# Impact of Airways Geometry on Transport of Gases to Blood

Ali Saab  Department of Mechanical Engineering, American University of Beirut,
Beirut, Lebanon  Leila Issa  Department of Computer Science & Mathematics,
Lebanese American University, Beirut, Lebanon  Salah Zeineddine  Department of
Internal Medicine, American University of Beirut, Beirut, Lebanon  Daniel M.
Tartakovsky  Department of Energy Resources Engineering, Stanford University,
367 Panama Street, Stanford, CA 94305, USA  Issam Lakkis  Department of
Mechanical Engineering, American University of Beirut, Beirut, Lebanon

###### Abstract

Topological structure of bronchial trees affects transport of gases and
aerosols in the respiratory system. We start by providing a quantitative
assessment of the alternative tree representations’ ability to predict
observable geometric and mechanistic characteristics, such as network
resistance, dead space volume, and path length. Then we present a model of
dynamic transport of oxygen and carbon dioxide along the airways, in the
alveoli, across the alveolar membrane, and along the pulmonary blood
capillaries. The model also accounts for the exchange of these two gases with
blood in the capillaries, as well as for age, gender and other in-species
characteristics. Our model’s predictions are compared with corresponding
observations, providing an additional venue to assess the validity of the
existing representations of the lung’s bronchial tree.

##  1 Introduction

Geometric structure of airways controls, to a large extent, transport of gases
and aerosols in the respiratory system. These airways form a complex branching
network, whose mathematical descriptions (e.g., connectivity, an airway’s
length-to-diameter ratio, etc.) underpin most quantitative studies in
dosimetry [16, 43] and in respiratory and obstructive diseases [7, 65].
Mathematical models of the bronchial tree in the human lung typically
postulate a scaling law that relates the (generation-averaged) characteristics
(diameter, length and branching angle) of each bronchial generation to those
of the previous and successive generations.

There is a general consensus that the human bronchial tree forms a bifurcating
network [65, 44]. The nature of bifurcations appears to be settled as well:
while earlier and/or highly idealized models assume bifurcation symmetry [43,
78, 35], more recent investigations allow bifurcations in bronchial trees to
be asymmetric [65, 44, 53]. Finally, guided by morphometric data [57, 52, 80],
most (if not all) network models assume certain geometric properties scale
across generations of the bronchial tree. Examples of such scaling laws are
Weibel’s [78] and Horsfield’s [27] models of bifurcation. The former posits
that, in symmetrically bifurcating trees, a branch size $S$ (e.g., its length
and diameter) scales with the branching order $n$ as $S(n)=S(0)\alpha^{n}$; in
other words, the knowledge of the trachea size, $S(n=0)$, and the scaling
factor $\alpha<1$ is sufficient to estimate the branch size in any generation
$n$ of the bronchial tree. The latter uses the same form of scaling law,
$S(m)=S(0)\beta^{m}$, but groups the airways in order to account for the
network asymmetry and starts the counting from the periphery upward, such that
$S(m=0)$ is the smallest size and $\beta>1$.

The parameters in these, and more evolved [84, 46], models are obtained by
optimizing the bronchial tree’s function. There is considerable debate about
which specific cost function of the lung is to be optimized. It has been
argued that the bronchial trees have evolved in a way that their bronchus’
length-to-diameter ratio minimizes the total pressure loss across the bronchi
[35], that their asymmetrically bifurcating structure minimizes either
dissipation of frictional energy [73] or a linear combination of the tree’s
resistance and volume [40], maximizes the volumetric flow rate [49], etc.
Natural in- and intra-species variability add a certain degree of randomness
to the bronchial tree’s structure [40], making it all but impossible to
determine which cost function is “correct”.

To assess the veracity of alternative tree geometries, we compare predictions
of mathematical models of air flow in the bronchial networks with relevant
observations. We start by providing a quantitative assessment of the existing
network and gas flow models’ ability to predict observable geometric and
mechanistic characteristics, such as network resistance, dead space volume,
and path length. The choice of these metrics is justified by the availability
of their (relatively) high-fidelity experimental measurements. Then we present
a model of dynamic transport of oxygen and carbon dioxide along the airways,
in the alveoli, across the alveolar membrane, and along the pulmonary blood
capillaries. The model also accounts for the exchange of these two gases with
blood in the capillaries, as well as age, gender and other in-species
characteristics. Our model’s predictions are compared with corresponding
observations, providing an additional means to assess the veracity of the
existing bronchial tree geometries. We are not aware of any other comparison
of the veracity of alternative bronchial tree models in terms of their ability
to predict blood gas transport observables.

The novelty of our work is twofold: we (i) develop a novel model of gas
transport in the lungs and (ii) use to build an an assessment tool, which
builds on that model to evaluate alternative airways geometries found in the
literature. To highlight the innovation of our model of gas flow in the lung,
we provide a brief overview of existing modeling strategies.

Models of gas transport in the lung can be subdivides into two categories:
lumped models and spatially distributed models, which are typically formulated
in terms of ordinary differential equations and partial differential
equations, respectively. Lumped models are derived from compartmental analysis
and vary in complexity. In its original formulation [60], a lumped model
divides the lung into three functional compartments: ideal, unventilated, and
unperfused. This model serves as the starting point for the development of
progressively more sophisticated models, such as VentPlan [62], VenSim [61],
and the Nottingham Physiology Simulator [23]. A hierarchy of lumped models of
the human lungs, which describe the mechanical behavior of the lung
(considered as a single compartment container) and gas exchange between the
airways (another compartment) and the blood (the third compartment) is
proposed in [3]. These and other lumped models are computationally efficient,
but have several limitations that preclude their use for the assessment of the
alternative airway geometries. First, they do not predict a spatial
distribution of gases in the airways and, hence, cannot be used to investigate
the impact of airways geometry (e.g., dimensions of the branches along the
tree) on gas exchange. Second, they rely on fitting parameters that calibrated
to match some selected quantities with measurements from patients; this
precludes their use for assessment of the degree to which a given airways
geometry is functionally similar to that of humans.

Distributed models, including the one presented in this study, resolve the
spatial variability in gas concentration along individual vessels forming the
airways. Most, if not all, of such models use advection-reaction equations to
model O2 transport during the respiratory process [12, 50, 69], without
accounting for the presence of CO2. The most recent study of this kind [38]
describes transfer of oxygen into the blood, enabling one to investigate the
impact of longitudinal variability in O2 concentration on the mean transfer
rate. Our model extends the latter study by incorporating both transport of
CO2 and biochemical reactions between O2 and CO2 in the blood in the
capillaries.

##  2 Materials and Methods

###  2.1 Review of bronchial tree geometries

Available data on the (generation-averaged) length $L(n)$ and diameter $D(n)$
of individual bronchus in the $n$th generation ($n=1,\cdots,24$) of the human
bronchial tree are collated in Tables 1 and 2, and presented graphically in
Figure 1. These data are separated by age, such that Table 1 and the top row
in Figure 1 present data for human adults, while Table 2 and the bottom row in
Figure 1 are for human infants. These data served to inform multiple models of
the human bronchial tree, six of which are described below.

