Fig 1.
Schematic view of peripheral lung tissue and model geometry.
From the left lower pulmonary lobe of the human lungs (a), a lung tissue segment with neighboring alveoli (b) is enlarged: the center alveolus is shown as the cross-section of a rhombic dodecahedron or Wigner-Seitz cell, with thin films as tissue walls (see main text for details and [15]). (c) Schematic cross section of a single alveolus in spherical form with alveolar radius RA, radius of the dephasing volume R and a set of spherical coordinates (r, θ, ϕ).
Fig 2.
Eigenvalues and expansion coefficients.
(a) Lowest eigenvalues, obtained from Eq (15), as a function of surface permeability ρ. The red arrow marks the typical surface permeability for peripheral lung tissue ρL ≈ 0.6 [43] (RA = 200 μm [40], D = 2.3 ⋅ 10−9 m2s−1 [41], η = 0.85 [42]). The ρ-values of the decisive decrease of the lowest eigenvalue are several orders of magnitude lower than ρL. (b) Eigenvalue spectrum for n ≥ 1 for the same parameters as in (a). The eigenvalues remain constant over the range of surface permeabilities ρ, thus, the assumption of absorbing boundary conditions ρ ≈ ρL imposes no significant constraint on the remaining eigenvalue spectrum. (c) Eigenvalues κn for absorbing boundary conditions as a function of volume fraction η. In the limit η → 1, the first eigenvalue κ1 approaches (see Eq (33)). (d) Expansion coefficients Gn from Eq (9). For η → 1, the first expansion coefficient takes the value G1 ≈ 0.7 and it can be verified that
.
Fig 3.
CPMG inter-echo relaxation rate dispersion.
(a) Dependence of CPMG relaxation rate ΔR2 on the inter-echo time and volume fraction as obtained from Eq (20). (b) Values of τ180/τ at the inflection points of the ΔR2 relaxation rate curve for different regional blood volumes fractions η. For η = 0.8, the inflection point possesses a value of τ180/τ = 0.001.
Fig 4.
Model CPMG relaxation rate as a function of inter-echo time τ180.
(a) Sketch of passively deflated lung tissue, modified from [23]. Air filled spaces or alveoli for passively deflated lung tissue are less numerous and prominent than in non-deflated lung tissue. (b) Relaxation rate R2 for passively deflated lung tissue (continuous line) in comparison with experimental data [23]. The analytical model is fitted to the experimental data points, with resulting fitted values of characteristic time τ = 0.56 ± 0.22 s (p = 0.088) and intrinsic relaxation rate R2,0 = 12.58 ± 0.96 s−1 (p = 9.72 ⋅ 10−4). With the use of Eq (5), the mean alveolar radius follows as RA = 31.46 ± 13.15 μm, which is in very good agreement with the expected value of ∼34 μm [41, 51]. (c) Model mean alveolar radius RA for different air volume fractions η (error bars represent the standard error of RA from the model fit; p-values never exceeded 0.088). Naturally, the mean alveolar radius increases with increasing air volume fraction and reaches a value of RA = 70.12 ± 28.04 μm for η = 0.85.
Fig 5.
Relaxation rate dispersion and quantification of mean air bubble radius for ageing hydrogel foam.
(a) Several experimental R2 values (symbols) measured at different imaging times of ageing hydrogel foam with a 0.5 T benchtop relaxometer [19] was used to fit the analytical model with R2 from Eq (23) and respective spectral parameters as determined above (solid lines; for further details, please see main text). Fit parameters for characteristic time τ and intrinsic relaxation rate R2,0 can be found in Table 1. (b) Mean air bubble radius as obtained through Eq (5) from the different values for τ of the fitted model. These values are compared to values obtained by triangulating μCT images of voxel size 19.4 μm of the same foam cross-sections that served to acquire the R2 dispersion curves [19]. In addition, the time evolution of radii by random walk simulations is shown as performed in [19]. The continuous lines are fits of second order polynomials to the data. The mean relative error of the model and random walk simulation data points to the fit curve of μCT data is 5.84 ± 1.28% and 14.36 ± 2.66%, respectively.
Table 1.
Fit parameters of intrinsic relaxation rate R2,0 and characteristic time τ in Fig 5a.
Fig 6.
Sensitivity analysis for varying relaxation rates.
(a) All measured relaxation rates for the 3.5h experimental data (Fig 5a) were varied within different ranges
with
. For multiple sets of such variations, the average of the resulting difference to and in proportion of the initially obtained radius RA is negligible for relative ranges < 0.01. (b) Scatter plot of the resulting radii vs. deviations for the example
for different strengths of variation; all other
values were also varied within their respective error ranges as in (a).