Ethics statement

The study was approved by the Ethics Committee of the Canton of Bern, Switzerland (KEK-Gesuchs-Nr: 181/03), and caregivers gave written informed consent.

Lung morphology

In our model, the airway morphology is represented by a generic dichotomous tree network of straight branching pipes [30], which terminates in trumpet-like compartments. The straight pipes represent the larger, non-compliant airways, where convective transport is dominant. The dimensions of the trachea and the relative dimensions (i.e. with respect to the trachea) of the airways in the first four generations are defined according to anatomical data from Weibel [26]. In addition, the entire airway network of pipes (all conducting airways including the trachea, without the trumpet-like compartments) was then scaled to meet a specific functional residual capacity (FRC) that is, the air volume that remains in the lung after tidal expiration. The length of the scaled trachea was defined as (Eq 1) where the subscript W indicates a reference quantity [26].

For airways past the 4^th generation, we applied the scheme for a regular branching asymmetry introduced by Majumdar et al. [25, 30]. In this scheme, each pipe-like airway bifurcates in a major and a minor daughter airway. In the airway network every parent, minor, and major daughter share a common node. The dimensions of the daughter airways, namely their diameter and length, are different fractions of the dimensions of their common parent pipe, (Eq 2) where d_z is the diameter of a pipe at generation z, and r and η denote the asymmetry parameter and reduction rate, respectively. The same scheme was used to define the airway lengths. Majumdar et al. proposed values r = 0.326 and η = 2.97 for which the resulting structure best represents the human lung on a statistical basis [25]. In the present model, this bifurcation scheme was applied recursively until the diameter of a pipe was less than a lower limit diameter d_lim = 1.8 mm. This limit defines the terminal pipes in the model and was chosen to represent bronchi at the 8^th-12^th generation. The generation of the individual terminal pipe-like airways depends on the limit diameter and varies within the model, due to the asymmetric bifurcation scheme. This limit diameter was chosen in order to complete the simulations with reasonable computational costs. A numerical experiment showed very small differences in the simulation outcomes for d_lim = 1.6,1.8,2.0 mm.

The computational unit distal to a terminal pipe constitutes a lobule and was modelled using a trumpet-like compartment (trumpet lobule). Please note that the “lobule” as defined here differs from the anatomical term lobule as defined by Miller [31], because it contains, apart from the acini that include respiratory bronchioles, alveolar ducts and alveoli, also the generations of conducting airways with diameter smaller than d_lim. Unlike the pipes, the trumpet lobule is a compartment with diverging, time-variable cross-section (Fig 1A). The number of these trumpet lobules is equal to the number of terminal pipes, as each terminal pipe leads to one lobule.

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Fig 1. Trumpet lobule geometry and lung model morphology.

(a) Schematic representation of the trumpet model used for the lobules with the cross-section of the terminal pipe S_t and the time-variable cross-section S(x,t) and volume V_lb(,t) of a trumpet lobule model. (b) Idealized lung model morphology based on bifurcation rule (Eq 2). For better visibility of the airway structure, all diameters were scaled with the factor 0.1. The length and diameter ratio of airways until the fourth generation are irregular and follow from empirical data by Weibel [26]. After the fourth generation, airways bifurcate in a minor and major daughter airway according to Eq 2. From this scheme an asymmetric network of airways with different total lengths results. With each generation, airways get smaller, both in terms of length and of cross-section. In the lung model, when the diameter of the airways falls below a limit diameter d_lim, a so-called terminal pipe is reached, and different model (the trumpet lobule) is used to represent smaller airways.

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The residual volume of the trumpet lobules was defined such that the total volume of the model (pipes and trumpet lobules) at the end of a tidal expiration equaled a predefined FRC. Fig 1B shows a sketch of a model lung illustrating the relative scales of a network of airways defined by the bifurcation rule (Eq 2).

Mathematical description of the trumpet lobule

For the trumpet model representing the lobules and their peripheral airways, a model for the total cross-section of the trumpet lobule S_lb(x,t), as well as for the mean advection velocity u_lb(x,t) had to be derived.

The flow rate at the inlet of each trumpet equals the flow rate in a terminal pipe and Q_t(t) was known from the results of a model for lung ventilation (described in the upcoming section) and was used to compute the total volume of the trumpet .

The initial volume of the trumpet lobule, V_lb(t = 0) = V_lb⁰, followed from FRC-based scaling of the lung model.

