J R Soc Interface

biophysics, medical physics, biomechanics

James D. Shemilt

# Non-local impact of distal airway constrictions on patterns of inhaled
particle deposition

James D. Shemilt1 Alex Horsley2 Jim M. Wild3 Oliver E. Jensen1 Alice B.
Thompson1 Carl A. Whitfield1,2 1Department of Mathematics, and 2Division of
Immunology, Immunity to Infection and Respiratory Medicine, University of
Manchester, Manchester, UK, 3POLARIS, Imaging Sciences, Department of
Infection, Immunity and Cardiovascular Disease, University of Sheffield,
Sheffield, UK
JDS, 0000-0002-9158-0930; AH, 0000-0003-1828-0058;
JMW, 0000-0002-7246-866; OEJ, 0000-0003-0172-6578;
ABT, 0000-0001-9558-1554; CAW, 0000-0001-5913-735X
<EMAIL_ADDRESS>
###### Abstract

Airway constriction and blockage in obstructive lung diseases create barriers
to effective drug deposition by altering how different regions of the lungs
are ventilated. Established computational particle deposition models have not
accounted for these impacts of disease. We present a new particle deposition
model that calculates ventilation based on the resistance of each airway, such
that ventilation patterns respond to airway constriction. We incorporate
distal airway constrictions representative of cystic fibrosis, and assess the
resulting impact on deposition down to the single-airway scale. We demonstrate
how constriction reduces deposition in the airways directly distal and
proximal to the affected airways. When multiple constrictions are clustered
together, deposition in the central airways proximal to the constrictions is
more strongly reduced, and deposition in the other central airways is
generally increased. This results in more uneven deposition in both the
central and distal airways, even when constrictions affect only the distal
airways. We use our model to calculate lung clearance index (LCI), a clinical
measure of ventilation heterogeneity, in lungs with constrictions of various
severities localised to one lobe. We find an increase in LCI coinciding with a
significant drop in deposition throughout the affected lobe.

###### keywords:

Particle deposition, network modelling, inhaled therapies, cystic fibrosis,
airway disease

††journal: rsif

##  1 Introduction

Inhaled therapies, including mucolytics and antibiotics, are commonly used to
treat cystic fibrosis (CF) [1, 2]. Recent evidence has also shown that inhaled
gene therapies can also halt or slow decline of lung function [3]. Airway
constriction or blockage caused by thickened mucus layers or mucus plugging
can create barriers to the effective application of inhaled therapies in CF.
Typically, it is desired for inhaled drugs to be deposited evenly throughout
the lungs’ conducting airways, including in the distal conducting airways.
These small airways can be susceptible to constriction or blockage even in
early disease [4], so delivering a sufficient dose to them, particularly in
more diseased and poorly ventilated regions of the lungs, can be challenging.
Barriers to effective application of inhaled therapies are not unique to CF;
airway blockage can also alter patterns of inhaled particle deposition in
chronic obstructive pulmonary disease (COPD) or severe asthma [5, 6].
Experimental studies have shown increased particle deposition in obstructive
lung diseases, potentially caused by increased flow rates through narrowed
airways [7], but measuring how airway constriction affects the detailed
spatial patterns of deposition is difficult. Computational particle-deposition
modelling has the potential to provide detailed spatial information but, to do
so, the heterogeneous ventilation induced by airway constriction must be
accounted for.

Early whole-lung particle-deposition models, such as the trumpet models (e.g.,
[8]) and single-path models (e.g., [9]), modelled the airway tree as a one-
dimensional structure, with physical properties that vary by depth in the
lungs but without any other spatial heterogeneity. Attempts have been made to
use similar models to investigate the effects of bronchoconstriction, where a
uniform reduction in radius was applied to all airways at certain depths,
leading to increased deposition in those airways [10]. More realistic,
spatially heterogeneous patterns of airway constriction cannot be simulated
with these models.

Multiple-Path Particle Dosimetry (MPPD) models [11, 12, 13, 14] are well-
established computational tools that calculate regional particle deposition in
an asymmetric airway tree. They have been shown to predict total deposition
well in healthy adult lungs [14]. The MPPD model assumes that the flow rate
through each airway is proportional to the volume of lung subtended by that
airway. This means that constricting an airway would not generally reduce the
predicted flow rate through that airway since the volume of airways and acini
subtended by it would not necessarily change. A more sophisticated ventilation
model has previously been incorporated into MPPD [15, 16], which took into
account lung compliance and resistance as well as the volume and capacity of
the region distal to each airway when calculating the flow through it. The
resistance of each airway was not directly related to its diameter, suggesting
that constricting an airway may not have induced a significant change to the
resistance or flow rate. The more complex ventilation model did allow for more
heterogeneous ventilation of the lungs, but did not have a significant impact
on lobar deposition rates in healthy lungs, leading the authors to recommend
continued use of the simpler uniform ventilation model in subsequent versions
of MPPD [15, 16]. It has been acknowledged that using a uniform ventilation
model is not likely to accurately predict deposition in diseased lungs [16,
17].

