Ideally, the detection and evaluation of early stage pulmonary emphysema, fibrosis or carcinogenic alteration with small and/or subtle tissue changes should not rely on a radiologists inspection of CT or MR images alone but also on quantifiable physical parameters. These microstructural parameters for a local tissue environment within one imaging voxel can be linked to the Carr-Purcell-Meiboom-Gill (CPMG) sequence relaxation rate and the variation of the time between subsequent 180° pulses¹. This allows quantifying the mean local alveolar radius and the air-tissue ratio.

For a small enough local environment, neighboring alveoli in peripheral lung tissue can be thought of as being uniformly arranged in a Wigner-Seitz cell geometry with dodecahedral alveolar air volumes that are encompassed by respective tissue dodecahedra, see also Fig. 1. The standard single sphere approximation assigns spherical volumes to both dodecahedra with radius $$$R_A$$$ and $$$R$$$, respectively, and allows reducing further analysis on one single alveolus in analogy to Krogh's capillary supply model². The strong susceptibility gradient at the air-tissue interface generates, in an external $$$B_0$$$ field, a local dipole field with strength $$$\delta\omega$$$. Dephasing of spin-bearing particles in the thin tissue films around the spherical alveoli is prone to diffusion effects and thereby characteristically influences MR signal decay. Since surface relaxation is not negligible, Smoluchowski boundary conditions are assumed, see Fig. 2. Spin fluctuations that are induced by the local dipole field can be described through a correlation function that can be related to relaxation rate $$$R_2$$$_, inter-echo time $$$\tau_{180}$$$ for CPMG sequences, and correlation time $$$\tau=\frac{R_A^2}{D}$$$ ($$$D$$$ being the diffusion coefficient with $$$D\approx1\,\mu\text{m}^2/\text{ms}$$$).

A detailed mathematical analysis allows expressing the diffusion-dependent part of the CPMG relaxation rate, $$$\Delta R_2$$$, with several microstructural parameters as $$ \Delta R_2 = \frac45\eta\tau_{\text{C}}\delta\omega^2 - \frac{2[\tau\delta\omega]^2}{\tau_{180}}\sum_{n=1}^{\infty}\frac{G_n}{\kappa_n^4}\tanh\left(\frac{\kappa_n^2\tau_{180}}{2\tau}\right)$$ with $$G_n = \frac{24 \eta}{5 [1-\eta ]\kappa_n^2}\,\frac{ [j_2(\kappa_n\eta^{-\frac13})-\eta j_2(\kappa_n)]^2}{ \eta^{\frac13} [j_2(\kappa_n)]^2-[j_2(\kappa_n\eta^{-\frac13})]^2 } \,,$$ where $$$\eta=\frac{R_A^3}{R^3}$$$, and eigenvalues $$$\kappa_n$$$ that suffice the eigenvalue equation $$$j_2(\kappa_n) y_2\left(\frac{\kappa_n}{\sqrt[3]{\eta}}\right) = j_2\left(\frac{\kappa_n}{\sqrt[3]{\eta}}\right) y_2(\kappa_n) $$$ ($$$j$$$ and $$$y$$$ being spherical Bessel functions of the first and second kind, respectively). To test this analytical relation, dispersion experiments for $$$R_2$$$ were carried out for lung-tissue-like ageing hydrogel foam on a 0.5 T benchtop relaxometer at different times of the expanding foam for $$$D=1.062\,\mu\text{m}^2/\text{ms}$$$ and $$$\eta=1/1.1677$$$³, see Fig. 3a. Fig. 3b demonstrates the increase of alveolar fit radii $$$R_A$$$ over time and compares them to results from µCT measurements and random walk (RW) simulations that are based on the triangulated geometry as obtained from the µCT images. The mean relative error for RW simulation radii against the polynomial fit curve for µCT radii (solid line) is $$$14.36\pm2.66$$$%, and $$$5.84\pm1.28$$$% for the CPMG model. To further test the model's reaction to $$$R_2$$$ measurement variations, all recorded $$$R_2^{(i)}$$$ ($$$i=1,\ldots,18$$$) for the experimental data at 3.5h in Fig. 3a were varied within different ranges $$$\delta R_2^{(i)}$$$ such that $$$\forall i:\; \delta R_2^{(i)}/R_2^{(i)}=\text{const}$$$, see Fig. 4a. Furthermore, for $$$R_2^{(10)}=R_2(17\,\text{ms})$$$, a scatter plot was done to evaluate the influence of the strength of $$$\delta R_2^{(i)}$$$ and the obtained fit parameter $$$R_A$$$, where again all $$$R_2^{(i)}$$$ were varied within their respective error ranges, see Fig. 4b. It can be seen that errors are negligible for $$$\delta R_2^{(i)}/R_2^{(i)}<1/50$$$.

The excellent agreement of theoretical predictions and µCT measurements as well as a robust sensitivity analysis for $$$R_2$$$ variations that are feasible in MR imaging, support the validity of the model. Also, the model does not need the inhalation of helium gas, which is an advantage towards other established quantification models in clinical MR imaging.