Assuming a uniform homothety ratio of κ = 0.85 [19] for airways lumped in a trumpet lobule, the total change of cross-section along the streamwise coordinate x can be described as (Eq 3) where S_t is the cross-section of the terminal pipe, and z(x) is the generation at position x with respect to the inlet of the trumpet lobule where z = x = 0.

Considering l_t to be the length of the terminal pipe, the cumulative length at generation z (with respect to the inlet of the lobule) would be .

The limits of this sum for z→∞ are (Eq 4)

From these relations, an expression for the generation in function of the distance to the inlet of the trumpet can be computed, (Eq 5)

Using Eq 3 together with Eq 4 for further treatment of the lobule model becomes a rather cumbersome task. We therefore sought a model S_lb(x,t), which approximates Eq 5 but allows to derive an analytical expression for u_lb(x,t).

To this end, we used a power law of the form (Eq 6) with n₁, n₂ = 20, 2, where the coefficients p₁ and p₂ were defined such that the lobule cross-section S_lb intersects with Eq 3 at a chosen generation z*, and the prescribed initial volume V_lb⁰ of the trumpet lobule is obtained for a given length l_lb of the trumpet lobule. The mathematical expression for this parameter definition as well as a graphic comparison between the formulation Eq 3 and the model Eq 6 are provided in S1 Appendix (Section 1).

The expression given in Eq 6 determines the shape of the trumpet lobule at any time.

An important feature of this model is the major contribution of peripheral airway (where x is close to l_lb) to the overall expansion of the lung. Furthermore, the differentiation with respect to time of Eq 6 as well as the integration along the trumpet centre line coordinate x is straightforward and therefore allows to formulate a modified advection diffusion equation for the trumpet lobule (see S1 Appendix, Section 1)

Apart from the constraint on the total volume, the geometrical and mechanical properties of the stiff (pipes) and the compliant parts (trumpet lobules) can be modified individually to study systematically the effects of structural lung inhomogeneity.

Ventilation

Lung ventilation was simulated by a lumped parameter (0-dimensional) model (Fig 2) based on the model morphology described above. For the pipe-like airways, purely resistive elements were used. Womersley's theory for pulsatile flow in tubes [32] was applied to account for inertial effects during normal breathing, which have a considerable effect on the flow resistance in bigger airways until the fifth generation [33] (see also S1 Appendix). In general, the pressure difference between two subsequent nodes with index i and j in the network of pipe-like airways reads p_i−p_j = R_ijQ_ij. Here Q_ij is the flow rate from node i to node j, and the hydrodynamic resistance R_ij depends on the radius r_ij and the length l_ij of the conducting airway between two nodes, and on the breath period T_B. More information on the pressure-flow relation can be found in S1 Appendix.

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Fig 2. Ventilation lumped parameter model schematics.

(a) Network of pipe- and trumpet-like elements used for the upper and lower airways, respectively. (b) Corresponding lumped parameter model composed of resistances for the conducting airways R_ij and for the trumpet lobule R_lb and compliance elements which relate the trans-lobular pressure to the volume of the trumpet lobule V_lb(t) (see Eq 7).

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The trumpet lobule model, mainly representing compliant airways, is composed of a nonlinear compliance element (elastic pressure, p_el) and a resistance element (viscous pressure loss, p_diss), acting in series between a node i, corresponding to a terminal pipe, and the pleural gap with pressure p_pl. This representation is based on the pressure-volume relation as presented by Bates [34]. However, instead of the regular linear elastic law, we used an empirical exponential expression to represent p_el. The corresponding pressure difference was defined as (Eq 7) where R_lb, and Q_lb denote the total flow resistance of the lobule, its volume and the flow rate into the trumpet lobule, respectively; is the lobule volume at FRC and is the dynamic volume during breathing [34]. The shape parameters β and γ were used to define and modify the degree of the compliance parameter φ (i.e. the distensibility of the lobule, described below in details) and the non-linearity (in the pressure-volume relation) of the trumpet lobule, respectively. There is a single β-γ pair for each lobule, and both are defined indirectly by fixed intersection points in the pressure-volume curve for the elastic pressure (see Fig 3A). More information on the mechanical properties of the model can be found in the Section “Structural and mechanical modifications of the model”.

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Fig 3. Pressure-volume curve for trumpet lobule.

Lobular volume V_lb in function of the elastic pressure p_el as defined in Eq 3. The purple area indicates the reference pleural pressure range and the orange area indicates the nominal lobule volume range . In panel (a) the influence of the compliance modification parameter ϕ∈[0.5,1.5] is shown. For lobules with normal compliance (ϕ = 1, solid line) the elastic pressure curve intersects in the point . In panel (b) the effect of the non-linear parameter γ is shown. The parameter was chosen such that the elastic pressure curve intersects with the point (black dot). Note that ϕ = 1 was used for all graphs in panel (b).