Whole-lung models that combine three-dimensional computational fluid dynamics
(CFD) simulations in the central airways with simpler one-dimensional models
for the distal airways and acini have recently been developed [18, 19, 20].
The distal airways are treated either as several smaller trumpet [18] or
multiple-path models [20]. These approaches provide models of deposition
throughout the whole lung, including detailed patterns of deposition in the
central airways. Air flow into each of the lungs’ lobes was inferred from
experimental measurements. However, within the distal airways, the same simple
ventilation models used in traditional trumpet or MPPD models were employed,
so the effects of ventilation heterogeneity induced by distal airway
constrictions could not easily be explored. Grill et al. [21] have recently
developed a patient-specific particle-deposition model which uses a more
sophisticated ventilation model. They validated their model’s outputs against
experimental particle-deposition data from healthy subjects, showing good
agreement, and they simulated one example of deposition in diseased lungs by
increasing the stiffness of a region of lung to represent localised fibrosis.
Whilst more sophisticated particle-deposition models are being developed, the
impacts of distal airway constriction representative of obstructive lung
disease on deposition at the whole-organ scale have not yet been investigated.

Other studies have used just CFD to model particle deposition (see, e.g., the
review by Longest & Holbrook [22]). This is typically limited to modelling a
relatively small number of airways. Some attempts have been made to
incorporate bronchoconstriction in these models, but they have largely focused
on simulating particle deposition in only two or three generations of airways
[23, 24]. Walenga & Longest [25] explored the effects of airway constriction
in a geometry representative of the whole lungs, in which each lobe was
modelled by a single path of airways, with the other airways branching off
that single path being truncated. They applied a uniform constriction to all
airways in their model; thorough exploration of heterogeneous patterns of
distal airway constriction are beyond the current capabilities of CFD models.

Whole-lung models that can capture ventilation heterogeneity induced by airway
constriction have been developed to simulate gas transport without particle
deposition. Foy et al. [26] investigated the effects of airway constriction on
multiple-breath washout (MBW), a clinical measurement of ventilation
heterogeneity. They showed that MBW indices are sensitive primarily to airway
constriction severity corresponding to a reduction in radius of between $80\%$
and $90\%$. It has been shown that variability in simulated MBW indices is
also elevated when airways are constricted by similar amounts [27].

In this study, we present a whole-lung particle-deposition model, in which
ventilation of the lungs is derived based on the resistance of each conducting
airway and is driven by the expansion and contraction of individual acini.
Transport of an inhaled gas and deposition of particles from that gas
throughout the lungs are then calculated. An overview of the model setup is
provided in §2, with full details in the supplementary material. The model can
predict significant changes to patterns of ventilation and particle deposition
throughout the lungs when there is bronchoconstriction. We investigate these
effects by applying spatially heterogeneous patterns of distal airway
constrictions, representative of airway disease. We apply several
qualitatively different patterns of constriction, including constricting
airways in one or two lobes only, constricting a number of localised clusters
of distal airways, or constricting a number of airways distributed randomly
throughout the lungs. We simulate associated clinical measures of ventilation
heterogeneity (lung clearance index, LCI) and regional dosing (gamma
scintigraphy).

##  2 Methods

Figure 1: (a) Conducting airways in one lung network geometry, with the five
lobes (Right Upper, Right Middle, Right Lower, Left Upper, Left Lower)
highlighted. The network in 3D (left) is represented schematically as a planar
graph (right). Path lengths from trachea (centre) to periphery are
proportional to number of generations. (b) Acini, illustrated as spheres
connected to the terminal conducting airways, for the same geometry.

###  2.1 Lung networks

The lungs contain a large, asymmetric network of conducting airways, with
acini attached to the terminal bronchioles. We model every airway
individually, and every acinus. The conducting airways are treated as rigid
tubes, while the acini expand and contract as the lungs are ventilated. We use
lung network geometries generated from computed tomography (CT) images of
$n=4$ adolescents with CF but normal-range forced expiratory volume in one
second. The CT images were collected clinically aligned with a previous lung
imaging study [28]; local research governance was in place to use anonymised
CT scans retrospectively for imaging research purposes. The dimensions and
positions of the largest central airways are taken directly from CT, and the
remaining conducting airways are generated using a volume-filling algorithm
[29]. Each of the four geometries has approximately $60,000$ conducting
airways.