Table 1: Length ($L$) and diameter ($D$), both in mm, of the $n$th bronchi generation, predicted with tree models A–G for human adults. Model A [46] infers $L\sim n^{-1.38}$ and $D\sim n^{-1.26}$ from data; Model B [79] $L\sim\exp(-0.92n)$ and $D\sim\exp(-0.39n)$ for $n\leq 3$, and $L\sim\exp(-0.17n)$ and $D\sim\exp(-0.29n+0.01n^{2})$ for $n>3$; Model D [27] $D\sim a^{25-n}$ with $a>1$ varying between three groups of bronchi characterized by three intervals of $n$; and Model E [51] identifying a linear relation, for each generation, between bronchi’s length & diameter and human height $H$, $L\sim H$ and $D\sim H$. The remaining tree models, C [29], F [85] and G [49], fit no curves to the data reported therein.  |  A [46] |  B [79] |  C [29] |  D [27] |  E [51] |  F [85] |  G [49]
---|---|---|---|---|---|---|---
$n$ | $L$ | $D$ | $L$ | $D$ | $L$ | $D$ | $L$ | $D$ | $L$ | $D$ | $L$ | $D$ | $L$ | $D$
1 | 120 | 18 | 120 | 18 | 100 | 16 | 100 | 16 | 89 | 18.2 | 120 | 16.8 | 120 | 20
2 | 46.1 | 7.5 | 47.8 | 12.21 | 50 | 12 | 40 | 12 | 38.3 | 13.0 | 47.8 | 11.4 | 42.2 | 18
3 | 26.4 | 4.5 | 19.1 | 8.3 | 22 | 11 | 26 | 10.3 | 14.7 | 9.0 | 19.1 | 7.1 | 30.3 | 13
4 | 17.7 | 3.1 | 7.6 | 5.6 | 11 | 8 | 18 | 8.9 | 10.5 | 6.7 | 7.6 | 4.4 | 23.4 | 9.4
5 | 13.0 | 2.4 | 12.7 | 4.5 | 10.5 | 7.3 | 14 | 7.7 | 8.1 | 4.0 | 12.7 | 3.4 | 18.4 | 7.2
6 | 10.2 | 1.9 | 10.7 | 3.5 | 11.3 | 5.9 | 11 | 6.6 | 6.91 | 3.16 | 10.7 | 2.7 | 14.6 | 5.7
7 | 8.2 | 1.6 | 9.0 | 2.8 | 11.3 | 5.9 | 10 | 5.7 | 5.0 | 2.6 | 9.0 | 2.1 | 10.7 | 4.5
8 | 6.8 | 1.3 | 7.6 | 2.3 | 9.7 | 5.4 | 10 | 4.9 | 3.6 | 1.7 | 7.6 | 1.7 | 9.8 | 3.6
9 | 5.8 | 1.1 | 6.4 | 1.9 | 10.8 | 4.3 | 10 | 4.2 | 2.9 | 1.2 | 6.4 | 1.4 | 7.8 | 3
10 | 5 | 1.0 | 5.4 | 1.5 | 9.5 | 3.5 | 10 | 3.5 | 2.9 | 1.0 | 5.4 | 1.1 | 6.5 | 2.4
11 | 4.4 | 0.9 | 4.6 | 1.3 | 8.6 | 3.5 | 9.6 | 3.3 | 2.5 | 0.8 | 4.6 | 1.0 | 5.3 | 2
12 | 3.9 | 0.8 | 3.9 | 1.1 | 9.9 | 3.1 | 9.1 | 3.1 | 2.3 | 0.6 | 3.9 | 0.8 | 4.3 | 1.6
13 | 3.5 | 0.7 | 3.3 | 1.0 | 8 | 2.9 | 8.6 | 2.9 | 2.1 | 0.6 | 3.3 | 0.7 | 3.6 | 1.3
14 | 3.1 | 0.7 | 2.7 | 0.8 | 9.2 | 2.8 | 8.2 | 2.8 | 1.9 | 0.6 | 2.7 | 0.6 | 2.8 | 1.1
15 | 2.9 | 0.6 | 2.3 | 0.7 | 8.2 | 2.7 | 7.8 | 2.6 | 1.7 | 0.5 | 2.3 | 0.5 | 2.5 | 0.9
16 | 2.6 | 0.6 | 2.0 | 0.7 | 8.1 | 2.5 | 7.4 | 2.4 | 1.6 | 0.5 | 2.0 | 0.5 | 2.0 | 0.7
17 | 2.4 | 0.5 | 1.7 | 0.6 | 7.7 | 2.4 | 7 | 2.3 |  |  | 1.7 | 0.4 | 1.6 | 0.6
18 | 2.2 | 0.5 | 1.4 | 0.5 | 6.4 | 2.2 | 6.7 | 2.2 |  |  | 1.4 | 0.4 | 0.8 | 0.5
19 | 2.1 | 0.4 | 1.2 | 0.5 | 6.3 | 2 | 6.3 | 2 |  |  | 1.2 | 0.4 | 0.8 | 0.7
20 | 1.9 | 0.4 | 1.0 | 0.5 | 5.2 | 1.8 | 5.7 | 1.8 |  |  | 1.0 | 0.3 | 1 | 0.8
21 | 1.8 | 0.4 | 0.8 | 0.5 | 4.8 | 1.6 | 5 | 1.5 |  |  | 0.8 | 0.3 | 1 | 0.4
22 | 1.7 | 0.4 | 0.7 | 0.4 | 4.2 | 1.4 | 4.4 | 1.3 |  |  | 0.7 | 0.3 | 1 | 0.4
23 | 1.6 | 0.4 | 0.6 | 0.4 | 3.6 | 1.1 | 3.9 | 1.1 |  |  | 0.6 | 0.3 | 0.8 | 0.4
24 | 1.5 | 0.3 | 0.5 | 0.4 | 3.1 | 1.0 | 3.5 | 0.9 |  |  | 0.5 | 0.3 | 0.6 | 0.4
25 |  |  |  |  |  |  | 3.1 | 0.8 |  |  |  |  |  |

Geometry A. The fractal lung model [84, 46] assigns a probabilistic density
function to scaling factors for the dimensions of parent and daughter bronchi
and employs the renormalization group theory to account for the heterogeneity
in the asymmetric airway branching. The model predicts the average dimensions
(diameter, length, and volume) of the airways to obey power laws in the
generation number $n$ (see the caption of Table 1), with the model parameters
inferred by fitting the model to the data from [78, 57].

Geometry B. The symmetrically bifurcating model of Weibel [78, 80] predicts
the diameter and length of a bronchus to follow an exponential law in the
generation number $n$ (see the caption of Table 1). The model was inferred
from, and parameterized by, a complete set of measurements of the first six
generations ($1\leq n\leq 7$) of airways’ diameters and lengths, some
measurements from generations $8\leq n\leq 11$, and morphometric estimates of
the lengths and diameters of respiratory bronchioles and alveolar airways
($n\geq 12$).

Geometry C. This model also assumes a fractal structure of the branching, with
a self-similarity distribution that is preserved up to the 24th generation.
The dimensions of the airways are inferred from prior observations [20, 24,
39, 82, 35, 78].

Geometry D. The asymmetric bifurcation model [27, 28] orders airways by their
relation to the periphery, as opposed to the trachea, groups them by similar
size, and subdivides them into zones according to their degree of asymmetry.
This ensures that the airways of the same order are more uniform in size than
the airways of the same generation in model B [78, 80]. Measurements of the
complete human bronchial tree [27, 28] are used to linearly correlate bronchi
length $L$ with their diameter $D$.

Table 2: Length ($L$) and diameter ($D$), both in mm, of the $n$th bronchi generation, predicted with the tree geometries A [46], E [51], and H [68] for human infants. | A [46] | E [51] | H [68]
---|---|---|---
$n$ | $L$ | $D$ | $L$ | $D$ | $L$ | $D$
1 | 26.5 | 5.69 | 26.5 | 5.69 | 26.5 | 3.28
2 | 11.06 | 2.19 | 15.79 | 4.34 | 15.79 | 2.8
3 | 6.64 | 1.25 | 5.96 | 3.28 | 5.96 | 1.91
4 | 4.62 | 0.84 | 3.89 | 2.59 | 3.89 | 1.51
5 | 3.49 | 0.62 | 3.71 | 1.85 | 3.71 | 1.47
6 | 2.77 | 0.48 | 2.78 | 1.41 | 2.78 | 1.21
7 | 2.28 | 0.39 | 2.22 | 1.14 | 2.22 | 1
8 | 1.93 | 0.32 | 1.98 | 0.85 | 1.98 | 0.89
9 | 1.66 | 0.27 | 1.92 | 0.7 | 1.92 | 0.87
10 | 1.46 | 0.24 | 1.75 | 0.64 | 1.75 | 0.78
11 | 1.29 | 0.21 | 1.64 | 0.58 | 1.64 | 0.72
12 | 1.16 | 0.18 | 1.57 | 0.51 | 1.57 | 0.68
13 | 1.05 | 0.17 | 1.5 | 0.5 | 1.5 | 0.64
14 | 0.95 | 0.15 | 1.43 | 0.47 | 1.43 | 0.6
15 | 0.87 | 0.14 | 1.36 | 0.45 | 1.36 | 0.55
16 |  |  | 1.3 | 0.43 | 1.3 | 0.51

Geometry E. The model [51] accounts for the bronchi orientation to gravity and
angles at branching points, and establishes linear relationships between a
bronchus’ length/diameter and the body height. This information is absent in
both symmetric [78, 80] and asymmetric [27, 28] models.The model is derived
from a set of observations (casts) collected in humans, dogs, rats and
hamsters. This data set seems to support the notion that airway branching in
humans is overall symmetric.

Geometry F. The symmetrical dichotomous branching model [85] generalizes the
model B [78, 80] by incorporating the mechanisms involved in airway narrowing
due to shortening of the smooth muscle.

