DF,dep = 1.5. When DI < 1 deposited particles are more regularly spaced than expected under CSR, and knowing
the location of one particle increases the probability to find an adjacent particle at some distance further away from it.
Hence, a fractal dimension that ranges between 1 (a homogeneous distribution) and 1.5 suggests a non-random pattern
that is more uniform than a CSR. Table 3 reveals that all the fractal dimensions calculated in this study range between
1.1 and 1.3, suggesting that the variability of any of these attributes is smaller than the variability that characterizes
random patterns.
5. Discussion
This study examines the variation in functional properties of the lung due to structural variations resulting from
deterministic asymmetrical tracheobronchial morphology. A major consequence of the morphological model of the
lung is the non-uniform distribution of path lengths leading from the trachea to the terminal branches. Hence, unlike
studies in which assumptions regarding the flow division among daughter airways force geometrical relationships
between their dimensions, flow division in the current work is based on pressure difference disparity over the two subtree segments distal of any bifurcation. This approach corresponds to the true “selection criteria” for flow partitioning
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D.M. Broday, Y. Agnon / Aerosol Science 38 (2007) 701 – 718
in an existent network, in contrast with the heuristic criteria that allocate flows to daughter airways in random (cf.
Glenny & Robertson, 1995; Kitaoka & Suki, 1997; Parker, Cave, Ardell, Hamm, & Williams, 1997), without a reference
to the lumped morphological variations of the tree structure distal of the bifurcation. As such, the variation in ventilation
reported here cannot be separated into independent components that reflect (i) structural induced variation and (ii)
inequalities in airflows among daughter airways. The contributions resulting from early termination of airway branches,
the difference in number of equal-type airways in each sub-tree segment, and the geometry of the sub-tree airways,
simultaneously induced by the asymmetry of the tree structure, are all lumped together and manifest the computationally
observed heterogeneous ventilation. This explains why identical partitioning of airflows occurs only in fully symmetric
airway trees, inducing homogeneous spatial flow distribution (DF,vent = 1).
The somewhat different DF ’s obtained from fractal analysis of the variation of distinct structural and functional
properties can reflect high sensitivity of the calculation (Kitaoka & Suki, 1997), since slight changes in the spatial
variation result in a difference in the regressed slopes and, therefore, affect the calculation of the fractal dimension. Our
results suggest that the differences between DF ’s that characterize different properties and processes are relatively small
(< 6%), in agreement with the conjecture of Glenny and Robertson (1995) that the fractal dimension is insensitive to
small changes in the local heterogeneity since it represents an overall scaling. Moreover, model results agree with the
common notion that lung morphologies that are more symmetric are characterized by a functional fractal dimension
closer to 1, representing more uniform ventilation. In any case, DF found from the model simulations does not match
the theoretical structural fractal dimension DF,struct =3 that represents the recursive hierarchical geometry of the airway
tree model (Eq. (9)). Nonetheless, structural properties of the system as a whole (e.g. distal volume and surface area) do
show scale-free behavior that is characterized by fractal dimensions close to those calculated for functional properties.
In fact, Mandelbrot (1983) estimated that the lung’s surface area has a fractal dimension of 2.17, in reasonable agreement
with our results (max = 2, Table 3) after correcting the units digit to reflect the Euclidean dimension. On the other
hand, the fractal dimension calculated from the spatial variation of Horsfield’s (1990) empirical flow distribution is
DF,vent = 1.59, which is rather larger than obtained from our simulations.
We found that the spatial variations of ventilation and particle deposition are related to structural properties of the
tracheobronchial tree, and are therefore evident in microgravity (isogravitational) conditions. Clearly, we expect gravity
to be a relatively minor player in directing airflows and inducing ventilation variability. Nonetheless, inertial effects
occurring at certain bifurcations of the upper lobes (due to sharp turns) are expected to influence the distribution of
airflows (and particle deposition, due to the 103 difference in the particle-air density). Gravitational effects are expected
to add some variability to particle deposition, with its effect increasing with the particle size.
Model results reveal that DF,vent and DF,dep range between 1.1 and 1.3, depending on the asymmetry of the lung
morphology. These results are in a general agreement with the values obtained experimentally for a range of laboratory
animals. Exact comparison, however, is impossible due to lack of human data and the variation encountered among
small and large mammals (Table 2).
Dose–response relationships depend on the bioavailability, activation, and clearance rate of the deposited matter.
