Magnetic Resonance Imaging by Synergistic Diffusion-Diffraction Patterns

Inferring on the geometry of an object from its frequency spectrum is highly appealing since the object could then be imaged noninvasively or from a distance (as famously put by Kac, “can one hear the shape of a drum?”). In nuclear magnetic resonance of porous systems, the shape of the drum is represented by the pore density function that bears all the information on the collective pore microstructure. So far, conventional magnetic resonance imaging (MRI) could only detect the pore autocorrelation function, which inherently obscures fine details on the pore structure. Here, for the first time, we report on a unique imaging mechanism arising from synergistic diffusion-diffractions that directly yields the pore density function. This mechanism offers substantially higher spatial resolution compared to conventional MRI while retaining all fine details on the collective pore morphology. Thus, using these unique synergistic diffusion-diffractions, the “shape of the drum” can be inferred.

Figure 1

(a) Two different representative pores having similar dimensions but different shapes. (b) The pore density function, $\rho (\mathit{r})$, for each geometry. Note that the probability of finding spins outside the pore boundaries is zero. (c) The structure functions, $\tilde{\rho }(\mathit{q})$, which are obtained by FT of $\rho (\mathit{r})$. Note that these represent the pore spectrum. (d) The averaged displacement propagator $\overline{P}(\mathit{R},\infty )$ for the pores, which, for fully restricted diffusion, are the pore ACFs. (e) The power spectra of the pores, $|\tilde{\rho }(\mathit{q}){|}^2$, obtained by FT of $\overline{P}(\mathit{R},\infty )$. In sPGSE MR, these power spectra are inherently detected [Eq. (3)]. Inset: enhancement of the diffraction pattern up to $q=1000\mathrm{c}{\mathrm{m}}^{-1}$.

Figure 2

(a) Raw data for both s- and dPGSE MR experiments. (b) Enhancement of a box in (a). (c) Enhancement of a box in (b), showing the zero crossings more clearly. (d) The $N(\mathit{q})$ data, obtained by division of the two raw data signal decays according to Eq. (7). (e) The FT of $N(\mathit{q})$ clearly yields $\rho (\mathit{r})$, the pore density function. (f) The FT of $E(\mathit{q})$ for sPGSE MR data; the expected triangular function is observed, which does not contain direct information on the pore shape. The divided data suffer from artifacts near some of the zero crossings due to division of noise; in these cases, the points were extrapolated and replaced. For sPGSE MR, $\Delta /\delta =110/4\mathrm{ms}$; for dPGSE MR, ${\Delta }_1={\Delta }_2=110\mathrm{ms}$ and${\delta }_1={\delta }_2={\delta }_3=4\mathrm{ms}$.

Figure 3

Theoretical fit of the experimental $N(\mathit{q})$. The size extracted was $22.5\pm 0.6\mu \mathrm{m}$, in very good agreement with the nominal inner diameter of the pores. This data set was used to produce the image shown in Fig. 4a.

Figure 4

(a) Two-dimensional backprojection of the pore density function. (b) Two-dimensional backprojection of the pore ACF. The one-dimensional projections of each function are shown in the insets. Note that an exact MR image of the collective pore microstructure is obtained in (a), while, in (b), an image of the ACF is obtained. Clearly, the image of the cylinder cross section is obtained only in (a). The in-plane resolution is $\sim 1.8\mu \mathrm{m}$.