Geometry G. The asymmetric model [49] postulates that the bronchial tree
possesses an optimal topology that serves to provide a best physiological
function. The chosen function is the lung’s ability to move the respiratory
gases. The subsequent studies, some of which are referenced in the
introduction, suggest a number of alternative lung functions as optimization
target.

Geometry H. The asymmetric model [68] provides a statistical representation of
the tracheobronchial tree. The model consists of generation-specific
probability density functions for airway diameters, airway lengths, branching
angles, and gravity angles. In each generation, the vessel diameter is related
to its length $D=2.115\ln L+0.267$.

Figure 1: Dependence of the length (left column) and diameter (right column),
normalized with the corresponding length and diameter of the trachea, on the
generation order $n$, in human adults (top row) and infants (bottom row).

###  2.2 Model of Oxygen and Carbon Dioxide Transport

Respiration is a complex process that starts with ventilation via airways that
is governed by lung mechanics, followed by species (mainly oxygen O2 and
carbon dioxide CO2) transport across the alveolar-capillary barrier between
the alveoli and the circulating blood in the pulmonary capillaries,
constrained by the O2-CO2 binding properties of blood. Our model accounts for
these phenomena, as shown in Figure 2. The following assumptions underpin our
analysis:

(i) the airways branches are cylinders of circular cross section,

(ii) all the branches belonging to a certain generation are identical,

(iii) all the alveoli are identical,

(iv) the blood capillaries on the circumference of the alveolus are the same
for all alveoli, and

(v) the O2 and CO2 concentrations at the capillary inlet are the same for all
capillaries.

Figure 2: Model workflow showing connections between constituent sub-models,
as well as data streams / model inputs.

####  2.2.1 Model of air flow

During inspiration/expiration, the lung expands/contracts as air is drawn
from/to the atmosphere through the conducting airways to/from the alveolar
space [21]. The expansion and contraction take place predominantly in the
alveoli owing to the negligible contribution of the airways to the lung
compliance. Pulmonary ventilation, quantified in terms of the volumetric flow
rate $Q(t)$ of the air exchange between the atmosphere and the alveoli, is
driven by the time-varying negative intrapleural pressure established by the
chest muscles and the recoil of elastic tissues. On the other hand, the
resistance to air flow, $\mathcal{R}$, is dominated by viscous forces in the
airways.

We use a circuit representation of flow in airways (Fig. 3a), such that each
branch is conceptualized as a resistor in a series to represent friction; and
inertance and shunt capacitance are used to represent the effects of inertia
and storage, respectively. The capacitive behavior arises when bronchus wall
is compliant. If all bronchi of the like generation are identical and if the
lung tissue resistance and compliance are uniformly distributed over the
alveoli, then all the nodes in a branch have the same potential (pressure) as
the corresponding nodes in the all the branches belonging to the same
generation. In that case, the general airways circuit model in Fig. 3a
simplifies to its counterpart in Fig. 3b, such that the $(i,j)$ nodes of the
general circuit ($j=1,\ldots,2^{i-1}$) are represented by a single $i$ node in
the simplified circuit.

Figure 3: Alternative circuit models of gas flow in the bronchial tree: a
general model (a) and its successive simplifications (b–d).

If bronchus wall compliance is small, then bronchi can be approximated as
rigid vessels and all the capacitors in the network can be replaced by open
circuits. This assumption is justified because the lung tissue compliance,
associated with expansion and contraction of the alveoli, is much larger than
that associated with expansion and contraction of the bronchi [41]. The tissue
resistance is typically much smaller than the airways resistance [17] and,
hence, can be neglected. Under these assumptions, the circuit model in Fig. 3b
simplifies to that in Fig. 3c. Finally, if the inertia effects are negligibly
small, then the lung mechanics is represented by the circuit in Fig. 3d.

For the latter model, the relationship between the (known) pressure difference
between the airways inlet and the intrapleural fluid, $\Delta p(t)$, and the
(unknown) lung volume, $V_{\text{lu}}(t)$, is [8]

$\displaystyle{\cal
R}\frac{\text{d}V_{\text{lu}}}{\text{d}t}+\frac{V_{\text{lu}}-V^{*}}{{\cal
C}}=\Delta p,$ (1)

where ${\cal C}$ is the compliance of the lung tissue; ${\cal R}$ is the
airways resistance; and $V^{*}=\text{FRC}-{\cal C}p_{\text{ip,max}}$, where
the functional residual capacity (FRC) and the maximal intrapleural pressure
$p_{\text{ip,max}}$ are model parameters. For normal breathing of an adult,
$\mathcal{R}=0.56$ cmH2O s/l and ${\cal C}=93.66$ ml/cmH2O [48], and
$p_{\text{ip,max}}=-5$ cmH2O [21]. Values of the FRC, collected from multiple
prior studies, are shown in Fig. 4 (left) as function of body height which
serves as a proxy for age. The pulsating pressure signal $\Delta p(t)$ is
generated via a relation

$\Delta
p=-p_{\text{ip,max}}+\frac{p_{\text{ip,max}}-p_{\text{ip,min}}}{2}\begin{cases}1+\sin\left(-\dfrac{\pi}{2}+\dfrac{\pi
ft^{\prime}}{i_{f}}\right)&\quad\text{for }0\leq t^{\prime}\leq i_{f}T\\\ \\\
1+\sin\left(\dfrac{\pi}{2}+\dfrac{\pi
ft^{\prime}}{e_{f}}\right)&\quad\text{for }i_{f}T<t^{\prime}<T\end{cases}$

where $i_{f}=(I/E)/(1+I/E)$, $e_{f}=1-i_{f}$, $t^{\prime}=t-T\lfloor
t/T\rfloor$ and $T=1/f$, with $\lfloor\cdot\rfloor$ denoting the floor
function. In the simulations reported below, we set $p_{\text{ip,min}}=-10$
cmH2O [21], the adult normal breathing frequency to $f=1/T=0.25$ Hz (15 bpm),
and the inspiration-to-expiration ratio to $I/E=$0.5 [67]. At the beginning of
inspiration ($t=0$), $V_{\text{lu}}(0)=\text{FRC}$ and $\Delta
p(0)=-p_{\text{ip,max}}$. With these inputs, a solution to the ordinary
differential equation (1) provides the lung volume response,
$V_{\text{lu}}(t)$, to the pressure signal $\Delta p(t)$ (Fig. 5). It also
enables us to compute $Q(t)$ $={\text{d}V_{\text{lu}}}/{\text{d}t}$, the total
volumetric flow rate of air through the airways.

Figure 4: Dependence of functional residual capacity FRC (left) and dead space
DS (right) on body height, which serves as a proxy for age. The correlation
between FRC and lung volume/age reported in [74] were used to convert total
lung volume into FRC. When needed, growth charts were used to express height
as function of age. [a]: Engstom et al. (1956), [b]: Cook and Hamann (1961),
[c]: Engstrom et al. (1962), [d]: Hart et al. (1963), [e]: Doershuk et al.
(1970), [f]: Wood et al. (1971), [g]: Hatch and Taylor (1976), [h]: Stocks and
Godfrey (1977), [i]: Greenough et al. (1986), [j]: Roberts et al. (1991), [k]:
Tepper et al. (1993), [l]: Merth et al. (1995), [m]: Stocks and Quanjer
(1995), [n]: McCoy et al. (1995), [o]: Neder et al. (1999), [p]: Castile et
al. (2000), [q]: Hulskamp et al. (2003), [r]: Hislop et al. (1986), [s]: Angus
and Thurlbeck (1972), [t]: Davies and Reid (1970), [u]: Auld et al. (1963),
[v]: Hart et al. (1963). $\star$: Stocks and Quanjer (1995) using Tepper et
al. (1993); Hanrahan et al. (1990); Merth et al. (1995); Taussig et al.
(1977); Greenough et al. (1986). Figure 5: Temporal variation of the pressure
signal $\Delta p(t)$ (the model’s input) and the lung volume
$V_{\text{lu}}(t)$ (the model’s output) under normal breathing conditions for
an adult.

