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 St and the time-variable cross-section S(x,t) and volume Vlb(,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 dlim, a so-called terminal pipe is reached, and different model (the trumpet lobule) is used to represent smaller airways.
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 Rij and for the trumpet lobule Rlb and compliance elements which relate the trans-lobular pressure to the volume of the trumpet lobule Vlb(t) (see Eq 7).
Fig 3.
Pressure-volume curve for trumpet lobule.
Lobular volume Vlb in function of the elastic pressure pel 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).
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 sIII 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.
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.
Table 1.
Parameter settings of the baseline model.
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 sIII 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.
Fig 7.
Phase III slopes sIII 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 sIII indicated for each breath (a). In (b) the values for the normalized phase III slopes for each breath are shown.
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 sIII 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.
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 sIII for regional and local type modifications of the lobular compliance.
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.
Table 2.
Demographic characteristics and lung volumes for four MBW tests from healthy control adolescents.
Table 3.
Multiple-breath washout (MBW) outcomes measured in reality and simulated in the lung model for four MBW tests from healthy control adolescents.