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To ensure mass conservation, the flow rates in and out of each node i satisfy the balance ∑Q = 0. At the inlet, i.e. upstream node of the trachea, the flow rate is defined by the predefined boundary conditions Q_in(t), and for the trumpet lobule the flow rate of the terminal pipe Q_t equals the change of volume of the lobule, (Eq 8) This results in a system of differential algebraic equations (DAE). Together with appropriate boundary conditions, the DAE system governs the flow distribution in the lumped parameter model. A detailed description of the numerical implementation of the lumped parameter model is provided in S1 Appendix.

During inspiration, gas is transported from the mouth and the nose toward the alveolar membrane, as a result of the motion of the diaphragm and the thoracic cavity, causing a volume increase in the lung and a pressure decrease in the pleural gap. This negative (relative) pressure in the pleural gap causes a pressure gradient across the peripheral lung tissue and along the airways to the mouth and nose. Therefore, the pleural pressure and the pressure at the mouth would be obvious choices for the boundary conditions for the lumped parameter model (LPM). However, the pleural pressure is in general not measurable during clinical routine. Instead, the pleural pressure was determined computationally such that a prescribed flow rate Q_in(t) was attained in the trachea. This flow rate could be easily determined during clinical testing.

Gas transport

Gas transport in the lung occurs by convection and diffusion [35, 36]. To model the distribution of MBW tracer gases in the lung, typically N₂ or sulfur-hexafluoride [1], a one-dimensional advection-diffusion transport equation for a scalar variable representing the normalized gas concentration c = c(x,t)∈[0,1] was solved with a finite difference method (detailed description in S1 Appendix). Simulation of MBW involved the computation of gas concentration in all model airways over multiple breath periods. We considered only inert gases for which the dominant transport mechanisms are advection by the carrier gas (ambient air) and diffusion. The transport process was modelled along the centerlines of each pipe-like airway and along the axis of each lobule trumpet. The 1D approximation entails consideration of averaged (lumped) quantities within a cross-section, or in case of the trumpet model, within all airways represented by the cross-section of the trumpet at a given axial position. Advection-diffusion processes in a pipe-like geometry are subject to high radial velocity gradients, which can lead to radial concentration gradients such that the average effective diffusivity within a cross-section increases. In the human lung, these phenomena can take place at different scales: In the upper airway, where the Reynolds number Re_d (based on the airway diameter d) is about 10'000, turbulent flow strongly enhances mixing. In smaller airways (Re≪2000), Poiseuille flow with a parabolic velocity profile can be assumed. Enhanced diffusion due to high velocity gradients was modelled based on the concept of Taylor dispersion, where the local diffusion coefficient is a function of the local Peclet number [37], (Eq 9) where and D are the effective and molecular diffusion coefficients, respectively, and Pe = ud/D is the Peclet number defined with the local mean velocity u and the local airway diameter d. This is an approximate measure. Taylor dispersion is a concept, which strictly applies only to developed flow in straight tubes. Hence, the modification of the diffusion coefficient (Eq 9) does not account for local changes in diffusivity due to secondary flow phenomena occurring at bifurcations and curved airways, and likely underestimates the level of diffusion enhancement.

The advection-diffusion equation for gas transport was solved separately in each airway, applying interface conditions at the bifurcation nodes to couple the transport between different airways. We considered a transport equation of the following general form (Eq 10) with the flux (Eq 11) Here, we distinguished between the total cross-section S and the cross-section S_ad in which advection with the carrier gas velocity u(x,t) takes place. This assumption is important, because the airway geometry becomes increasingly complex (i.e. non-tubular, alveolar ducts and alveolar trees) for smaller airways, and tracer gas advection does not occur in the entire lumen [35, 38, 39]. At bifurcations, the concentration flux was conserved by enforcing ∑F = 0. In S1 Appendix, a model for the trumpet lobule is derived which allows a more specific form of Eq 10 to be stated for the gas transport within trumpet lobules with non-constant cross-section. In addition, the finite difference scheme for spatial derivatives and time-integration method used for the numerical solution of Eq 10 are explained.