Next, we partition the total volumetric flow rate $Q(t)$ among individual
bronchus of the bronchial tree. The latter is assumed to branch dichotomously
and consist of $N_{\text{gen}}$ generations. That implies that the $i$th
generation ($i=1,\ldots,N_{\text{gen}}$) comprises $N_{i}=2^{i-1}$ bronchi.
Each bronchus of the $i$th generation is modeled as a cylinder of diameter
$D_{i}$ and cross-sectional area $A_{i}=\pi D_{i}^{2}/4$. When applied to such
a dichotomously branching tree, mass conservation requires the cross-
sectionally averaged flow velocity in the $i$th-generation bronchus,
$U_{i}(t)$, to be

$U_{i}(t)=\frac{Q(t)}{N_{i}A_{i}},\qquad i=1,\ldots,N_{\text{gen}}.$ (2)

This expression implies that the average velocity in the last generation,
$U_{N_{\text{gen}}}=Q/(N_{N_{\text{gen}}}A_{N_{\text{gen}}})$, is small but
non-zero. The terminal velocity $U_{N_{\text{gen}}}$ is the velocity of air
through the terminal bronchioles as air fills/empties the
expanding/contracting alveoli (Fig. 6).

Figure 6: Each terminal branch is connected to a number of alveoli. Our model
assumes that the bronchiole is connected to the center of each compliant
spherical alveolus.

Each bronchiole (terminal airways branch) is connected to
$N_{\text{al}}^{\text{tb}}=N_{\text{al}}/2^{N_{\text{gen}}-1}$ alveoli (Fig.
6). The total number of alveoli depends on age, sex, and lung volume, ranging
from $N_{\text{al}}\approx 3\cdot 10^{7}$ in a preterm infant to
$N_{\text{al}}\approx 5\cdot 10^{8}$ in a twenty year old adult [74, 48, 26]
(Fig. 7 and Table 3). For an adult, assuming $N_{\text{gen}}=24$, this yields
the number of alveoli per bronchiole to be approximately 60. We treat alveoli
as compliant spheres of radius $R_{\text{al}}(t)$; their total volume is
$N_{\text{al}}V_{\text{al}}(t)$, where the volume of a single alveolus is
$V_{\text{al}}=4\pi R_{\text{al}}^{3}/3$. At the beginning of inspiration
($t=0$), $R_{\text{al}}(0)=100$ $\mu$m [48]. Since the lung volume
$V_{\text{lu}}(t)=N_{\text{al}}V_{\text{al}}(t)+\text{DS}$ is the sum of the
total alveolar volume, $N_{\text{al}}V_{\text{al}}(t)$, and the time-invariant
dead space volume, DS, the change of $V_{\text{al}}$ with time satisfies

$\frac{\text{d}V_{\text{al}}}{\text{d}t}=\frac{Q(t)}{N_{\text{al}}}.$ (3)

At the end of expiration, the total alveolar volume is
$N_{\text{al}}V_{\text{al}}(t=0)=\text{FRC}-\text{DS}$. Values of DS,
collected from multiple prior studies, are shown in Fig. 4 (right) as function
of body height, which serves as a proxy for age. With this initial condition,
a solution of (3) yields $V_{\text{al}}(t)$.

Figure 7: Dependence of the number of alveoli, $N_{\text{al}}$, on age of
adults (left) and infants (right).

###  2.3 Model of transport of dissolved species

####  2.3.1 Gas transport in airways

At any time $t>0$ and location $x\in(0,L_{i})$ along the $i$th bronchus,
cross-sectionally averaged volumetric concentration of gases (O2 or CO2),
$C_{\text{aw}}^{(i)}(x,t)$, satisfies an advection-dispersion equation [88]

$\displaystyle\frac{\partial C_{\text{aw}}^{(i)}}{\partial
t}+U_{i}(t)\frac{\partial C_{\text{aw}}^{(i)}}{\partial
x}=\mathcal{D}_{i}\frac{\partial^{2}C_{\text{aw}}^{(i)}}{\partial
x^{2}},\qquad i=1,\cdots,N_{\text{gen}},$ (4)

where
$\mathcal{D}_{i}=\mathcal{D}_{\text{aw}}+D_{i}^{2}U_{i}^{2}/(192\mathcal{D}_{\text{aw}})$
is the hydrodynamic dispersion coefficient, and $\mathcal{D}_{\text{aw}}$ is
the coefficient of molecular diffusion of the corresponding species (O2 or
CO2). The equations for adjacent bronchi are coupled to ensure mass
conservation. In the inspiration phase, this coupling gives rise to boundary
conditions

$\begin{split}&C_{\text{aw}}^{(1)}(0,t)=C_{\text{aw}}^{\text{in}};\quad\text{$C_{\text{aw}}^{(i+1)}(0,t)=C_{\text{aw}}^{(i)}(L_{i},t)$,
$i=1,\ldots,N_{\text{gen}}-1$};\quad
C_{\text{aw}}^{(N_{\text{gen}})}(L_{N_{\text{gen}}},t)=C_{\text{al}}^{\text{in}}\\\
&\mathcal{D}_{i}A_{i}\frac{\partial C_{\text{aw}}^{(i)}}{\partial
x}(L_{i},t)=2\mathcal{D}_{i+1}A_{i+1}\frac{\partial
C_{\text{aw}}^{(i+1)}}{\partial
x}(0,t),\;i=1,\ldots,N_{\text{gen}}-1;\end{split}$ (5)

where $C_{\text{aw}}^{\text{in}}$ is the (prescribed) concentration of O2 or
CO2 at the inlet to trachea ($i=1$), and $C_{\text{al}}^{\text{in}}$ is the
concentration of O2 or CO2 at the interfaces between the terminal bronchi
($i=N_{\text{gen}}$) and the alveoli. Since the latter is unknown, Eq. (4)
with $i=N_{\text{gen}}$ is supplemented with a Robin boundary condition at
$x=L_{N_{\text{gen}}}$, Eq. (7), which accounts for the gases exchange between
the alveolar space, the terminal bronchioles, and the blood (see below).

In the expiration phase, the first of the boundary conditions (5) is replaced
with $\partial_{x}C_{\text{aw}}^{(1)}(0,t)=0$.

####  2.3.2 Gas transport in alveoli

The alveoli adjacent to a bronchiole are interconnected via “pores of Kohn”
[13], and form a sac (Fig. 6). These interconnections justify modeling gases
in all the alveoli belonging to a single sac as well mixed, so that
$C_{\text{al}}=C_{\text{al}}^{\text{in}}$ for any single alveolus. This
assumption is appropriate because the time scale associated with diffusive
transport over the alveolar radius ($\sim 0.7$ ms for adults, and $\sim 0.07$
ms for infants) is much smaller than the respiration time scale. Mass
conservation of dissolved gases in the alveolar space yields a mass-balance
equation for the average concentration of the species in the alveolar space,
$C_{\text{al}}(t)$,

$\frac{\text{d}C_{\text{al}}V_{\text{al}}}{\text{d}t}={Q}_{\text{al}}^{\text{in}}C_{\text{al}}^{\text{in}}+A_{\text{al}}^{\text{in}}J_{\text{diff}}-A_{\text{ca}}q_{\text{al}\rightarrow\text{bl}},$
(6)

where $Q_{\text{al}}^{\text{in}}=Q(t)/N_{\text{al}}$;
$A_{\text{al}}^{\text{in}}=A_{N_{\text{gen}}}/N_{\text{al}}^{\text{tb}}$ is
the cross-sectional area of the terminal bronchiole per connected alveolus;
$A_{\text{ca}}$ is the capillary-alveolus contact area;
$q_{\text{al}\rightarrow\text{bl}}$ is the mean species flux from the alveolar
space into the blood; and
$J_{\text{diff}}=-\mathcal{D}_{N_{\text{gen}}}\partial_{x}C_{\text{aw}}(L_{N_{\text{gen}}},t)$
is the diffusive flux of the dissolved species from the terminal bronchiole
into the alveolus. Since
$Q_{\text{al}}^{\text{in}}N_{\text{al}}^{\text{tb}}=Q/N_{\text{gen}}$,
equation (6) yields the Robin boundary condition at the end of the terminal
bronchioles

$\mathcal{D}_{N_{\text{gen}}}A_{N_{\text{gen}}}\frac{\partial
C_{\text{aw}}^{(N_{\text{gen}})}}{\partial
x}(L_{N_{\text{gen}}},t)+\frac{Q}{N_{\text{gen}}}C_{\text{aw}}^{(N_{\text{gen}})}(L_{N_{\text{gen}}},t)=N_{\text{al}}^{\text{tb}}\left(\frac{\text{d}C_{\text{al}}V_{\text{al}}}{\text{d}t}+A_{\text{ca}}q_{\text{al}\rightarrow\text{bl}}\right).$
(7)

This completes the specification of the auxiliary conditions for (4), which
couple transport of O2 and CO2 to the physiological processes in pulmonary
membrane and blood through (as yet unspecified) quantities $C_{\text{al}}$ and
$q_{\text{al}\rightarrow\text{bl}}$.