Figure 1(a) shows the conducting airways in one of the geometries. We denote
the generation of an airway as one more than the number of bifurcations
between it and the trachea, so that the trachea is denoted generation $1$.
Central conducting airways, are those with generation less than $10$, and
distal conducting airways are the remaining conducting airways (figure 1a).
The lungs are composed of five lobes which are indicated on figure 1(a). We
also represent the geometry as a planar graph in figure 1(a); we will present
data from simulations below in this way to enable visualisation of deposition
patterns across the entire network. We generate the planar graph
representation of the network as follows. In each generation, the distal ends
of the airways are placed on a circle with radius $(g-1)R_{0}$, where $g$ is
the generation and $R_{0}$ is a constant spacing. Given the angular position,
$\theta_{p}$, of the distal end of an airway in generation $g$, the angular
positions of its daughter airways’ distal ends are $\theta_{p}\pm 2^{-g}\pi$,
which ensures that all airways in the same generation have the same length and
that daughter airways remain adjacent to each other and connected to their
parent.

To enable direct comparison of our model with MPPD, and to minimise
differences purely due to lung size, we rescale the four lung geometries so
that they are representative of an adult’s lungs with functional residual
capacity (FRC) of $3.3\mathrm{L}$. Given the measured FRC, $V_{\rm FRC}$, from
each geometry, we uniformly increase the length and radius of each airway in
that geometry by a factor of $(3.3\mathrm{L}/V_{\rm FRC})^{1/3}$. This results
in a range of airway dead-space volumes between $101\mathrm{mL}$ and
$145\mathrm{mL}$, which is realistic for adult lungs, noting we do not include
the oral cavity [30]. In figure 2, we present data from simulations in all
four geometries and compare these to MPPD to validate the model. Subsequently,
we present results from simulations in one of the four geometries, chosen
arbitrarily, to focus on comparing deposition in the unconstricted geometry
with deposition after applying constrictions to the distal airways.

###  2.2 Ventilation

We assume that particle transport and deposition do not affect ventilation, so
we first simulate ventilation and then use the calculated flow rates in each
airway to simulate particle transport and deposition. The ventilation model is
based on a previous model from Whitfield et al [31]. We provide an overview
here and details in the supplementary material, §S2. We represent each lung
geometry as a network of edges and vertices embedded in 3D space, with each
edge representing an airway and vertices being placed at the ends of each
edge. The resistance to air flow of each edge is calculated via Poiseuille’s
law, providing a linear relation between the difference in pressure along each
airway and the flow rate through it. Whilst Poiseuille’s law is not likely to
accurately capture the complex dynamics in the large airways, it provides a
computationally efficient simple approximation. We have tested a nonlinear
resistance model that accounts for the formation of flow boundary layers at
bifurcations [32], but found it had minimal impact on total deposition.

When simulating ventilation, each terminal vertex of the network represents an
acinus. Figure 1(b) illustrates these acini as individual spheres. We model
each acinus as a viscoelastic bag, defining a relation between the pressure in
the acinus, its volume and the pleural pressure. We assume that the breathing
rate at the top of the trachea and the pleural pressure both vary sinusoidally
in time, and that the pleural pressure is spatially uniform. We assume a
breath time of $T_{b}=5\mathrm{s}$ and a tidal volume of
$V_{T}=625\mathrm{mL}$. All acini are assumed to have the same initial volume
and compliance. The total lung elastance, combining the contributions from all
acini, is taken to be $6.82\mathrm{\,cmH_{2}OL}^{-1}$, which is derived from
the measured value of $5\mathrm{\,cmH_{2}OL}^{-1}$ for lungs with FRC of
$4.5\mathrm{L}$ [33], rescaled proportionally to the FRC of $3.3\mathrm{L}$.
We tested the model after incorporating the effects of gravity on acinar
dynamics by imposing a pleural pressure gradient and non-linear compliance
(adapted from [34]); whilst this had some effect on the fraction of particles
deposited in different lobes, the effects were relatively minor and the impact
on total deposition was minimal. Incorporating these gravitational effects
significantly increased computational time, so they were not used.

###  2.3 Particle transport and deposition

We use the calculated air flow rates in each airway to solve an advection-
diffusion equation for the transport of inhaled particles through the lungs.
We solve for the concentration of inhaled particles throughout the lung
network. Transport of particle concentration along an airway is governed by

$\frac{\partial\bar{c}}{\partial t}=\frac{\partial}{\partial
z}\left(-\bar{u}\bar{c}+D_{\rm eff}\frac{\partial\bar{c}}{\partial
z}\right)-s,$ (1)

where $z$ is the axial coordinate, $\bar{c}(z,t)$ is the cross-sectionally
averaged concentration, $\bar{u}(t)$ is the mean velocity, $D_{\rm eff}$ is an
effective diffusivity that takes into account axial diffusion and dispersion,
and $s$ represents loss due to particle deposition. To solve (1), we first
discretise each airway into several edges, thus defining a modified network.
Then, we recast (1) using the machinery of discrete calculus, defining a
discrete analogue of the advection-diffusion equation that can be efficiently
solved on the network; this combines a finite difference approximation of (1)
within each airway with mass conservation for inhaled gas and particles at
bifurcations between airways. Full details are given in the supplementary
material, §S3.