Structural and mechanical modifications of the model

The proposed multi-scale model was designed to study the effects of functional and structural inhomogeneities of the peripheral airways on the N₂ washout procedure. To illustrate the capabilities of the model, several parameters of the trumpet lobules were systematically modified. The parameters were defined in a way that their effect on lung geometry and mechanics was physically meaningful and intuitively clear:

1. Lobule Compliance: Lung compliance, as a measure of distensibility of the lung, is not uniform within the healthy lung in vivo [40]. Moreover, it is found impaired in many lung diseases, e.g. increased in emphysema, and decreased in pulmonary fibrosis [41–43]. We tuned tissue compliance with the parameter ϕ, which is a non-dimensional factor that has a stiffening or softening effect on the elastic pressure. In the following the meaning of ϕ is explained in more detail: We considered a mean tidal volume per lobule, where TV is the tidal volume, and N_lb is the total number of trumpet lobules. We then defined the nominal lobule volume range . Furthermore, we considered a reference pressure amplitude Pa, which corresponds to a typical pressure range due to pleural pressure changes [44, 45]. For a given pressure load , the elastic component of the lobule mechanics can be considered stiff (reduced compliance) if its maximum dynamic volume is smaller than (ϕ<1), and soft (increased compliance, ϕ>1) otherwise. Fig 3A shows the elastic pressure curve (Eq 7) for a lobule for different values of ϕ, which effectively modify the curve via the shape parameter β as (Eq 12) The remaining shape parameter γ accounts for the non-linearity of the elastic pressure curve (Fig 3B). It was computed numerically for all the lobules (using a minimization approach) such that the elastic pressure curve intersects with the point in the volume-pressure plane. This point was chosen because the resulting curve qualitatively lies between a linear (γ→0) and an almost right-angled curve (γ→∞).
2. Lobule Volume: Volume differences between the respiratory units (i.e. the anatomical lobules) but also within each unit have been described in healthy lungs [6, 46] and are more pronounced in obstructive lung disease [47]. We used the modification parameter θ to modify the residual volume of a trumpet lobule as , such that θ<1 resulted in reduced residual trumpet lobule size.
3. Lobule Resistance: Differences in airway resistance [34] between separate airspaces are known to affect the washout process [48]. Such differences were introduced via a modification of the pressure loss (see Eq 7) in the trumpet lobule p_diss,mod = τR_lbQ_lb, where the modification parameter τ>1 directly modifies the lobular flow resistance R_lb. This can be used to model the obstruction of airways in a lobule, e.g. due to mucus plugging.

Comparison of model simulations with in vivo MBW measurements in healthy children

The aim of the comparison of model results with in vivo data was to demonstrate that the model can relate microscale modifications at the lobular level to clinically observed MBW metrics (MBW washout envelope and phase III slope analysis [1]).

We used data from healthy adolescents recruited for lung function studies in the Inselspital Children’s University Hospital, Bern, Switzerland. N₂MBW measurements (N = 4) were collected according to the recent ERS/ATS consensus guidelines [1] using the ultrasonic flowmeter (Exhalyzer D, Eco Medics AG, Duernten, Switzerland) and the software provided by the manufacturer (Spiroware 3.1.6) as previously described [49]. During the test, the subject sat in an upright position wearing a nose-clip and was asked to breathe regularly through a snorkel-like mouthpiece connected to a bacterial filter and a dead space reducer.

Baseline configuration vs. modified lobular compliance

A baseline simulation has been performed with the parameter settings listed in Table 1, but without any added asymmetries with respect to lobular compliance, lobular residual size, or lobular resistance. We compared this reference simulation with results for modified lobular compliance where two regions (each accounting for 25% of all trumpet lobules) were modified using ϕ = 0.5 and ϕ = 1.5, respectively. Fig 4 shows the results for both simulations (baseline and compliance modifications). The washout curve (Fig 4C and 4D) can be analyzed in different ways: Each expiration starts with zero N₂ concentration (phase I), which corresponds to the washout of the dead space (the pipes in the model), and then a rise in N₂ concentration, which is first very quick (phase II) and later slow (phase III) [50–52]. Phase III is the part of the concentration-expired volume curve that corresponds to 50% to 95% of the expired volume per breath [1] (Fig 5). The end-expiratory N₂ concentration diminishes from one breath to the next as the N₂ washout progresses. The ratio of two subsequent end-expiratory N₂ concentration values, i.e. the decay of the washout curve envelope, provides information about the gas washout efficiency of the whole lung, and indirectly about the ventilation of lung compartments. For the baseline model configuration, this exponential decay is uniform (linear envelope graph in a semi-logarithmic plot over time, in Fig 4D). In the model with regions of variable lobular compliance, the decay is non-uniform and delays towards the end. This can be associated with a non-uniform ventilation in different regions of the lung. To quantify these washout properties, an exponential function of the following form was fitted to the envelope of end-expiratory values. (Eq 13) Here, α₁ and α₂ (with α₁,α₂>0) are the decay rates of two superimposed processes, one fast and one slow [53], and A∈[0,1] represents the relative weight of the slow decay process (α₁<α₂). The decay of the process is considered uniform if A = 1 and non-uniform if A<1. Changing the compliance properties as described above resulted in A = 0.44, α₁ = 0.018, and α₂ = 0.045, compared to A = 1.0 and α₁ = 0.031 for the baseline.