####  2.3.3 Gas transport in pulmonary membrane

Exchange of gases between the alveolar air and the blood circulating in the
surrounding capillaries occurs by diffusion through a pulmonary membrane,
whose thickness is $h_{0}\approx 0.6$ $\mu$m [81]. Diffusion across the
membrane is driven by a difference in partial pressure between the blood side
and the alveolar side. Partial pressure of O2 is typically higher in the
alveolus, while partial pressure of CO2 is higher on the blood side. Hence, O2
diffuses from the alveoli into the capillaries, and CO2 diffuses from the
capillaries into the alveoli (see Fig. 6). Since the alveolus radius is two
orders of magnitude larger than the membrane thickness, the membrane is
assumed to be locally planar, so that the species concentration in the
membrane, $C_{\text{mem}}(r,t)$, is described by a diffusion equation
$\partial_{t}C_{\text{mem}}={\cal
D}_{\text{wat}}\partial_{r}^{2}C_{\text{mem}}$, where ${\cal D}_{\text{wat}}$
is the diffusion coefficient in the membrane, which is set to that in water.
Since the diffusion time scale, $h_{0}^{2}/{\cal D}_{\text{wat}}\sim 1$ ms for
both O2 and CO2, is much smaller than the respiration period (1.5 – 4 s), we
assume diffusion through the membrane to be quasi-steady so that the species
flux through the membrane is uniform

$q_{\text{al}\rightarrow\text{bl}}(\xi,t)=\frac{{\cal
D}_{\text{wat}}}{k_{\text{H}}}\frac{{C}_{\text{al}}(t)-P_{\text{bl}}(\xi,t)/(RT)}{h(t)}.$
(8)

Here $P_{\text{bl}}(r,t)$ is the species (O2 or CO2) partial pressure in the
blood at point $\xi$ along the capillary, $k_{\text{H}}$ is Henry’s constant,
$R$ is universal ideal gas constant, and $T$ is the temperature. The alveolar
membrane of thickness $h(t)$ is viscoelastic and responds to any change in
alveolar pressure in a way that conserves the volume of the membrane,
$V_{\text{mem}}$:

$h(t)=\sqrt[3]{\frac{3}{4\pi}\left[{V}_{\text{mem}}+\frac{4}{3}\pi{R}_{\text{al}}^{3}(t)\right]}-{R}_{\text{al}}(t).$
(9)

On the blood side, the partial pressure $P$ and concentration $C$ are related
by the dissociation curves, presented in the next section.

####  2.3.4 Gas transport in blood

Each alveolus is surrounded by several pulmonary capillaries, in which the red
blood cells travel at the blood speed ${U}_{\text{bl}}$. The diameter of the
capillary, $D_{\text{cap}}=10$ $\mu$m, equals the size of a red blood cell.
Nevertheless, as in [37], our blood perfusion model treats blood in the
capillaries as a uniform homogeneous phase (masking the discrete constituents:
plasma and erythrocytes). The model assumes equilibrium conditions for the
reactions, thus allowing the use of an empirical dissociation curves to relate
the species’ partial pressure to its concentration. If the speed of blood
along the capillary is $U_{\text{bl}}$, then the concentrations of both O2 and
CO2 in blood, at any space-time point $(\xi,t)$ along a capillary, are
described by

$\frac{\partial C_{\text{bl},k}}{\partial t}+U_{\text{bl}}\frac{\partial
C_{\text{bl},k}}{\partial\xi}=\frac{W}{{A}_{\text{cap}}}{q}_{\text{al},k\rightarrow\text{bl}},\qquad
k=\text{O${}_{2}$ and CO${}_{2}$}.$ (10)

Here $A_{\text{cap}}=\pi{D}_{\text{cap}}^{2}/4$ is the cross-sectional area of
the capillary, and $W$ is the contact area (per unit length of the capillary)
between the capillary and the alveolar membrane. Gas exchange between the
alveolar space and the blood circulating in the surrounding capillaries is a
function of the blood speed in the capillaries, the contact area between an
alveolus and the surrounding capillaries, and the capillaries distribution.

The blood speed in a single pulmonary capillary, $U_{\text{bl}}$, is estimated
as

$U_{\text{bl}}=\frac{\dot{V}_{\text{bl}/\text{al}}}{N_{\text{cap}/\text{al}}{A}_{\text{cap}}}.$
(11)

The number of capillaries per alveolus is

$N_{\text{cap}/\text{al}}=\frac{1}{A_{\text{cap}}L_{\text{cap}}}\left(\frac{V_{\text{bl}}}{N_{\text{al}}}\right),$

where, assuming that a single capillary extends over half of the alveolar
perimeter, $L_{\text{cap}}=\pi R_{\text{al},0}$; and, assuming that each
alveolus is surrounded by a spherical shell of thickness equal to
$D_{\text{cap}}$, the blood volume in the pulmonary capillaries is
$V_{\text{bl}}=\beta V_{\text{bl, max}}$, with $V_{\text{bl,
max}}=N_{\text{al}}4\pi{R}_{\text{al},0}^{2}D_{\text{cap}}$ providing the
upper bound on the blood volume in the capillaries, and $\beta$ denoting the
percent contact area between the alveoli and the capillaries. Finally, in 11,
the capillaries blood flow rate per alveolus is
$\dot{V}_{\text{bl}/\text{al}}=\dot{V}_{\text{bl}}/N_{\text{al}}$, where
$\dot{V}_{\text{bl}}=V_{\text{bl}}/\tau_{\text{r}}$ is the volume flow rate of
blood in the pulmonary capillaries and $\tau_{\text{r}}$ is the residence time
of blood in a pulmonary capillary, with a typical value of
$\tau_{\text{r}}\approx 1$ s [77].

Equations 10 are coupled through the interdependence between O2 and CO2, which
arrises due to the blood chemistry. We use the O2-hemoglobin dissociation
curve [64], derived from a kinetic study of several chemical reactions between
the respiratory gases and hemoglobin, to relate the O2 concentration,
$C_{\text{O}_{2}}$, to the O2 partial pressure, $P_{\text{O}_{2}}$; the
reaction rate constant in this relation depends on pH and the CO2 partial
pressure. We also use the CO2 dissociation curve [42] to relate the CO2
concentration, $C_{\text{CO}_{2}}$ to both its partial pressure,
$P_{\text{CO}_{2}}$, and the O2 partial pressure, $P_{\text{O}_{2}}$.

These two dissociation curves capture both Bohr’s and Haldane’s effects in
blood chemistry, both of which stem from the coupling between O2 and CO2.
Bohr’s effect refers to the observation that, for a fixed value of
$P_{\text{O}_{2}}$, the hemoglobin saturation and, hence, $C_{\text{CO}_{2}}$
increase as $P_{\text{CO}_{2}}$ decreases. Haldane’s effect refers to the
observation that, for a fixed value of $P_{\text{CO}_{2}}$, both the
hemoglobin saturation and $C_{\text{O}_{2}}$ decrease as $C_{\text{CO}_{2}}$
increases.

The boundary conditions for Eqs. (10) at the inlet to the capillaries
($\xi=0$) are set to $P_{\text{O}_{2}}(\xi=0,t)=40$ mmHg and
$P_{\text{CO}_{2}}(\xi=0,t)=46$ mmHg.

##  3 Numerical Implementation

Equations (4) and (5) are solved using the finite volume method [45], which
conserves mass by matching the gases’ fluxes at the faces of the grid cells.
For a fixed control volume $\Omega_{j}$, e.g., the $j$th grid cell in Fig. 8,
the advection-dispersion equation (4) is expressed as

$\displaystyle V_{j}\frac{\partial\bar{C}_{j}}{\partial
t}=\sum_{f\subset\partial\Omega_{j}}A_{f}\left(Q_{f}^{\leftarrow}C_{f}+\mathcal{D}_{f}\frac{\partial
C_{f}}{\partial n_{f}}\right),$ (12)

where $\bar{C}_{j}$ is the average concentration in the control volume
$\Omega_{j}$, whose size is $V_{j}=A_{j}\Delta x_{j}$ and whose surface is
denoted by $\partial\Omega_{j}$; the summation is over all the faces $f$
comprising the control surface $\partial\Omega_{j}$; $Q^{\leftarrow}_{f}$ is
the volumetric flow rate into the control volume across the face $f$; $A_{f}$
is the area of the face $f$; $C_{f}$ is the concentration at the face $f$; and
the derivative in the diffusion flux is in the direction of the normal vector
to the face $f$.

