Nuclear magnetic resonance diffusion pore imaging: Experimental phase detection by double diffusion encoding

Diffusion pore imaging is an extension of diffusion-weighted nuclear magnetic resonance imaging enabling the direct measurement of the shape of arbitrarily formed, closed pores by probing diffusion restrictions using the motion of spin-bearing particles. Examples of such pores comprise cells in biological tissue or oil containing cavities in porous rocks. All pores contained in the measurement volume contribute to one reconstructed image, which reduces the problem of vanishing signal at increasing resolution present in conventional magnetic resonance imaging. It has been previously experimentally demonstrated that pore imaging using a combination of a long and a narrow magnetic field gradient pulse is feasible. In this work, an experimental verification is presented showing that pores can be imaged using short gradient pulses only. Experiments were carried out using hyperpolarized xenon gas in well-defined pores. The phase required for pore image reconstruction was retrieved from double diffusion encoded (DDE) measurements, while the magnitude could either be obtained from DDE signals or classical diffusion measurements with single encoding. The occurring image artifacts caused by restrictions of the gradient system, insufficient diffusion time, and by the phase reconstruction approach were investigated. Employing short gradient pulses only is advantageous compared to the initial long-narrow approach due to a more flexible sequence design when omitting the long gradient and due to faster convergence to the diffusion long-time limit, which may enable application to larger pores.

Figure 1

Gradient schemes and pore imaging phantoms used for this work. 180° RF pulses were inserted into the gradient profiles to realize spin echo sequences. (a) $q$-space profile $G_{11}\left(t\right)$. (b) Double diffusion encoding profile $G_{121}\left(t\right)$. (c) Long-narrow profile $G_{\mathrm{LN}}\left(t\right)$. The gradient durations $\delta , {\delta }_{\mathrm{N}}$ and ${\delta }_{\mathrm{L}}$ are composed of the gradient flat top time plus ramp up time. In the phantoms, closed pores are aligned as an array. (d) Phantom type I: Pores of equilateral triangular shape with edge length $L=3400\mu \mathrm{m}$ or 5750 μm. (e) Phantom type II: Pores shaped as half-moons ($L=3450\mu \mathrm{m}$), fish ($L_1=3450\mu \mathrm{m}, L_2=2350\mu \mathrm{m}$), and stars ($L_1=3100\mu \mathrm{m}, L_2=3000\mu \mathrm{m}$).

Figure 2

One-dimensional pore imaging experiments for the equilateral triangle obtained by method 1. $q$-space and DDE signals were acquired radially in $q$ space and were used to calculate the form factor $\tilde{\rho }\left(q\right)$ followed by an inverse Fourier transform to obtain the pore space function $\rho \left(x\right)$ in $x$ space. The vector $\mathit{G}$ indicates the gradient direction of the radial acquisitions. $\delta$ is the gradient pulse duration, $T$ is the diffusion time. Experimental results are indicated by symbols and simulations by lines. (a) For the vertical gradient direction, the DDE signal $S_{121}\left(q\right)$ is complex, which allows estimating the phase of $\tilde{\rho }\left(q\right)$. The magnitude of $\tilde{\rho }\left(q\right)$ is obtained from the $q$-space signal $S_{11}\left(q\right)$. $\rho \left(x\right)$ corresponds to the projection of the triangle domain onto the gradient direction. (b) For the horizontal gradient direction, the projection is mirror-symmetric resulting in the real DDE signal. Longer gradient pulse durations were used in (c, d) and a shorter diffusion time was used in (e). Both entails a worse reconstruction of the pore shape. For a better visualization of the imaginary part of $S_{121}\left(q\right)$, the complex conjugate signals were plotted. The edge length of the domain was $L=3400\mu \mathrm{m}$. Eighteen $q$ values were measured. The maximum $q$ value $q_{\max }=8\mathrm{m}{\mathrm{m}}^{-1}$ corresponds to a nominal resolution of $\mathrm{\Delta }x=0.39\mathrm{mm}$.

Figure 3

One-dimensional pore imaging experiment for the equilateral triangle by double diffusion encoding only (method 2) using the same phantom as described in the caption of Fig. 2. The DDE signal was acquired in the vertical direction as indicated by the vector $\mathit{G}$ and was used to calculate both magnitude and phase of $\tilde{\rho }\left(q\right)$ with subsequent inverse Fourier transform $\rho \left(x\right)$ in $x$ space. Experimental results are indicated by symbols and simulations by lines. Eighteen $q$ values were measured. For better visualization of the imaginary part, the complex conjugate of $S_{121}\left(q\right)$ was plotted. The maximum $q$ value $q_{\max }=8\mathrm{m}{\mathrm{m}}^{-1}$ corresponds to a nominal resolution of $\mathrm{\Delta }x=0.39\mathrm{mm}$.

Figure 4

Diffusion pore images of equilateral triangles. Experimental results in the upper row and simulations below. (a–d) Pore images obtained by method 1 with the combination of $q$-space and DDE measurements and subsequent phase estimation. (e) Use of method 2: magnitude and phase estimation employing the DDE measurement only. (f) Pore images acquired with method 3 using the long-narrow gradient profile. Additional parameters for (a)–(f): edge length $L=3400\mu \mathrm{m}$, 37 directions, 9 $q$ values per spoke, $q_{\max }=8\mathrm{m}{\mathrm{m}}^{-1}$, nominal resolution of $\mathrm{\Delta }x=0.39\mathrm{mm}$.

Figure 5

Pore images obtained using (a) the phase estimation method (method 1), (b) the magnitude estimation method (method 2) and (c) the long-narrow approach (method 3) in a phantom with larger triangular pores (edge length $L=5750\mu \mathrm{m}$). Additional parameters: 37 directions, 10 $q$ values per spoke, $q_{\max }=6.5\mathrm{m}{\mathrm{m}}^{-1}$, nominal resolution of $\mathrm{\Delta }x=0.48\mathrm{mm}$.

Figure 6

Diffusion pore images of domains with further pore shapes. Pores were obtained using method 1 (left column), method 2 (middle column), and method 3 (right column). For each pore shape, the experimental results are shown in the upper row and simulations below. Additional Parameters: 37 directions, (a) $q_{\max }=7\mathrm{m}{\mathrm{m}}^{-1},8q$ values per spoke, nominal resolution of $\mathrm{\Delta }x=0.45\mathrm{mm}$, (b) $q_{\max }=10\mathrm{m}{\mathrm{m}}^{-1}$, 10 $q$ values, $\mathrm{\Delta }x=0.31\mathrm{mm}$, (c) $q_{\max }=12\mathrm{m}{\mathrm{m}}^{-1}$, 12 $q$ values, $\mathrm{\Delta }x=0.26\mathrm{mm}$.