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Fig 4. Overview of different simulation results for baseline model configuration and modified compliance.

Results of the simulated N₂ gas washout for the baseline configuration (uniform and constant lung model parameter, black) and for altered trumpet lobule compliance using ϕ = 0.5, 1.5 in two different regions, respectively, each accounting for 25% of all trumpet lobules (purple color). Normalized phase III slopes s_III are shown for the first (a) and last (b) breath. The washout profile (N₂ concentration at the entrance of the trachea) in (c) linear scale and (d) logarithmic scale for 50 simulated breaths with a uniform concentration decay for the baseline model, and a non-uniform decay for the regionally modified model (slow-fast washout profile). Panels (e) and (f) illustrate further the model with modified lobular compliance at the end of the fifth breath: (e) spatial N₂ concentration distribution where the trumpet lobules parametrized with lower and higher compliance are located on the left and right side of the airway tree, respectively. In (f) the mean lobular N₂ concentration is shown for each trumpet lobule.

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Fig 5. Schematic presentation of the N₂ concentration-volume curve of a single breath.

Expired N₂ concentration is expressed as % of the initial N₂ concentration. Phase III is defined between 50% and 95% of the expired volume. Slope III: the slope of the concentration-volume curve during phase III.

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Table 1. Parameter settings of the baseline model.

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The spatial distribution within the lung model after the fifth breath (Fig 4E and 4F) provides further insight to the non-uniform (i.e. heterogeneous) washout process. The concentration remains high in lobules with decreased compliance (ϕ = 0.5) and is reduced more rapidly in those with increased compliance (ϕ = 1.5). Later in the washout process, the relative concentration difference between the two regions is therefore higher than in the baseline configuration and the washout slows down due to the increasingly dominant contributions of the slowly washed-out units.

The concentration profile can be further analyzed and interpreted on a per-breath basis. The slope of the concentration-volume curve during phase III (slope III) indicates whether the gas mixtures from different lung regions have different gas concentrations [35, 50] (Fig 5). Therefore, slope III has been linked to inhomogeneous gas transport dynamics, and parameters derived from this analysis are of increasing clinical relevance [54–56]. To quantify the phase III profile in our study, the parameter s_III is defined, which is the slope of a linear function fitted to the N₂ concentration values corresponding to the phase III, normalized with the mean N₂ concentration during phase III (Fig 5): (Eq 14) Comparing the baseline washout with the washout of modified compliance, s_III was higher (steeper slope) in the modified lung already for the first breath (Fig 4A), and this difference in s_III increased further until the last breath (Fig 4B).

Outcome parameters for comparison

For the comparison of other types of lung model modifications that represent a specific kind of structural and functional lung asymmetry, we used the weight parameter A and the decay rate α₁ (and α₂) from the fitting function (Eq 9) approximating the washout envelope, as well as the clinically important parameter s_III (Eq 14).

Other types of regional lobular modifications

In separate simulations, the residual volume of trumpet lobules was altered using θ = 0.5 for a subset of lobules accounting for 25% of all trumpet lobules. Note that the model is designed such that it scales the residual volume of the remaining 75% of lobules automatically to preserve the prescribed FRC of the lung. In another simulation, the resistance was increased (τ = 8) in 25% of the lobules, which corresponds to a local reduction of airway diameters by approximately 40%.