Figure 8: A schematic of the mesh used in the finite volume method for a
branch in generation $i$ and the two daughter branches in generation $i+1$.

We use a second-order time integration scheme. The advection and diffusion
terms are discretized using a second-order upwind and a second-order central
difference schemes, respectively. Since O2 and CO2 are coupled through the
oxyhemoglobin dissociation, the numerical solution over each time step is
obtained by iterating between the airways and the blood until the
concentrations and the fluxes match at the interface. Although not shown here,
the grid size and the time step are chosen based on a standard convergence
study.

##  4 Results

The input data needed for our model are summarized in Table 3 for a human
newborn and a twenty-year-old healthy male. Their height and mass are at the
50th percentile line of their respective growth charts. Additional input for
our model is the alternative bronchial tree representations reported in Tables
1 and 2. In the simulations reported below we compare the model predictions
based on these tree representations to ascertain their compatibility and
ability to predict key observable physiological quantities. (We excluded the
tree geometries [29] and [27] from our simulations because of their
unphysically high dead space volumes and unphysically small resistance, as
shown in Table 4).

Table 3: Values of model parameters for two baseline cases. Parameter | infant | adult
---|---|---
Age (yrs) | 0 | 20
Gender | male | male
Height (cm) | 49.99 | 176.85
Mass (kg) | 3.53 | 70.6
Respiration frequency (Hz) | 0.75 | 0.25
Tidal volume (ml) | 21.7 | 468.3
Functional residual capacity (ml) | 61.9 | 3228.27
Initial alveolar radius ($\mu$m) | 35 | 100
Number of alveoli (million) | 152 | 499
Minimum intrapleural pressure (cmH2O) | -10 | -10
Maximum intrapleural pressure (cmH2O) | -5 | -5
Compliance (ml/cmH2O) | 4.34 | 93.66
Blood flow rate in capillaries (l/min) | 0.518 | 11.2

Since the time scale of diffusion across the alveolar membrane is proportional
to the the square of the membrane thickness, it is much smaller than the time
scales of the other transport mechanisms. Consequently, gas exchange between
the alveolar space and the blood occurs very quickly, so that its partial
pressures on both sides remain close to each other, as long as O2 in the blood
is below saturation level. Figure 9 shows that this is indeed the case: when
the blood is not saturated with oxygen, $P_{\text{O}_{2}}$ in the alveolar
space is close to but smaller than $P_{\text{O}_{2}}$ in the blood, indicating
O2 transport into the blood. This figure also demonstrates that
$P_{\text{CO}_{2}}$ in the blood is close to but larger than
$P_{\text{CO}_{2}}$ in the alveolar space, indicating CO2 transfer into the
alveolar space.

Figure 9: Temporal variation of the mean partial pressure of oxygen (left) and
carbon dioxide (right) in the blood and in the alveolar space of adults, as
predicted with our model for two bronchial tree representations.

Figure 10 exhibits temporal variations (over 10 breathing cycles) of the mean
$P_{\text{O}_{2}}$ and $P_{\text{CO}_{2}}$ in the blood for adults and
infants, respectively. Oxygen is quicker to reach saturation in the blood than
carbon dioxide is. That is because while O2 is stored only in the blood
hemoglobin, CO2 can be stored in the blood in different ways. Since the
solubility of CO2 is much higher than that of O2, carbon dioxide is
transported easier in the dissolved state. More than 60$\%$ of CO2 is stored
in the bicarbonate ionic form [59], with 30$\%$ being stored in the hemoglobin
and 7$\%$ in the plasma.

Fgure 10 shows that the predictions of the mean $P_{\text{O}_{2}}$ and
$P_{\text{CO}_{2}}$ based on the airways geometry [46] have a fluctuation
amplitude that is higher than those based on all the other 24-generation
models. This is attributed to the overestimated lengths of the generations
14–25 (see Figure 1 or Table 1). Increasing the lengths $L_{i}$ of the
generations where diffusion is dominant results in a smaller mass fluxes at
the alveolar inlet, since $\partial_{x}C(x,t)\sim 1/\sum L_{i}$. This damping
effect results in smaller fluctuation amplitudes and in overall reduction in
gas exchange.

Figure 10: Temporal variation of the mean partial pressure of oxygen (left)
and carbon dioxide (right) in blood capillaries of adults (top) and infants
(bottom), as predicted with our model for two bronchial tree representations.

The model predictions of $P_{\text{O}_{2}}$ and $P_{\text{CO}_{2}}$ in Figs. 9
and 10 demonstrate their sensitivity to the selection of a bronchial tree
conceptualizations. Consequently, we use our model to ascertain the veracity
of these alternative tree representations (Models A–H in Tables 1 and 2). This
is done by comparing the model predictions of key physiological quantities
(total resistance, dead space volume, total path length, and min and max mean
$P_{\text{O}_{2}}$ and $P_{\text{CO}_{2}}$ in the capillaries) with their
observed values reported in the literature. For each airway geometry, we
computed $P_{\text{O}_{2}}$ and $P_{\text{CO}_{2}}$ distributions along the
capillaries every one eighth of a period over a whole period in the quasi-
stationary state. In most cases, the quasi-stationary state was reached after
ten breathing cycles. The minimum and maximum values of $P_{\text{O}_{2}}$ and
$P_{\text{CO}_{2}}$ are obtained by averaging these distributions over the
length of the capillary. The results are summarized in Table 4 together with
the number of generations, and length and diameter of the terminals branches.
The measured values of the dead space DS for adults and infants are taken as
averages of those reported in [18, 4, 25, 36, 86, 71] and [32, 15, 19, 63, 34,
47, 83, 10], respectively. The measured values of the resistance $\mathcal{R}$
for adults and infants are taken as averages of those reported in [14, 1, 76,
22, 30, 33, 75] and [19, 83, 9, 70, 54, 31, 55, 6, 56, 87, 58, 66, 72, 2],
respectively.

Table 4: Key physiological characteristics (dead space, DS; resistance, $\cal R$; total path length $L$; time needed to reach saturation of oxygen in blood, $T_{r}$; and mean partial pressures, $\bar{P}_{\text{O}_{2}}$ and $\bar{P}_{\text{CO}_{2}}$) computed with our model for alternative airways geometries of adults and infants. These predicted values are compared with their independently measured counterparts (M). $\dagger$ corresponds to spatial averages along the capillary by assuming that $P_{\text{O}_{2}}$ increases from 40 to 100 and $P_{\text{CO}_{2}}$ decreases from 46 to 40 over one third of the capillary length. Partial pressures are in units of cmH2O. Airways | DS | $\cal R$ | $L$ | $T_{r}$ | $\bar{P}_{\text{O}_{2}}$ | $\bar{P}_{\text{CO}_{2}}$
---|---|---|---|---|---|---
geometry | (mL) | (cmH2O.s/L) | (cm) | (sec) | (min) | (max) | (min) | (max)
adults
[46] | 2681.62 | 1.49 | 29.46 | 6 | 100.7 | 100.9 | 27 | 28.9
[79] | 1656 | 0.184 | 27.28 | 1.3 | 100.4 | 100.8 | 17.3 | 22.4
[29] | 77537 | 0.037 | 33.93 | - | - | - | - | -
[27] | 103141 | 0.037 | 34.53 | - | - | - | - | -
[51] | 71.07 | 0.357 | 19.4 | 21.4 | 96.2 | 101 | 42.9 | 45.4
[85] | 830 | 0.56 | 27.27 | 1.3 | 100.5 | 100.9 | 21.8 | 28.7
[49] | 1759 | 0.049 | 31.17 | 1.2 | 100.6 | 100.8 | 23.7 | 27.5
M | 127-190 | 1.08-2.6 | 32 | - | 90† | 90† | 41† | 41†
infants
[46] | 1.48 | 98.8 | 13.5 | 2 | 97.6 | 101.5 | 40.3 | 43
[51] | 16.58 | 2.06 | 15.06 | 0.52 | 100.4 | 101.3 | 32 | 36.4
[68] | 22.75 | 6.09 | 15.06 | 0.64 | 100.4 | 101.2 | 31 | 34.7
M | 4.15-6.7 | 22.6-52 |  |  | 90† | 90† | 41† | 41†
Table 5: Computed average air flow-rate ($V$) entering the alveoli, blood flow-rate ($Q$) and ventilation to perfusion ratio($V/Q$) for the different airways geometries. These predicted values are compared with measurements reported in the literature (M).  Airways | $V$ | $Q$ | $V/Q$
---|---|---|---
geometry | mm3/s ($l/min$) | mm3/s ($l/min$) |
adults
[46] | 0.00065 (19.46) | 0.00037 (11.28) | 1.75
[79] | 0.0007 (20.96) | 0.00037 (11.28) | 1.86
[51] | 0.0007 (20.96) | 0.00037 (11.28) | 1.86
[85] | 0.00069 (20.66) | 0.00037 (11.28) | 1.86
[49] | 0.00069 (20.66) | 0.00037 (11.28) | 1.86
M [5] |  |  | 0.67-1.7
infants
[46] | 0.00044 (0.40) | 0.000057 (0.52) | 0.77
[51] | 0.000318 (2.9) | 0.000057 (0.52) | 5.58
[68] | 0.000311 (2.84) | 0.000057 (0.52) | 5.46
M[11] |  |  | 0.8-1.1

