Dephasing and diffusion on the alveolar surface

We propose a surface model of spin dephasing in lung tissue that includes both susceptibility and diffusion effects to provide a closed-form solution of the Bloch-Torrey equation on the alveolar surface. The nonlocal susceptibility effects of the model are validated against numerical simulations of spin dephasing in a realistic lung tissue geometry acquired from synchotron-based $\mu \mathrm{CT}$ data sets of mouse lung tissue, and against simulations in the well-known Wigner-Seitz model geometry. The free induction decay is obtained in dependence on microscopic tissue parameters and agrees very well with in vivo lung measurements at 1.5 Tesla to allow a quantification of the local mean alveolar radius. Our results are therefore potentially relevant for the clinical diagnosis and therapy of pulmonary diseases.

Figure 1

(a) Synchrotron image of healthy lung tissue with an isotropic resolution of $1.625\mu \text{m}$. (b) Frequency of different air volume fractions $\eta$ in subvolumes with side length $208\mu \text{m}$. The mean volume fraction is $\eta =0.851\pm 0.045$.

Figure 2

Frequency distribution $p\left(\omega \right)$ for different air volume fractions $\eta$.

Figure 3

Simulation of lung microstructure in the Wigner-Seitz model [32, 33]. Alveoli are represented in white, tissue with spin-bearing particles in black. The air volume fraction $V_{\text{air}}/V_{\text{tot}}\approx 0.9$ agrees with values from the literature [34, 35]. The tissue model is calculated using Eq. (8): the thickness of the small layer containing water molecules between alveoli is approximately given as the alveolar septum gap $\epsilon$.

Figure 4

Geometrical model of lung tissue. In human lung tissue (a), alveoli are randomly distributed and the fraction of air volume to total volume is very large. (b) Synchrotron-based $\mu \mathrm{CT}$ image without filtering of healthy lung tissue in a mouse with an isotropic resolution of $325\text{nm}$ and a field of view of $325\mu \text{m}\times 325\mu \text{m}\times 162.5\mu \text{m}$. In the well-established Wigner-Seitz model, water molecules are located at the interfaces of Wigner-Seitz cells [32, 33]. Such a cell can be approximated by a sphere (c). In the alveolar surface model (see main text), local lung tissue is approximated as an ensemble of spherical surfaces with average radius $R$. Due to the susceptibility difference $\mathrm{\Delta }\chi ={\chi }_{\text{air}}-{\chi }_{\text{tissue}}$, the free induction decay in human lung tissue shows a large deviation from a monoexponential decay.

Figure 5

Frequency distributions $p\left(\omega \right)$ as obtained from the alveolar surface model [blue solid line obtained from Eq. (13)], numerical simulations of the Wigner-Seitz geometry (black dotted line) and from numerical simulations of synchrotron-based $\mu \mathrm{CT}$ images for $\eta =90–95\%$ (red dashed line, see Fig. 3). All curves show a clear asymmetry of the line shape, the line shape always peaks at a negative Larmor frequency. However, there is a clear quantitative difference between the curves.

Figure 6

(a) Real and (b) imaginary part of the eigenvalues ${\lambda }_{2k,0}\left(\sqrt{3i\tau \delta \omega }\right)$. In the motional narrowing limit $\tau \delta \omega \to 0$, the eigenvalues become purely real: ${\lambda }_{2k,0}\left(0\right)=2k[2k+1]$. The dotted lines show the approximations given in Eq. (B7) and Eq. (B8). Contrary to the case of a two-dimensional dipole field, the eigenvalues exhibit no symmetries and brunch cuts [45].

Figure 7

(a) Real and (b) imaginary part of the expansion coefficients $c_k$ according to Eq. (21) in dependence on $\tau \delta \omega$. With increasing $\tau \delta \omega$, an increasing number of coefficients contribute (see also Table 2 in Appendix pp2). In the limit $\tau \delta \omega \to 0$, the coefficients become: ${\lim }_{\tau \delta \omega \to 0}{c}_k=\sqrt{2}{\delta }_{k,0}$.

Figure 8

Local magnetization $m_x(\theta ,t)=\mathrm{Re}(m(\theta ,t))$ and $m_y(\theta ,t)=\mathrm{Im}(m(\theta ,t))$ in dependence on the normalized time $t/\tau$ and polar angle $\theta$ for $\tau \delta \omega =10$. With increasing time, the $x$ component of the local magnetization decreases while the $y$ component increases, especially for $\theta \approx \pi /2$. Both components remain symmetric around $\theta =\pi /2$ for all times.

Figure 9

Transverse components of the local magnetization $m(\theta ,t)$ for large diffusion coefficients [(a) and (b)], and small diffusion coefficients (large value of $\tau \delta \omega$) [(c) and (d)] in the exact solution [black lines, obtained from Eq. (19)] and in the Gaussian approximation [red lines, obtained from Eq. (24)]. The solid lines in (a) and (b) show the local magnetization at $t/\tau =0.1$, the dashed lines at $t/\tau =0.2$. The blue lines in (c) and (d) show the local magnetization in the static-dephasing limit obtained from Eq. (10).

Figure 10

Comparison of measured free induction decay with theoretical fits from Eq. (23) for eight voxels, see also Table 1.

Figure 11

Absolute value of the free induction decay $M\left(t\right)$ given in Eq. (23) compared with the strong collision approximation given in Eq. (27). With increasing value of the parameter $\tau \delta \omega$, the decay becomes faster and shows a larger deviation from a monoexponential decay.

Figure 12

Dependency of the zero $v$ on the parameter $\tau \delta \omega$ according to Eq. (28). For small values of $\tau \delta \omega \ll 1$ the real part of $v$ takes the constant value $\text{Re}\left(v\right)=1/3$ and the imaginary part diverges like $\text{Im}\left(v\right)=2/\left[\tau \delta \omega \right]$ [see Eq. (E7)]. In the opposite limit $\tau \delta \omega \gg 1$, the real part of $v$ decreases like $\text{Re}\left(v\right)={\pi }^2/{\left[\tau \delta \omega \right]}^2$ and the imaginary part decreases like $\text{Im}\left(v\right)=4{\pi }^2/{\left[\tau \delta \omega \right]}^3$ as shown in Eq. (E6).