Fig 6 shows the simulation results for these three modified cases together with the baseline configuration. Note that the baseline results (black line in Fig 6) are barely visible, because some of the simulations in the model with increased resistance yielded very similar results. The compliance modifications showed the strongest effect and yielded a non-uniform, delayed washout (Fig 6C and 6D). While the resistance modification did not lead to any notable difference from the baseline result (A = 1.0), heterogeneous residual lobular volume also yielded a non-uniform washout (A = 0.84) similar but weaker than the model with modified compliance. Interestingly, the washout was delayed in later breaths in the case of modified residual lobular volume, and of compliance modifications. The modified residual volume caused in addition a faster decay for early breaths (Fig 6C before t = 60 s). This was also reflected in the slow (α₁ = 0.028) and fast (α₂ = 0.074) decay rates, e.g. when compared to the decay rate for the baseline configuration (α₁ = 0.031). A reason for this could be that in smaller and shorter lobules, the flow of pure oxygen replaces a larger fraction of N₂ and thus the washout per breath is more efficient (faster). Although the residual lobule size is different, a similar flow rate of pure oxygen is preserved in all lobules as long as the compliance properties and the pleural pressure distribution remain uniform.

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Fig 6. N₂ washout for baseline model configuration and modified trumpet lobule properties.

Results of the simulated N₂MBW for the baseline configuration (uniform and constant lung model parameters) and for three cases where different trumpet lobule parameters were modified regionally: 1) Lobular compliance was altered using ϕ = 0.5,1.5 in two regions (same as for results shown in Fig 4), 2) the lobular residual volume was decreased using θ = 0.5, 3) the hydrodynamic resistance was amplified by a factor of 8 for 25% of all lobules. Normalized phase III slopes s_III are shown for the first (a) and last (b) breath. (c, d) The washout (only the envelope of the N₂ end-expiratory concentrations (symbols) and the corresponding fitting functions (dashed line, Eq 13) are shown) for 50 simulated breaths.

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The comparison of normalized phase III slopes s_III also clearly discriminated the compliance modifications from the other cases. In a per breath analysis, a flat plateau-like profile resulted for the baseline, as well as for the cases with altered lobule size and resistance. For regionally altered lobule compliance, however, the gas concentration increased nearly linearly in phase III. This difference is already visible from the first breath, but becomes more prominent as the washout progresses (Fig 6A and 6B). For lobular resistance and volume modifications, the phase III concentration profile remained a flat plateau during the whole washout. In the case of modified lobule compliance s_III increased from breath to breath. These different trends in s_III are illustrated in Fig 7. Increased slopes s_III indicate that concentration differences at airway bifurcations are increasing in phase III, such that contribution from high concentration units becomes more and more dominant. This was only the case for compliance modifications, where unequal rates of pure oxygen feed into different lobules. In case of modified volume differences, the spatial concentration distribution was not uniform, but did not change during phase III. In the next section, the relation between spatial concentration distribution in the lung and the phase III concentration profile is further discussed.

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Fig 7. Phase III slopes s_III for baseline model configuration and modified trumpet lobule properties.

Simulated N₂ washout profile for baseline and different types of regional modifications with phase III slopes s_III indicated for each breath (a). In (b) the values for the normalized phase III slopes for each breath are shown.

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In summary, the compliance modifications had the strongest impact on gas washout in terms of non-uniform concentration decay and s_III values. A heterogeneity in residual lobule volume has a notable influence on the washout profile during early and later breaths, while the phase III concentration profile does not differ from the baseline. Compared to these two cases, the changes in hydrodynamic resistance have a negligible effect on gas washout profile. This is in accordance to a previous report [7].

Regional vs. local compliance modifications

To study the capabilities of the model in reflecting the effect of different spatial distributions of structural inhomogeneities, we look again at the example of compliance modifications. Instead of applying these modifications to a subset of neighboring lobules (regional distribution), we distributed the modifications over 50% of the lobules regularly distributed in the entire lung domain (local distribution). This distribution intends to mimic lung diseases such as cystic fibrosis that do not spread in a locally organized manner [57, 58]. To this end, we performed simulations where for every other lobule the compliance was alternately reduced (ϕ = 0.5) or increased (ϕ = 1.5).

Local and regional compliance modifications yielded similar results for the whole washout (Fig 8). Both caused a non-uniform decay with A = 0.52 (local) and A = 0.44 (regional). Minor differences were found in the slow and fast decay rates (α₁ = 0.020 vs. 0.018 and α₂ = 0.044 vs. 0.045 (local vs. regional)). However, the washout curve differed considerably with respect to phase III slope s_III (Fig 8A and 8B). Starting with only small differences in the first breath, s_III increased with every breath for regional modifications, while it decreased for local compliance modifications, reaching slightly negative values after approximately 35 simulated breaths (Fig 9D).

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Fig 8. N₂ washout for regional and local distribution of compliance modifications.