The values of the total path length, $L$, total resistance, $\mathcal{R}$, and
dead space DC in Table 4 for various bronchial tree geometries are estimated
as

$\displaystyle L=\sum_{k=0}^{N_{\text{gen}}}{L_{k}},\quad{\cal
R}=\sum_{k=0}^{N_{\text{gen}}}\frac{128\mu L_{k}}{\pi
D_{k}^{4}2^{k}},\quad\text{DS}=\sum_{k=0}^{N_{\text{gen}}}2^{k}\frac{\pi
D_{k}^{2}L_{k}}{4}$ (13)

where $\mu$ is the viscosity of air. In Table 4, we compare thus computed
values of dead space DS with their measured counterparts (see also Fig. 4);
the latter are used in $N_{\text{al}}V_{\text{al}}(t=0)=\text{FRC}-\text{DS}$
to estimate the total alveolar volume. The expression for total resistance
${\cal R}$ is based on the assumption of Poiseuille flow in each channel. Due
to losses at the branching junctions, the actual resistance is expected to be
larger than ${\cal R}$, especially during expiration because the mixing loss
associated with two channel flows merging into one is larger than that of a
single channel flow splitting into two.

Table 4 reveals that the resistance $\mathcal{R}$ estimated for the various
airways geometries for adults is well below the range of reported
measurements, except for the fractal model [46] that results in the resistance
that falls within the observed range. In general, the smaller the airways
resistance, the sooner the mean partial pressures of oxygen and carbon dioxide
reach their quasi-stationary state, i.e., the smaller $T_{r}$ (see, also, Fig.
10). The only exception is the model [51], whose estimate of the resistance
$\mathcal{R}$ is well below the range of reported measurements but estimate of
$T_{r}$ is an order of magnitude larger than the estimates derived from the
other models.

In infants, the resistance $\mathcal{R}$ predicted with the airways geometry
[46] is well above the observed range, while the estimates based on the
geometries [51] and [68] are significantly below. As a consequence, airways
geometries [51] and [68] predict $T_{r}<1$ s, while the geometry [46] results
in $T_{r}>2$ s. Both estimates are almost two orders of magnitude smaller than
the observed value. The predictions of quasi-steady state $P_{\text{CO}_{2}}$
in the blood, based on the tree models [51] and [68], oscillate in the range
appreciably below their nominal range (40–46 mmHg) (Fig. 10). As noted in
[51], abnormally elevated CO2 levels in the blood are typically caused by
hypoventilation, whereas reduced levels are caused by hyperventilation. This
finding enables us to categorize the airways geometries in Table 4 as either
hyperventilating or hypoventilating.

In assessing the adults (infant) airways geometries, the blood flow-rate in
the capillaries, $Q$, was set to its physiological value (Table 3). The mean
ventilation flow-rate, $V$, was computed as the ratio of the intake volume of
air in one breathing cycle to the period of the cycle at the quasi-stationary
state. Mechanical models $\dot{V}_{\text{lu}}=\dot{V}_{\text{lu}}(\Delta p)$,
including the one in Eq. (1), suggest that $V$ is primarily a function of the
pressure signal, compliance of the lung tissue and resistance of the airways.
In assessing the alternative airways geometries, the same compliance and
pressure signal was used for all the reported adult (infant) geometries, while
the airways resistance varied with the airways geometry according to Eq. (13).
This strategy yields a one to one correspondence between the resistance (Table
4) and the ventilation to perfusion ratio (Table 5).

Table 4 also reveals that a higher dead space volume does not always lead to a
smaller airways resistance, and vice versa: compare, e.g., the predictions
based on the tree models [51] and [68] or [46] and [79]. For a given circular
channel of length $L$ and diameter $D$, both the channel volume and the
resistance would increase if $L$ increases by a factor $\alpha>1$ and $D$
increases by a factor $\beta$ that satisfies the condition
$\alpha^{-1/2}<\beta<\alpha^{1/4}$.

Impact of the lung resistance and compliance on gas exchange with the blood is
captured through the time scale of the mechanical model, $\tau=\cal{RC}$. This
characteristic time, listed in Table 6 for the alternative airways geometries,
characterizes how fast the system responds to changes in the driving pressure
difference in term of the air volume exchange. The compliance $\mathcal{C}$, a
property of the lung tissue, is taken to be the same in all the geometric
models for adults (infants), since these differ only in terms of airways
geometry. That is in contrast to the resistance $\mathcal{R}$, which depends
on the airways geometry. This implies that the resistance is a sufficient
metric to compare the speed of volume exchange for the various adults (or
infants) cases.

Table 6: Model predictions of characteristic time $\tau=\mathcal{C}\mathcal{R}$; the number of airways generations, $N_{\text{gen}}$; length and diameter of the last generation channels, $L_{N_{\text{gen}}}$ and $D_{N_{\text{gen}}}$; the ratio of the airways’ inlets diameter per alveolus to the alveolar diameter, $R_{\text{in}}/R_{\text{al}}$; the Reynolds number in the last generation, $\mathtt{Re}_{N_{\text{gen}}}=\rho\bar{u}_{N_{\text{gen}}}D_{N_{\text{gen}}}/\mu$; and the Péclet number in the last generation, $\mathtt{Pe}_{N_{\text{gen}}}=L_{N_{\text{gen}}}\bar{u}_{N_{\text{gen}}}/\cal D$. Here $\rho$ and $\mu$ are the density viscosity of blood, respectively; ${\cal D}$ is the diffusion coefficient of species in air; and the mean speed $\bar{u}_{N_{\text{gen}}}=(2TVf)/2^{N_{\text{gen}}-1}$ is estimated by assuming that the lung changes its volume by the tidal volume, $TV$, over half a breath period, $1/(2f)$; and $f$ is the breathing frequency. | $\tau$ | $N_{\text{gen}}$ | $L_{N_{\text{gen}}}$ | $D_{N_{\text{gen}}}$ | $R_{\text{in}}/R_{\text{al}}$ | $\mathtt{Re}_{N_{\text{gen}}}$ | $\mathtt{Pe}_{N_{\text{gen}}}$
---|---|---|---|---|---|---|---
Model | (sec) |  | (mm) | (mm) |  |  |
adults
[46] | 0.14 | 24 | 1.49 | 0.328 | 0.045 | 0.01 | 0.04
[79] | 0.017 | 24 | 0.501 | 0.418 | 0.073 | 0.01 | 0.01
[29] | 0.003 | 24 | 3.1 | 0.96 | 0.387 |  |
[27] | 0.003 | 25 | 3.9 | 0.79 | 0.525 |  |
[51] | 0.033 | 16 | 1.55 | 0.454 | 0.0003 | 1.81 | 5.82
[85] | 0.052 | 24 | 0.501 | 0.29 | 0.035 | 0.01 | 0.02
[49] | 0.005 | 24 | 0.6 | 0.35 | 0.051 | 0.01 | 0.01
infants
[46] | 0.43 | 15 | 0.87 | 0.13 | 0.0011 | 0.54 | 3.15
[51] | 0.009 | 16 | 0.429 | 0.3 | 0.012 | 0.28 | 0.76
[68] | 0.026 | 16 | 0.429 | 0.507 | 0.035 | 0.23 | 0.53

