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The authors have declared that no competing interests exist.

The multiple-breath washout (MBW) is a lung function test that measures the degree of ventilation inhomogeneity (VI). The test is used to identify small airway impairment in patients with lung diseases like cystic fibrosis. However, the physical and physiological factors that influence the test outcomes and differentiate health from disease are not well understood. Computational models have been used to better understand the interaction between anatomical structure and physiological properties of the lung, but none of them has dealt in depth with the tracer gas washout test in a whole. Thus, our aim was to create a lung model that simulates the entire MBW and investigate the role of lung morphology and tissue mechanics on the tracer gas washout procedure. To this end, we developed a multi-scale lung model to simulate the inert gas transport in airways of all size. We then applied systematically different modifications to geometrical and mechanical properties of the lung model (compliance, residual airway volume and flow resistance) which have been associated with VI. The modifications were applied to distinct parts of the model, and their effects on the gas distribution within the lung and on the gas concentration profile were assessed. We found that variability in compliance and residual volume of the airways, as well as the spatial distribution of this variability in the lung had a direct influence on gas distribution among airways and on the MBW pattern (washout duration, characteristic concentration profile during each expiration), while the effects of variable flow resistance were negligible. Based on these findings, it is possible to classify different types of inhomogeneities in the lung and relate them to specific features of the MBW pattern, which builds the basis for a more detailed association of lung function and structure.

Obstructive lung diseases, like cystic fibrosis or primary ciliary dyskinesia, lead to inhomogeneous ventilation. The degree of observed inhomogeneity represents a clinical measure for the progression of the disease. The multiple-breath washout (MBW) is a lung function test that measures this inhomogeneity in the lung. However, the factors that influence the results of the test and differentiate between health and disease are not well understood. Computational models help us to understand better the relation between anatomical structure and physiological properties of the lung, but none of them has dealt in depth with the MBW test in whole. Our aim was to create a lung model that simulates the entire MBW test and study the role of lung structure and tissue mechanics on the washout procedure. We developed a multi-scale lung model to simulate the inert gas transport in all airways including the gas exchange area. Our model offers the opportunity to understand the ventilation distribution in the healthy lung. It can also mimic certain patterns of lung disease by applying modifications in mechanical properties out of the physiological limits. Thus, it can be used to study MBW characteristics in health and disease.

The multiple-breath washout (MBW) is a lung function test that measures the degree of ventilation inhomogeneity [_{2}), the test is simplified, as no washin phase is needed. In the washout phase the subject inhales a gas other than the tracer gas (e.g. pure O_{2} in case of N_{2}MBW), so that the lungs wash out the tracer gas gradually by each expiration [

The anatomical diversity in the airways as well as the physiological properties of the lung tissue (e.g. compliance, resistance) influence the respiratory function in a complex way [

The aim of this article is to introduce a computational model that simulates gas transport in the entire lung during the MBW test, taking into account transport phenomena at different scales. It should allow relating physiological phenomena at the smallest scales of the lung to gas concentrations at the mouth, which can be measured clinically. Such simulations can provide detailed insight in the gas transport dynamics during the MBW in different airways and help to understand better the effect of lung morphology and tissue mechanics on tracer gas washout. To this end, structural data from human lungs have been used to construct a fractal lung model, taking into account morphological and physiological asymmetries in lung anatomy [

Such models inherently lack suitable validation methods [_{2}MBW) tests from healthy controls (N = 4). Although this comparison is too small to serve as a statistically solid validation of the model, the good agreement between computational results and in vivo data illustrates the potential of the proposed model to reproduce clinical test data.

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.

In our model, the airway morphology is represented by a generic dichotomous tree network of straight branching pipes [

For airways past the 4^{th} generation, we applied the scheme for a regular branching asymmetry introduced by Majumdar et al. [_{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 _{lim} = 1.8 ^{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 _{lim} = 1.6,1.8,2.0

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 [_{lim}. Unlike the pipes, the trumpet lobule is a compartment with diverging, time-variable cross-section (

(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 (_{lim}, a so-called terminal pipe is reached, and different model (the trumpet lobule) is used to represent smaller airways.