Results of the simulated N₂ gas washout for regional and local trumpet lobule compliance modifications. In both cases, the lobular compliance was altered using ϕ = 0.5, and ϕ = 1.5 each for 25% of all trumpet lobules. Normalized phase III slope s_III are shown for the first (a) and last (b) breath. The washout (c, d) for 50 simulated breaths, where only N₂ end-expiratory concentrations (symbols) and the corresponding fitting function (dashed line, Eq 13) are shown for better visibility.

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Fig 9. Spatial N₂ concentration distribution after the fifth breath for regional and local type compliance modifications.

Spatial concentration distributions for regional (a) and local (b) type modifications at the end of the fifth breath. In trumpet lobules with lowered compliance (ϕ = 0.5) the concentration remains high, whereas the lobules with increased compliance (ϕ = 1.5) are washed out more efficiently. In panel (c), the corresponding washout profile is shown. The different profiles in the phase III (slope vs. plateau) for regional and local type modifications correlate with the pattern of the concentrations gradients at airway bifurcations. Panel (d) shows the temporal evolution (trend) of the normalized phase III slopes s_III for regional and local type modifications of the lobular compliance.

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The spatial concentration distribution at the end of the fifth breath is depicted for both local and regional compliance modifications in Fig 9A and 9B. The lobular concentrations were clearly different for lobules with altered compliance properties. Trumpet lobules with increased compliance properties showed higher (normalized) N₂ concentration and vice versa for reduced compliance. Furthermore, the comparison of the concentration distributions for regional and local compliance heterogeneity suggest the following mechanisms governing the phase III slopes: 1) The mixing of gas due to concentration gradients at airway bifurcations occurred at different levels: For regional modifications, the mixing took place at several generations of large airways, starting from the main bronchi to smaller conducting airways, where gas from different regions comes together, causing an increasingly non-uniform concentration distribution within the network. For local modifications, the mixing took place at already small airways and only over a few generations immediately above the lobules, which leads to a more uniform concentration distribution in the large airways. 2) Spatial concentration differences throughout the airway tree increased over time for regional modifications, while they remained approximately constant for local modifications (see S1 Video and S2 Video). This may be an explanation for the increasingly different phase III slopes (Fig 9C and 9D).

Model for the healthy lung

In order to mimic the small degree of inhomogeneous ventilation that exists physiologically in the healthy lungs [6], we used a combination of regional modifications of lobular size (θ = 0.5 in two regions each accounting for 12.5% of all trumpet lobules) and lobular compliance (ϕ = 0.5,ϕ = 1.5 in two regions respectively, each accounting for 12.5% of all trumpet lobules) to approximate normal structural and mechanical inhomogeneities of the lung. The modifications parameters were determined through an iterative process: First, the total set of lobules was split in sub-regions (each 12.5%) accounting for modified lobular compliance and size. From the previously conducted modification parameter study it was known that a region with smaller lobule volume (θ<1) would account for faster washout during early breaths, while a region of less compliant lobule (ϕ<1) would cause the washout to be delayed (less efficient) towards later breaths. In a first step, the modification parameter θ was altered until the washout envelope of the measured and the simulated N₂MBW curve were in agreement for the first 5–10 breaths (qualitatively, by visual inspection). Subsequently, the modification parameter ϕ was altered, which did not change the results from the first fitting step, until the washout envelopes also approximately matched for the last 30–50 breaths. In a MBW test simulation using this model, the lung clearance index (LCI) [1] was with 6.1 in the normal range reported in the literature [56]. S3 Video (supporting material) visualizes the evolution of the spatial concentration distribution over 50 breaths in such a model for a sinusoidal inlet flow profile. It shows the asymmetric concentration distribution due to regional differences in lobule size and lobule compliance, and how these differences evolve over time.

Comparison between simulations and real MBW measurements

Demographics about the healthy subjects and lung volumes for each test simulation are provided in Table 2. For each simulation, we used the model for the healthy lung and applied the flow rate profile and the FRC as measured during the MBW test. The FRC was used to scale each lung model, using Eq 1 with the reference value FRC_W = 3.5 l [26]. Fig 10 and Table 3 show measured concentration curves and MBW outcomes, respectively, for N₂MBW measured in healthy subjects together with corresponding simulations.

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Fig 10. Measurements and simulation of N₂MBW in healthy controls.