Since both length and diameter of the airways channels decrease with the
generation number $n$, air flow and gases transport become increasingly
dominated by diffusion, approaching the Stokes flow regime in the last
generation, $n=N_{\text{gen}}$, as indicated by the low values of Reynolds
($\mathtt{Re}_{N_{\text{gen}}}$) and Péclet ($\mathtt{Pe}_{N_{\text{gen}}}$)
numbers in Table 6. This implies that the diffusion term
$A_{\text{in}}J_{\text{diff}}$ in Eq. 6 plays a significant role in the
transfer of O2 and CO2 into and out of the alveolar space. The magnitude of
this term depends on two quantities, which are related directly to the tree
geometry: the concentration gradient, $\partial_{x}C(x_{\text{in}},t)$, and
the area of the airways inlet to the alveolus, $A_{\text{in}}$. The former is
affected by the spatial distribution of the species along the airways, while
the latter is estimated as $A_{\text{in}}=2^{N_{\text{gen}}-1}\pi
D_{N_{\text{gen}}}^{2}/(4N_{\text{al}})$. Among the cases presented in Fig.
10, the airways geometry [79] results in the lowest CO2 levels and the highest
$A_{\text{in}}=\pi R_{\text{in}}^{2}$ and $R_{\text{in}}/R_{\text{al}}$ (Table
6). The airways geometry [51] yields the highest CO2 levels and the lowest
$A_{\text{in}}$ and $R_{\text{in}}/R_{\text{al}}$. The difference in CO2
removal from the blood, predicted with the tree models [79] and [51], is due
to the fact the former consists of 24 generations while the latter comprises
16 generations only, with both sharing nearly similar values of the terminal
branch diameter. The impact of the small alveolar inlet in the tree model [51]
is also seen in Fig 10, wherein the small flux at the alveolar inlet causes
the oxygenation to be very slow, so that it takes 22 s (6 respiration cycles)
for the blood to reach saturation.

##  5 Discussion

Quantitative description of the tracheobronchial airway morphometry is
essential for the investigation of toxicologic effects of inhaled aerosols and
gases under a variety of conditions. Limitations in measurements prohibit the
accurate estimation of the dimensions of the generations beyond the first few.
As a result, some studies opted to include only the first few generations in
their design, while others relied on inferred dimensions for the rest of the
generations, as discussed in the Introduction and Materials and Methods
sections. Inaccurate measurements and corresponding quantitative descriptions
of the airway anatomy undermine the ability of the associated dosimetry models
in animal toxicology to accurately predict particle transport and deposition
dynamics within the respiratory tract.

A major goal of this study was to assess the adequacy of alternative lung
airways geometries in terms of their ability to predict both observed
geometric characteristics and observed lung function. Such an assessment is
important because accurate predictions of the exchange of inhaled aerosols and
gases with the blood in the pulmonary capillaries are not possible without a
physically sound representation of the airways geometry. Such predictions
require both an adequate pulmonary tree representation and a model to predict
the dynamics of air and gases transport all the way down to the blood
circulating in the pulmonary capillaries. A properly constructed mathematical
representation of these processes not only enhances our understanding of
pulmonary diseases, but also enables investigation of many what-if scenarios.

To assess the veracity of alternative airways geometries in terms of their
impact on predictions of the exchange of O2 and CO2 with the blood, we
developed a spatially distributed model of unsteady transport of these gases
throughout the bronchial tree. This model advances the state of the art of
distributed models by accounting for the coupling between O2 and CO2 (the
oxyhemoglobin dissociation), and transport of these gases in the alveoli,
across the alveolar membrane, and in the blood circulating the capillaries. In
our model, gas exchange with the blood is not only controlled by the
ventilation to perfusion ratio (which is a purely mechanistic indicator), but
is also limited by the conditions in the blood as constrained by the blood
chemistry (dissociation curves), by diffusion-limited transport both in the
small vessels (characterized by small Reynolds and Péclet numbers) and at the
entrance to the alveoli, and by the gases concentrations at the inlets to the
trachea and the capillaries (boundary conditions). The utility of our model
extends beyond assessment of _in vivo_ networks to analysis and design of _in
silico_ networks.

We posit that a physically sound representation of the airways geometry should
meet topological, mechanical, and gas transport observables. Any geometric
representation of the tracheobronchial airway morphometry should conform to
high-fidelity observable geometric quantities, such as the dimensions of the
first few generations, dead space volume, and path length. Such a
representation should also reproduce high-fidelity observables that
characterize the mechanical behavior such as resistance and inductance. The
associated lung mechanics model must also reproduce high-fidelity observables,
including the tidal volume under normal breathing conditions. The degree to
which alternative airways geometries reproduce these observables form a basis
of a multi-faceted assessment tool to discriminate between them. Specifically,
a proper representation of the airways geometry must ensure that the predicted
dynamics of oxygenation and carbon dioxide removal in the blood agrees with
physiologically significant observables, such as time to reach oxygen
saturation in blood and mean partial pressure of O2 and CO2. We found that
overestimation of the lengths of the bronchial generations wherein diffusion
is dominant underpredicts mass fluxes at the alveolar inlet. This damping
effect results in smaller fluctuation amplitudes and in overall reduction in
gas exchange. We also found the area of the airways inlet to alveolus
(terminal bronchiole diameter) to play a key role in predictions of the gases
exchange. Thus extra care must be taken when reporting these dimensions.

A suitable computational and/or experimental model that employs a physically
sound representation of the airways dimensions can be ultimately used to
design patient customized ventilation modalities, where the model is
periodically updated with in vivo observations. We should not, however, ignore
the fact the computational and/or experimental methods that incorporate the
airways geometry used should be sufficiently accurate in a way that is
compatible with the objectives and scope of a particular study.

## Conflict of Interest Statement

The authors have no conflict of interest.

## Acknowledgements

IL was supported by the American University of Beirut Research Board under
award number 103371. LI funded by the Lebanese American University Research
Fund under award number SRDC-r-2017-21. DMT was supported in part by National
Science Foundation under award number DMS-1802189.

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## Figure Legends

Figure 1.

Dependence of the length (left column) and diameter (right column), normalized
with the corresponding length and diameter of the trachea, on the generation
order $n$, in human adults (top row) and infants (bottom row).

Figure 2.

Model workflow showing connections between constituent sub-models, as well as
data streams / model inputs.

Figure 3.

Alternative circuit models of gas flow in the bronchial tree: a general model
(a) and its successive simplifications (b–d).

Figure 4.

Dependence of functional residual capacity FRC (left) and dead space DS
(right) on body height, which serves as a proxy for age. The correlation
between FRC and lung volume/age reported in [74] were used to convert total
lung volume into FRC. When needed, growth charts were used to express height
as function of age. [a]: Engstom et al. (1956), [b]: Cook and Hamann (1961),
[c]: Engstrom et al. (1962), [d]: Hart et al. (1963), [e]: Doershuk et al.
(1970), [f]: Wood et al. (1971), [g]: Hatch and Taylor (1976), [h]: Stocks and
Godfrey (1977), [i]: Greenough et al. (1986), [j]: Roberts et al. (1991), [k]:
Tepper et al. (1993), [l]: Merth et al. (1995), [m]: Stocks and Quanjer
(1995), [n]: McCoy et al. (1995), [o]: Neder et al. (1999), [p]: Castile et
al. (2000), [q]: Hulskamp et al. (2003), [r]: Hislop et al. (1986), [s]: Angus
and Thurlbeck (1972), [t]: Davies and Reid (1970), [u]: Auld et al. (1963),
[v]: Hart et al. (1963). $\star$: Stocks and Quanjer (1995) using Tepper et
al. (1993); Hanrahan et al. (1990); Merth et al. (1995); Taussig et al.
(1977); Greenough et al. (1986).

Figure 5.

Temporal variation of the pressure signal $\Delta p(t)$ (the model’s input)
and the lung volume $V_{\text{lu}}(t)$ (the model’s output) under normal
breathing conditions for an adult.

Figure 6.

Each terminal branch is connected to a number of alveoli. Our model assumes
that the bronchiole is connected to the center of each compliant spherical
alveolus.

Figure 7.

Dependence of the number of alveoli, $N_{\text{al}}$, on age of adults (left)
and infants (right).

Figure 8.

A schematic of the mesh used in the finite volume method for a branch in
generation $i$ and the two daughter branches in generation $i+1$.

Figure 9.

Temporal variation of the mean partial pressure of oxygen (left) and carbon
dioxide (right) in the blood and in the alveolar space of adults, as predicted
with our model for two bronchial tree representations.

Figure 10.

Temporal variation of the mean partial pressure of oxygen (left) and carbon
dioxide (right) in blood capillaries of adults (top) and infants (bottom), as
predicted with our model for two bronchial tree representations.