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.

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, _{lb}(_{lb}^{0}, followed from FRC-based scaling of the lung model.

Assuming a uniform homothety ratio of _{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

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

Using _{lb}(x,t), which approximates _{lb}(x,t).

To this end, we used a power law of the form
_{1}, n_{2} = 20, 2, where the coefficients p_{1} and p_{2} were defined such that the lobule cross-section S_{lb} intersects with _{lb}^{0} 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

The expression given in

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

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.

Lung ventilation was simulated by a lumped parameter (0-dimensional) model (_{i}−_{j} = _{ij}_{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

(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

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 [_{el}. The corresponding pressure difference was defined as
_{lb}, _{lb} denote the total flow resistance of the lobule, its volume and the flow rate into the trumpet lobule, respectively;

Lobular volume V_{lb} in function of the elastic pressure p_{el} as defined in

To ensure mass conservation, the flow rates in and out of each node i satisfy the balance ∑_{in}(_{t} equals the change of volume of the lobule,

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 _{in}(

Gas transport in the lung occurs by convection and diffusion [_{2} or sulfur-hexafluoride [_{d} (based on the airway diameter d) is about 10'000, turbulent flow strongly enhances mixing. In smaller airways (

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
_{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 [

The proposed multi-scale model was designed to study the effects of functional and structural inhomogeneities of the peripheral airways on the N_{2} 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:

Lobule Compliance: Lung compliance, as a measure of distensibility of the lung, is not uniform within the healthy lung in vivo [_{lb} is the total number of trumpet lobules. We then defined the nominal lobule volume range

Lobule Volume: Volume differences between the respiratory units (i.e. the anatomical lobules) but also within each unit have been described in healthy lungs [

Lobule Resistance: Differences in airway resistance [_{diss,mod} = _{lb}_{lb}, where the modification parameter _{lb}. This can be used to model the obstruction of airways in a lobule, e.g. due to mucus plugging.

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 [

We used data from healthy adolescents recruited for lung function studies in the Inselspital Children’s University Hospital, Bern, Switzerland. N_{2}MBW measurements (N = 4) were collected according to the recent ERS/ATS consensus guidelines [

We use the above described model of the whole lung to study the effects of structural and mechanical modifications on MBW outcomes.

A baseline simulation has been performed with the parameter settings listed in _{2} concentration (phase I), which corresponds to the washout of the dead space (the pipes in the model), and then a rise in N_{2} concentration, which is first very quick (phase II) and later slow (phase III) [_{2} concentration diminishes from one breath to the next as the N_{2} washout progresses. The ratio of two subsequent end-expiratory N_{2} 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 _{1} and _{2} (with _{1},_{2}>0) are the decay rates of two superimposed processes, one fast and one slow [_{1}<_{2}). The decay of the process is considered uniform if _{1} = 0.018, and _{2} = 0.045, compared to _{1} = 0.031 for the baseline.

Results of the simulated N_{2} 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 (N_{2} 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 N_{2} 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 N_{2} concentration is shown for each trumpet lobule.

Expired N_{2} concentration is expressed as % of the initial N_{2} 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.

Breathing profile | Tidal volume | TV | [l] | 0.5 |

Breath period | T_{B} |
[s] | 3.2 | |

Inlet flow profile | Sine function | |||

Length of simulation | 50 breaths | |||

Model settings | Functional residual capacity | FRC | [l] | 3.0 |

Limit diameter pipe airways | d_lim | [mm] | 1.8 | |

Number of lobules | [–] | 304 | ||

Pipe generations | 7–11 |

* The number of pipe generations is not uniform for all model lobules, due to the asymmetrical branching scheme.