Simulated N₂ washout compared to data from N₂MBW tests from four healthy subjects (measurements (a)—(d)). For the simulations, the inlet flow profile and the FRC as measured during the N₂ washout test were used. To match the washout envelope in the measured data, both lobular residual volume and lobular compliance were partially modified in the lung model to mimic normal lung heterogeneity.

https://doi.org/10.1371/journal.pcbi.1007079.g010

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Table 2. Demographic characteristics and lung volumes for four MBW tests from healthy control adolescents.

https://doi.org/10.1371/journal.pcbi.1007079.t002

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Table 3. Multiple-breath washout (MBW) outcomes measured in reality and simulated in the lung model for four MBW tests from healthy control adolescents.

https://doi.org/10.1371/journal.pcbi.1007079.t003

Overall, the simulated washout envelopes for the four cases were in good agreement with the experimental data. On a breath-per-breath basis, differences were more prominent. For example, end-expiratory concentration values were moderately different for several breaths. A higher variability was found during phase III. Possible reasons for this could be: the complex structural and mechanical asymmetries in the healthy lung, which are not sufficiently modelled with the types of inhomogeneities used in this study (lobular compliance, residual size, and resistance); the non-uniform breathing pattern of the healthy subjects; and measurement errors. A detailed explanation or justification of these discrepancies was beyond the scope of this study.

Limitations of the model

The presented model has several limitations. For the sake of simplicity, several physiological and anatomical features were either idealized or neglected. The trumpet lobules include the last generations of the conducting airways, in order to keep the computational costs reasonable. We acknowledge that the acinar geometric properties are not entirely simulated in the lobule and that parameters like the homothety ratio are, in reality, not uniform throughout the airways contained in the lobule. It was beyond the scope of this study to investigate the effects of different cross-sectional development in the acinar region on the MBW washout curve. Although the dimensions of the model (airway diameter and length per generation, FRC, lobular size distribution) were derived from anatomical data, the overall geometry of the model is rather generic and not anatomically based. Spatial asymmetries between the right and the left lung were not modelled. The FRC-dependent scaling of the modeled airway tree allows using it for MBW simulations in a large FRC range, however anatomical differences between children and adults were not introduced in the model.

Pressure losses due to turbulent flow in the big airways and flow changes at the bifurcations and/or the heavily curved airways are not included in the model, as these phenomena require 3D anatomical data. However, the results in this and in previous studies [7] demonstrate that flow resistance plays a smaller role in ventilation inhomogeneity compared to differences related to the loading and constitution of respiratory units. Although there is a spatial pleural pressure gradient in real lungs, the pleural pressure was assumed to be spatially uniform, because, to our best knowledge, there is no realistic estimate for this gradient. We further assumed that the initial mechanical stress distribution is uniform between the lobules although variability should be expected in real lungs. It will be interesting to investigate the effect of this variability and effects due to gravity on future studies.

Next, regarding the model for the healthy lungs, we acknowledge that various other combinations of θ and ϕ, as well as introduction of other modifying parameters may lead to a similar MBW profile and outcomes. However, our aim was neither to mimic breathing characteristics on the individual level [59], nor to relate MBW outcomes to particular local anatomical characteristics, but to provide an example that resembles healthy lungs, based on principles of lung physiology. Finally, no specific model for the upper airways, the mouth, and the interface equipment has been used in the simulations. The additional pathway for gas transport introduced by these parts was represented by an elongation of the straight pipe representing the trachea.

Conclusion

We presented a multi-scale model of the whole lung that simulates the gas transport and washout in conducting and acinar airways, including non-linear tissue mechanics. In order to mimic a physiological degree of ventilation inhomogeneity as described in healthy lungs, we introduced modifications in mechanical and geometrical properties on a lobular level. This study demonstrates that regional and local alterations of airway properties have different effects on the expiratory phase III in the MBW. Phase III slope profiles were notably more pronounced and sensitive to the degree of modifications for regional type modifications compared with local type modifications. Furthermore, the study revealed the different functional relations between the MBW concentration curves and airway compliance, volume and flow resistance. Increased heterogeneity of lobular compliance and residual volume correlated with a delayed washout, while heterogeneous flow resistance had a negligible impact. Finally, the simulation results are in accordance with real MBW data obtained from healthy subjects, on a qualitative level.

The model can be used to study MBW characteristics in health and disease. It offers the opportunity to understand the ventilation distribution in the healthy lung, and to investigate more profoundly MBW features that extract localized information, like the slope III analysis. By applying modifications in mechanical properties that exceed the physiological limits, the model can also mimic certain patterns of lung disease. Thus, it can be used to study the effect of such diseases on MBW concentration curves. In addition, the model may also serve as a tool to visualize gas transport in the lung during a MBW test, which could support patient education.