We define $s$ in (1) by assuming that deposition occurs via three mechanisms:
inertial impaction, gravitational sedimentation and diffusion. Following the
approach of many other studies [17], we assume these mechanisms act
independently, and approximate deposition rates using semi-empirical or
derived formulae. For impaction, we use a formula from Zhang et al. [35]
derived from CFD simulations in airway bifurcation geometries. For
sedimentation, we use a formula from Pich [36], which was also used by Oakes
et al. [18]. For diffusion, we use a formula from Ingham [37], which has been
validated against experimental results and CFD for sub-micron nanoparticles
[38]. We do not model extrathoracic deposition, which is typically significant
primarily for particles at least $6\mu\mathrm{m}$ in diameter [39], focusing
instead on patterns of lung deposition. We mostly present simulations of
$4\mu\mathrm{m}$ diameter particles, as this is representative of typical
median particle sizes generated by nebulisers used to administer inhaled
therapeutics [40].

When simulating particle deposition, each acinus is modelled as a symmetric
tree of airways. Each acinar airway consists of a duct of fixed radius, and a
region of alveolar space with time-varying volume. The lengths and radii of
the ducts are based on data from [41], as is the relative volume of each
airway within an acinar tree. The total volume of each acinus at each moment
in time is taken from the ventilation simulation.

We model particle deposition during a single breath, assuming that the initial
concentration of particles is zero everywhere, and that the flux of
concentration into the trachea is fixed during inhalation. To solve the
discrete advection-diffusion equation, we approximate the time derivative
using a first-order backwards Euler finite-difference scheme, and solve the
resulting system of equations using the BiCGSTAB solver in the Eigen C++
library [42]. We use a time step of $\Delta t=0.01\mathrm{s}$, which is small
enough that total deposition is independent of the exact value. To discretise
the airways to solve (1), we ensure all edges within each airway have the same
length, every airway, including in the acini, contains at least eight edges
and edges are at most $200\mu\mathrm{m}$ long. The four discretised lung
networks have between 2.5 and 3 million edges each. This discretisation
ensures convergence of both conducting and acinar airway deposition; in the
simulation presented in figure 2(a), approximately doubling the total number
of edges changes the total deposition by only $0.2\%$. One typical simulation
takes around two hours on a single-core $3.0\mathrm{GHz}$ processor.

In figure 2, we validate the model’s output against MPPD simulations [43]. In
the MPPD simulations, we use their stochastic lung geometry with an airway
dead space of $113\mathrm{mL}$, which is within the range of dead-space values
of our four lung geometries. Breath time, tidal volume and FRC are all the
same as outlined in §2.2, although MPPD uses a constant rate of inhalation and
exhalation whilst we use a sinusoidal breathing profile.

###  2.4 Application of airway constrictions

We investigate the effects of airway disease by applying several different
patterns of constrictions to the distal conducting airways in one of the lung
geometries. We always constrict a subset of the generation 12-15 airways only,
and constrict these airways all with the same severity (a value between 0 and
1, the proportion by which we reduce the airways’ radii). The way in which we
select the airways for constriction, plus the constriction severity,
determines the pattern of constriction. These choices are made both to
illustrate key aspects of the physics of the system, and to represent typical
features of CF disease. CF affects the small airways first [4], motivating our
focus on distal airway constrictions. Ventilation MRI of patients with CF
typically shows multiple small patches of poorly ventilated lung [28, 44],
motivating us to apply several localised clusters of airway constrictions in
§3.3. We also investigate scenarios where constrictions are distributed
randomly throughout the lungs or localised to one lobe. We use the same breath
time and tidal volume before and after applying constrictions. There is
evidence that, in CF, the volume of air entering the lungs over a fixed time
of several breaths does not change significantly as disease severity increases
[45], although breathing may become faster and shallower in very severe
disease. Mild disease, which we focus on simulating here, is not likely to
significantly affect breathing rates, and keeping breathing parameters fixed
allows us to focus on the impacts of airway constriction.

###  2.5 Multiple-breath washout and gamma scintigraphy

We modify the model to also simulate multiple-breath washout (MBW). We
simulate many breaths instead of one, set the deposition term in (1) to zero,
set the diffusivity to the value for nitrogen gas in oxygen at
$37\degree\mathrm{C}$, and assume an initially uniform concentration of tracer
gas (nitrogen) throughout the lungs. We also assume that gas is well-mixed
within each acinus, as the diffusivity of nitrogen is much higher than that of
any sized aerosol we consider in the deposition model, so acinar mixing is
likely to be much stronger. This MBW model is effectively equivalent to that
of Foy et al. [26]. Lung clearance index (LCI), a clinical measure of
ventilation heterogeneity, is then calculated as follows. Suppose that after
$n_{\rm L}$ breaths, the concentration at the top of the trachea, $c_{0}(t)$,
at end-exhalation is less than $1/40$ of the initial concentration,
${c}_{0}(n_{\rm L}T_{b})<c_{0}(0)/40$, and that this is not the case for the
previous breath. Then

$\mathrm{LCI}=\frac{V_{\rm ce}}{V_{\rm tr}}\left[{c}_{0}(0)-{c}_{0}(n_{\rm
L}T_{b})\right],$ (2)

where $V_{\rm ce}$ is the cumulative expired volume of gas and $V_{\rm tr}$ is
the total volume of exhaled tracer gas [46].