All these processes are related to the surface density of the deposited matter, which has been associated with health
outcomes of inhalation of respirable particulate matter. Figs. 6a–e depict the variation of the deposition density (the
total deposition divided by the surface area beyond any airway segment). The key feature revealed is that a linear
relationship, which suggests a power law behavior, is apparent only for the smaller particles (0.01 and mainly 0.1 m).
The larger particles, in particular particles with diameter of 1 m which are known to be more airborne persistent due
to a smaller total deposition efficiency, show deposition density variation that is not scale free and hence cannot be
characterized by a fractal dimension. Hence, in accordance with clinical data, the deposition density of therapeutic
aerosols (∼ O(1 m)) is not expected to be scale independent but to show prominent deposition “hot spots”. Larger
particles tend to deposit by inertial mechanisms with intensity that is directly related to the particle’s Stokes number,
St = dp2 p u/18l, where dp and p are the particle diameter and density, respectively, u and are the air velocity and
dynamic viscosity, respectively, and l is the airway’s length. In contrast, ultrafine particles deposit mainly by diffusion
and the key parameter is · Scp = l/d 2 u, where is the kinematic air viscosity and d is the airway diameter (Broday,
2004; Broday & Georgopoulos, 2001). The different dependence of these governing parameters on the geometry of the
airways (and therefore on the recursive scaling) may explain why the deposition density does not show a power law
distribution and a scale-free (fractal) characteristics.
Normally, the fractal dimension is used for characterization and identification of possible related mechanisms and
not in forward modeling. However, the knowledge that fractal dimensions of different properties and phenomena are
D.M. Broday, Y. Agnon / Aerosol Science 38 (2007) 701 – 718
0
-2
-2
-3
-4
-3
-4
-5
-5
-6
-6
-7
-7
2
0
0
4
6
ln(ntot /n)
8
0
10
0
max=3
-1
0.01 um
0.1 um
1 um
10 um
-2
-4
-5
-6
-7
6
8
max=4
10
10
0.01 um
0.1 um
1 um
10 um
-4
-6
4
8
-3
-5
2
6
ln(ntot /n)
-2
-3
0
4
2
-1
ln(CV)
ln(CV)
0.01 um
0.1 um
1 um
10 um
max=2
-1
ln(CV)
ln(CV)
0
0.01 um
0.1 um
1 um
10 um
max=1
-1
715
-7
0
2
4
ln(ntot /n)
6
8
10
ln(ntot /n)
0
max=5
-1
0.01 um
0.1 um
1 um
10 um
ln(CV)
-2
-3
-4
-5
-6
-7
0
2
4
6
8
10
ln(ntot /n)
Fig. 6. Relative dispersion of the surface density of deposited particles. Linear relationship, which suggest power law behavior, is apparent only for
the smaller (0.01 and 0.1 m) particles. (a) max = 1, (b) max = 2, (c) max = 3, (d) max = 4, (e) max = 5.
similar suggests that these processes may be related. This, in turn, can serve for predicting one property given the other.
For example, since the variation in particle deposition has been shown to be related to the variation in ventilation, in
vivo measurements of lung ventilation are expected to provide information about the expected deposition variation of
size-specific inhaled particles. In fact, even the knowledge of the lung morphology, for example from detailed anatomic
studies, can provide a means to estimate the particle deposition profile and its variation. Hence, although the fractal
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D.M. Broday, Y. Agnon / Aerosol Science 38 (2007) 701 – 718
dimension by itself is a descriptive measure obtained from a diagnostic process, it has been shown to be related to other
processes and may possibly therefore be used within a prognostic tool.
6. Conclusions
The asymmetric branching pattern of the lungs offers a possible mechanism for explaining the observed heterogeneity
of pulmonary airflow in isogravitational planes. The model relates the structure and function of the tracheobronchial
tree and offers an explanation for the spatial distribution of ventilation. The variation of particle deposition within the
lung follows a power law behavior from which an almost identical fractal dimension can be calculated. Therefore,
fractal dimensions that characterize distinct structural and functional properties of the lung seem to be very close to
each other, within a calculation error from the mean value. Indeed, DF,vol , DF,surf and DF,vent differ within less than 4%
from each other for a given max . Moreover, even among different max the various fractal dimensions do not change
more than 6%, with the larger values corresponding to the more asymmetric lung morphology. The variation of the
surface density of the deposited matter does not follow a power law distribution, possibly due to simultaneous variation
of multiple properties that combine to a non-fractal-like overall variation of the deposition density.
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