Analytic determination of lung microgeometry with gas diffusion magnetic resonance

Through inhalation of, e.g., hyperpolarized ${}^3\mathrm{He}$, it is possible to acquire gas diffusion magnetic resonance measurements that depend on the local geometry in the vast network of microscopic airways that form the respiratory zone of the human lung. Here, we demonstrate that this can be used to determine the dimensions (length and radius) of these airways noninvasively. Specifically, the above technique allows measurement of the weighted time-dependent diffusion coefficient (also called the apparent diffusion coefficient), which we here derive in analytic form using symmetries in the airway network. Agreement with experiment is found for the full span of published hyperpolarized ${}^3\mathrm{He}$ diffusion magnetic resonance measurements (diffusion times from milliseconds to seconds) and published invasive airway dimension measurements.

Figure 1

Schematic illustration of the airway branching pattern in a pulmonary acinus. The line segments represent the center-line segments of the airways that make up the first (labeled 1 in the figure), second, third, and fourth of the nine generations of airways in the pulmonary acinus.

Figure 2

Illustration of how the sensitivity of $\mathfrak{D}$ to changes in $L, D_a$, and $R$ varies with $\mathrm{\Delta }$. The solid, dotted, and dashed curve shows the relative change in $\mathfrak{D}$ caused by a $20\%$ increase in $L, D_a$, and $R$, respectively, computed with the following parameter values as reference: $L=0.80\mathrm{mm}, D_a=0.48{\mathrm{cm}}^2{\mathrm{s}}^{-1}, R=0.35\mathrm{mm}, D_b=0.86{\mathrm{cm}}^2{\mathrm{s}}^{-1}, G\left(t\right)$ consisting of two rectangular pulses, and $\delta$ equal to $\mathrm{\Delta }$ if $\mathrm{\Delta }\le 10\mathrm{ms}$ and equal to $10\mathrm{ms}$ otherwise.

Figure 3

Fraction of the gas molecules in the pulmonary acinus that have not visited its exit as a function of Δ. Specifically, the curve shows $\frac{1}{EL}{\sum }_{i,i^{\prime }}{\int }_0^L{\int }_0^L\overline{K}{(x^{\prime },x,\mathrm{\Delta })}_{i^{\prime }i}dxd{x}^{\prime }$ as a function of $\mathrm{\Delta }$ computed with $L=0.80\mathrm{mm}$ and $D_a=0.48{\mathrm{cm}}^2{\mathrm{s}}^{-1}$, where the integrand, which can be obtained using Ref. [27], is the continuum-limit propagator for the diffusive motion of the gas molecules' projections on the geometric graph formed by the airways' center-line segments subject to the condition that the vertex located at ${\mathit{o}}_1$ is absorbing and all other vertices are nonabsorbing.

Figure 4

$\mathfrak{D}$ as a function of $\mathrm{\Delta }$ computed and experimentally estimated ($\mathrm{\Delta }$ varied in the range $1\mathrm{ms}$ to $7\mathrm{ms}$). The solid curve shows $\mathfrak{D}$ as a function of $\mathrm{\Delta }$ computed with $L=0.80\mathrm{mm}, D_a=0.48{\mathrm{cm}}^2{\mathrm{s}}^{-1}, R=0.35\mathrm{mm}, D_b=0.86{\mathrm{cm}}^2{\mathrm{s}}^{-1}, G\left(t\right)$ consisting of two rectangular pulses, and $\delta$ equal to $\mathrm{\Delta }$ if $\mathrm{\Delta }\le 10\mathrm{ms}$ and equal to $1\mathrm{ms}$ otherwise. The dashed curve shows the $D_a\mathrm{\Delta }/L^2\sim 1$ and $D_b\mathrm{\Delta }/R^2\sim 1$ approximation for $\mathfrak{D}$ as a function of $\mathrm{\Delta }$ computed with the same parameter values. The symbols show $\mathfrak{D}$ as a function of $\mathrm{\Delta }$ experimentally estimated based on the hyperpolarized ${}^3\mathrm{He}$ diffusion MR measurements on healthy adults reported in Refs. (each symbol and error bar represents the mean and standard deviation of measurements on $n$ healthy adults) [6] ($▵,n=14$), [7] ($▴,n=4$), [8] ($▹,n=8$, see also Ref. [9]), [10] ($▸,n=15$, see also Ref. [11]), [12] ($▿,n=8$), [13] ($▾,n=11$), [14] ($◃,n=5$), [15] ($◂,n=6$, see also Ref. [16]), and [17] ($◊,n=5$, the authors of [17] kindly made the raw gas diffusion MR data available).

Figure 5

$\mathfrak{D}$ as a function of $\mathrm{\Delta }$ computed and experimentally estimated ($\mathrm{\Delta }$ varied in the range $0.01\mathrm{s}$ to $1.6\mathrm{s}$). The solid curve shows $\mathfrak{D}$ as a function of $\mathrm{\Delta }$ computed with $L=0.80\mathrm{mm}, D_a=0.48{\mathrm{cm}}^2{\mathrm{s}}^{-1}, R=0.35\mathrm{mm}, D_b=0.86{\mathrm{cm}}^2{\mathrm{s}}^{-1}, G\left(t\right)$ consisting of two rectangular pulses, and $\delta$ equal to $\mathrm{\Delta }$ if $\mathrm{\Delta }\le 10\mathrm{ms}$ and equal to $1\mathrm{ms}$ otherwise. The symbols show $\mathfrak{D}$ as a function of $\mathrm{\Delta }$ experimentally estimated based on the hyperpolarized ${}^3\mathrm{He}$ diffusion MR measurements on healthy adults reported in Refs. (each symbol and error bar represents the mean and standard deviation of measurements on $n$ healthy adults) [18] ($▵,n=4$), [18] ($▴,n=2$), [19] ($▹,n=10$), [20] ($▸,n=17$), [6] ($▿,n=14$), [18] ($▾,n=3$), and [19] ($◃,n=10$).