We use particle-deposition predictions to simulate gamma scintigraphy, a
clinical measurement of regional dosing. To do so, we divide the 2D plane into
many pixels, sum the deposition that occurs within each pixel (across the
whole depth of the lungs), and use kernel density estimation in Matlab to
generate a deposition distribution that simulates a scintigraphy image.
Further details of this process are in the supplementary material, §S5.

##  3 Results

###  3.1 Particle deposition in the lungs without airway constriction

Figure 2: Data from simulations in the unconstricted lung geometries. In
(a)-(d), particle diameter is $4\mu\mathrm{m}$. (a) Deposited fraction within
each conducting airway in one geometry, shown as a planar graph. Deposition at
bifurcations is assigned to the parent airway. (b) Simulated scintigraphy plot
for the same example. (c) Deposited fraction in each lobe, showing mean $\pm$
one standard deviation across the four geometries. (d) Deposited fraction in
each generation, showing mean $\pm$ one standard deviation across the four
geometries and comparison with MPPD. (e) Total and conducting-airways
deposition, versus particle diameter.

We first simulate particle deposition in all four lung geometries without
applied bronchoconstriction, and compare result to MPPD for validation (figure
2). Deposition is typically much higher in larger central airways than in
smaller distal airways, but there is some heterogeneity in the spatial
distribution of deposition due to asymmetries in the lung geometry (figure
2a). There is a range of path lengths between the trachea and the terminal
conducting airways, with the shortest path terminating at generation 8, and
the longest paths extending beyond generation 20. Deposition is generally
higher in regions where path lengths are longer (figure 2a). Since each
terminal conducting airway connects to exactly one acinus, regions with a lot
of long paths have a larger number of acini, so more particles are typically
drawn through them and so more deposit. Simulated scintigraphy (figure 2b)
demonstrates how the model can reproduce features of healthy deposition
patterns: for example, deposition is relatively uniform spatially, with a few
hot spots at central-airway bifurcation points where significant inertial
impaction occurs. It shows higher deposition in the centre of the image where
the lung is thicker, so the cumulative deposition across the whole depth of
the lungs is higher. Figure 2(c) shows the distribution of deposited particles
between the lungs’ five lobes.

To validate the model in healthy lung geometries, we compare deposition in the
four lung geometries against results from MPPD (figure 2d,e). Deposition in
each airway generation shows generally good agreement for $4\mu\mathrm{m}$
diameter particles (figure 2d), despite MPPD using a lung geometry based on
different morphometric data. There is also good agreement with MPPD
predictions of total deposition over a range of particle sizes, with only
minor quantitative differences for particle diameters smaller than
$0.1\mu\mathrm{m}$ or larger than $1\mu{\mathrm{m}}$. Higher conducting-
airways deposition for large particles in our model is likely due to the
central airways being narrower on average than in the MPPD lung geometry,
causing higher impaction. We use different semi-empirical formulae than in
MPPD to determine deposition rates, but despite this, agreement is still good.
The MPPD model has been validated against experimental data [14], so good
agreement here suggests our model accurately captures deposition fractions in
healthy lungs.

###  3.2 Effects of varying airway constriction severity

Figure 3: Simulations in which all generation 12-15 airways in the left upper
lobe are constricted with varying severities (fraction by which airway radii
are reduced). In each simulation, the constriction severity is the same for
all left upper lobe airways. (a) Change in deposition in the affected lobe
(LU), and the other lobes, separated into central conducting-airways
deposition, distal conducting-airways deposition and acinar deposition. (b)
LCI, from MBW simulations in the same constricted geometries. (c) Change in
individual airway deposition in the simulation with constriction severity
0.825.