The spatial distribution within the lung model after the fifth breath (

The concentration profile can be further analyzed and interpreted on a per-breath basis. The slope of the concentration-volume curve during phase III (_{III} is defined, which is the slope of a linear function fitted to the N_{2} concentration values corresponding to the phase III, normalized with the mean N_{2} concentration during phase III (_{III} was higher (steeper slope) in the modified lung already for the first breath (_{III} increased further until the last breath (

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 _{1} (and _{2}) from the fitting function (_{III} (

In separate simulations, the residual volume of trumpet lobules was altered using

_{1} = 0.028) and fast (_{2} = 0.074) decay rates, e.g. when compared to the decay rate for the baseline configuration (_{1} = 0.031). A reason for this could be that in smaller and shorter lobules, the flow of pure oxygen replaces a larger fraction of N_{2} 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.

Results of the simulated N_{2}MBW 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 _{III} are shown for the first (a) and last (b) breath. (c, d) The washout (only the envelope of the N_{2} end-expiratory concentrations (symbols) and the corresponding fitting functions (dashed line,

The comparison of normalized phase III slopes _{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 (_{III} increased from breath to breath. These different trends in _{III} are illustrated in _{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.

Simulated N_{2} 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.

In summary, the compliance modifications had the strongest impact on gas washout in terms of non-uniform concentration decay and _{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 [

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 [

Local and regional compliance modifications yielded similar results for the whole washout (_{1} = 0.020 vs. 0.018 and _{2} = 0.044 vs. 0.045 (local vs. regional)). However, the washout curve differed considerably with respect to phase III slope _{III} (_{III} increased with every breath for regional modifications, while it decreased for local compliance modifications, reaching slightly negative values after approximately 35 simulated breaths (

Results of the simulated N_{2} 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 N_{2} end-expiratory concentrations (symbols) and the corresponding fitting function (dashed line,

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.

The spatial concentration distribution at the end of the fifth breath is depicted for both local and regional compliance modifications in _{2} 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

In order to mimic the small degree of inhomogeneous ventilation that exists physiologically in the healthy lungs [_{2}MBW curve were in agreement for the first 5–10 breaths (qualitatively, by visual inspection). Subsequently, the modification parameter

Demographics about the healthy subjects and lung volumes for each test simulation are provided in _{W} = 3.5 l [_{2}MBW measured in healthy subjects together with corresponding simulations.

Simulated N_{2} washout compared to data from N_{2}MBW tests from four healthy subjects (measurements (a)—(d)). For the simulations, the inlet flow profile and the FRC as measured during the N_{2} 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.

MBW test | Subject | Age (years) | Weight (kg) | Height (cm) | sex | FRC (l) | VT (l) |
---|---|---|---|---|---|---|---|

#1 | A | 15.9 | 70.6 | 172.7 | F | 3.00 | 0.88 |

#2 | B | 17.9 | 50.0 | 169.5 | M | 2.85 | 0.67 |

#3 | C | 15.7 | 67.5 | 187.5 | M | 3.75 | 1.10 |

#4 | C | 15.7 | 67.5 | 187.5 | M | 3.52 | 0.63 |

M: male, F: female, FRC: functional residual capacity, VT: tidal volume. Of note, tests #3 and #4 were performed in sequence by the same subject (C).

MBW test | LCI (TO) measured | LCI (TO) simulated | LCI washout breaths measured | LCI washout breaths simulated |
---|---|---|---|---|

#1 | 6.78 | 6.34 | 27 | 24 |

#2 | 6.37 | 6.22 | 31 | 30 |

#3 | 6.98 | 6.68 | 25 | 23 |

#4 | 7.12 | 6.75 | 43 | 40 |

LCI: lung clearance index, TO: turnovers. LCI washout breaths: number of washout breaths needed to reach the LCI

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.

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 [

Next, regarding the model for the healthy lungs, we acknowledge that various other combinations of

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.

Same model modifications as in

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Same model modifications as in

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Same lung model modifications as for the model for the healthy lungs and the comparison with experimental data from healthy controls.

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