To explore how the severity of airway constriction impacts deposition, we
first constrict all generation 12-15 airways in a single lobe, with a range of
constriction severities. The effect of increasing constriction severity on
deposition in the affected airways is non-monotonic (figure 3a). Distal-
airways deposition is not significantly impacted by mild constrictions but,
when the severity exceeds 0.4, it increases as impaction is significantly
enhanced in the narrowed airways. As the severity is increased further beyond
0.675, the increased resistance of the affected airways causes flow to be
reduced into the affected lobe to such a degree that distal deposition
decreases rapidly. Increased impaction in narrowed airways has been
highlighted previously in simpler models [10], but these have not captured the
subsequent decrease as airway resistance is further increased and ventilation
patterns are altered. Central and acinar deposition in the affected lobe
decrease as the constriction severity is increased and ventilation of the lobe
is reduced (figure 3a). In the other lobes, deposition increases due to a
higher proportion of the inhaled particles entering these lobes and the speed
of air flow through them being faster. The assumption of fixed tidal volume
means that these local changes in flow rate are solely due to redistribution
of lobar ventilation. The strongest increase in deposition in the
unconstricted lobes is in the central airways since impaction is the dominant
deposition mechanism there, and this is enhanced by faster flow. By contrast,
in the distal conducting airways, where sedimentation is the dominant
mechanism, there is minimal change to deposition as shorter residence times
for particles leads to reduced sedimentation, enough to largely offset any
increased impaction.

Figure 3(b) shows how lung clearance index (LCI) responds to the same applied
constrictions. In agreement with previous modelling studies [26, 27], we find
that LCI is sensitive to a narrow range of constriction severities, which
correspond to a reduction of airway radius of around $80\%$. Comparison with
the deposition results (figure 3a) indicates that the increase in LCI
coincides with where deposition is strongly reduced in the central and acinar
airways in the constricted lobe, and increased elsewhere. When constriction
severity is $0.825$, there is significantly reduced flow through the left
upper lobe, but many of the constricted airways still receive increased
deposition (figure 3c). The model predicts that when LCI is raised, particle
deposition is likely to be more heterogeneous due to altered ventilation
patterns, but constricted airways can still receive a significant dose.
However, airways proximal or distal to those severely constricted airways are
likely to receive a reduced dose, and a large fraction of the inhaled dose may
be diverted to unconstricted regions of the lung. In keeping with previous
modelling results, figure 3(b) suggests that LCI is mainly sensitive to severe
(but not total) airway constrictions. Deposition of $4\mu\mathrm{m}$-diameter
particles is, however, sensitive to a wider range of constriction severity,
with contrasting effects from mild and severe airway constrictions.

###  3.3 Effects of varying spatial patterns of airway constriction

Figure 4: Comparison of two patterns of distal airway constrictions. In (a),
constrictions (severity 0.9) are applied in twelve clusters. Each cluster is
generated by randomly selecting a generation-12 airway, then constricting all
generation-12 airways within a radius of $R_{C}=2.4\mathrm{cm}$ of it and all
of their descendants down to generation 15. We enforce that no clusters
overlap. In total, 322 generation-12 airways, and all of their descendants
down to generation 15, were constricted. In (b), 322 generation-12 airways
were chosen at random, and they and their descendants down to generation 15
were constricted (severity 0.9). (i) Constricted airways highlighted in the 3D
networks, and (ii) in the planar graph representations. (iii) Change in airway
deposition versus deposition in the unconstricted geometry. (iv) Change in
individual airway deposition as box plots for generations 1-9.

To investigate the impact of the spatial distribution of distal airway
constrictions, we simulate two patterns of applied airway constrictions:
clustered constrictions (figure 4a) and constrictions distributed randomly
throughout the lungs (figure 4b). To generate clusters of constriction, we
randomly select $N_{C}=12$ generation-12 airways, then constrict all
generation-12 airways within a radius of $R_{C}=2.4\mathrm{cm}$ of any of
these airways, and constrict all of their descendants down to generation 15.
We apply severe constrictions (severity 0.9) to all of these airways. The same
process is also used to generate clustered constrictions in figure 5, but the
number of clusters, $N_{C}$, and the radius, $R_{C}$, are different. We always
enforce that no two clusters overlap.

Both for clustered and randomly distributed constrictions, deposition is
reduced significantly in the constricted airways, and in the airways distal to
the constrictions, since flow through them is strongly reduced (figures
4aiii,biii). Deposition is also reduced in the airways directly proximal to
the constricted airways. This effect is stronger when multiple constricted
airways are clustered together (figure 4aiii), with decreases in deposition
extending more proximally and the decreases being stronger than when
constrictions are not clustered (figure 4biii). This difference is also
evidenced by comparison of figures 4(aiv) and 4(biv): more airways in
generations 5-9 have significantly reduced deposition when constrictions are
clustered than when they are not.

Both when the constrictions are clustered and when they are not, deposition
typically increases in airways which are not directly proximal or distal to a
constriction (figures 4c,g). This is because more of the inhaled gas is
transported through these open paths once other paths are blocked, and flow
through them is faster, so inertial impaction increases. When constrictions
are clustered, increases in deposition in the central airways are generally
larger than when constrictions are not clustered. This is evident in the
comparison of figures 4(aiii) and 4(biii), and of figures 4(aiv) and 4(biv).
Figure 4(aiv) shows positive, although relatively small, median increases in
airway deposition in several generations of central airways. Figure 4(biv)
shows almost exactly zero median change in the same generations when
constrictions are not clustered. Variability in deposition in almost every
generation of central airways is larger when constrictions are clustered. This
highlights how clustered constrictions in the distal airways can cause more
heterogeneous central-airways deposition.

Figure 5: Data from simulations with six clusters of distal airway
constriction (severity 0.9) in the upper lobes. In each simulation, six
generation-12 airways were selected at random in the upper lobes, and all
airways within a radius of $R_{C}$ of any of these were constricted, along
with all of their descendants down to generation 15. Four realisations were
simulated for each of $R_{C}=2.4\mathrm{cm}$, $R_{C}=3.2\mathrm{cm}$ and
$R_{C}=4\mathrm{cm}$. (a) An example with $R_{C}=2.4\mathrm{cm}$, and (b) an
example with $R_{C}=4\mathrm{cm}$, showing constricted airways (left) and
scintigraphy (right). (c) Percentage change in deposition versus the
unconstricted case, showing change in the upper lobes (left), and in the other
lobes (right). These data are also separated into central conducting, distal
conducting and acinar deposition. Mean $\pm$ standard deviation across the
four realisations is plotted for each case. (d) Change in individual airway
deposition for an example with $R_{C}=3.2\mathrm{cm}$.

To examine the impact of increasing the size of clusters of constriction, we
now apply $N_{C}=6$ clusters to the upper lobes, and vary the cluster radius,
$R_{C}$ (figure 5). The process of generating clusters of constrictions is
otherwise as described above. Simulated scintigraphy can be seen to clearly
detect large clusters of constriction where many airways are blocked (figure
5b). Where clusters of constrictions are smaller and fewer airways are
constricted, not all constricted regions are visible on scintigraphy (figure
5a). In figure 5(a), a patch of reduced deposition is clearly visible in the
right lung, where two clusters are close together, but small clusters on the
left lung, particularly those occurring in the centre of the image where lung
depth is thickest, are not clear on scintigraphy. This highlights potential
limitations of scintigraphy: it is more sensitive to severe blockages, and it
may be more sensitive to constrictions that occur on the edges of the image
field than those that occur in the centre.

Figure 5(c) quantifies how deposition changes in different regions of the
lungs as the size of cluster of constrictions is increased. In the upper
lobes, deposition decreases in the central, distal and acinar airways as the
clusters are made larger and more airways are constricted. Elsewhere,
deposition is increased as flow becomes faster through the unconstricted
lobes. Central-airways deposition increases significantly, whilst there is
almost no change to deposition in the distal conducting airways in the
unconstricted lobes. Impaction is the dominant mechanism in the central
airways, so deposition there is much more sensitive to increased flow rates.
In fact, figure 5(c) suggests that the increase in deposition in the central
airways in these lobes is at least as large as the decrease in the lobes with
constrictions. Within the constricted lobes, there are still some open paths,
and deposition in airways on these paths still tends to increase as long as
they are not directly proximal to a constriction (figure 5d). Therefore, some
inhaled particles continue to be deposited in a lobe even if many of its
airways are constricted, although deposition then strongly favours the open
airways that are not on the same path as any constriction. The lack of
increase in distal-airways deposition in the lobes without constrictions
(figure 5c) is due to the fact that increased flow rates mean shorter
residence times for most inhaled particles so decreased sedimentation, which
is the dominant deposition mechanism in the distal airways. In turn, this
means residence times in the acini are longer in these lobes, leading to
increased acinar deposition. In these lobes, the ratio of central to distal
conducting airway deposition is increased purely by application of
constrictions elsewhere in the lungs.

##  4 Discussion

We have developed a model of inhaled particle deposition which predicts
ventilation patterns based on the resistance of every airway in the lungs. We
have tested the model in several lung geometries, validating total deposition
results against the Multiple-Path Particle Dosimetry (MPPD) model [43],
showing good agreement even despite using different lung geometries. We
investigated the impact of applying various patterns of distal airway
constrictions, representative of features of CF disease. Strong changes in
deposition can occur in the constricted airways themselves, but also, when
changes to ventilation are taken into account, there can be significant
impacts on deposition in unconstricted airways elsewhere in the lungs.
Deposition in airways directly proximal or distal to constricted airways is
reduced as flow rates through these paths are decreased. The severity of
constriction at which this decrease becomes severe approximately coincides
with a spike in the predicted LCI. We demonstrated that when distal airway
constrictions are clustered together, their effects on deposition in the
central airways can be heightened, with deposition becoming more
heterogeneous. Central airways away from the constrictions received increased
deposition, driven by enhanced inertial impaction as flow rates were increased
through these open paths, since constricted airways received less flow and the
flow rate into the trachea was assumed to be the same before and after
applying constrictions. By contrast, the unconstricted distal conducting
airways tended not to receive an increased dose since gravitational
sedimentation, the dominant deposition mechanism in the small airways, is not
increased by faster flow.

Our results highlight the importance of accounting for altered ventilation
patterns when simulating particle deposition in lungs affected by airway
disease. Experimental studies have previously established that CF disease can
significantly alter sites of particle deposition [7]. We found that severe
constrictions in the lungs’ upper lobes, which are often affected early in CF
[47], can cause increased deposition in the central airways elsewhere in the
lungs. This may partially explain the increased central-airways deposition
observed in some severe CF patients [7]. However, other effects, such as
increased turbulence, which our model cannot account for, may also contribute.
Further work directly replicating the experiments of Anderson et al. [7] would
be required to confirm this. In severe disease, breathing may become faster
and shallower [45], which may also enhance central-airways deposition.

Our model provides detailed predictions of deposition down to the single-
airway scale, beyond the scope of current medical imaging techniques. The
physical insights we have provided, relating changes in ventilation and
deposition patterns, have the potential to improve estimates of regional
dosing of an inhaled therapeutic once ventilation patterns have been inferred
in a patient’s lungs from clinical tests of ventilation heterogeneity, such as
MBW or ventilation MRI [46, 44]. Dosing plans drawn up based on lung exposure
calculated from MPPD and healthy volunteer studies may not be accurate for
those with significant ventilation heterogeneity. Our data suggest there may
be particular risks around higher concentrations of inhaled drug being
delivered to central airways and lower doses to the most strongly affected
lung regions.

The model has some limitations. Whilst the calculation of ventilation takes
into account airway constriction, compliance of airways is not accounted for,
so the impact of transient airway collapse and reopening during a breath is
not described. This symptom of airway disease could impact deposition
patterns, and so should be investigated in future studies. The model assumes
that each acinus expands and contracts independently, so does not directly
take into account how the motion of one acinus may modify its neighbours’
motion; this effect has been examined in previous ventilation models [48] and
could be incorporated into future deposition models. Particle deposition in
each airway is approximated using semi-empirical formulae, as has been done in
many established whole-lung models [17]. CFD simulations may instead provide
more accurate predictions of particle deposition in the central airways [18,
20]. However, our approach keeps computational time low, meaning we have been
able to simulate many different airway constriction scenarios, and make
insights at the airway scale throughout the lungs. We have provided detailed
physical insight into the effects of simulated patterns of airway
constriction, representative of typical features of early CF disease. In
future, patterns of airway constriction could be inferred directly from
medical imaging, in order to move towards patient-specific predictions of the
effects of disease. In order to isolate the impacts of airway constriction, we
have not varied other parameters such as breathing rate or tidal volume.
Faster, shallower breathing rates should be incorporated in future studies if
modelling patients or patient-phenotypes where this is observed [45]; this
study has instead focused on features representative of mild disease, where
breathing rates are unlikely to be significantly altered.

Using a novel computational model that accounts for altered ventilation
induced by airway constrictions, we have established new understanding of key
impacts of airway disease on patterns of inhaled particle deposition. We have
demonstrated how airway constrictions localised to the distal airways can
affect deposition throughout the lungs. Spatial clustering of constrictions
can increase the impact on central-airways deposition, making it more
heterogeneous by reducing deposition in airways directly proximal to
constrictions and increasing deposition elsewhere. These results have
implications for understanding how ventilation defects and airway blockage in
obstructive lung diseases may affect how inhaled therapeutics deposit. This is
an important step towards developing realistic simulations of particle and
drug deposition that can be used to accurately model exposure in disease
states as well as healthy lungs.

Code used to simulate ventilation and particle transport and deposition is
available at xxxxxx. Code used to generate constriction patterns, and the
network files used in all simulations presented in this study are available at
xxxxxx. Code used to generate the scintigraphy plots, to calculate LCI, and to
generate planar graphs from the 3D networks are available at xxxxxx.
Generation of the lung networks used the same code as in [31].

J.D.S. led the investigation, wrote the paper, and contributed to the design
of the study and coding of the numerical algorithms. A.H. critically revised
the manuscript and contributed to the development of the study. O.E.J.
contributed to the conception, design and coordination of the study, and
critically revised the manuscript. A.B.T. contributed to the design and
coordination of the study and critically revised the manuscript. J.M.W.
collected and collated the CT data used in the study. C.A.W. carried out the
CT image processing, and contributed to the conception, design and
coordination of the study, the coding of the numerical algorithms, and
revision of the manuscript.

We declare we have no competing interests.

J.D.S. was supported by an EPSRC doctoral training award. A.H., C.A.W., and
O.E.J. are supported by the NIHR Manchester Biomedical Research Centre
(NIHR203308). The numerical algorithms developed extend on previous work by
CAW funded by the UK Medical Research Council (MR/R024944/1)

The views expressed are those of the authors and not necessarily those of the
NIHR or the Department of Health and Social Care.

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