Diffusion pore imaging enables direct imaging of arbitrarily shaped closed pores filled with an NMR-visible diffusing medium. The initial diffusion pore imaging approach uses a long-narrow gradient profile ¹. This method has been shown in recent experiments to be capable of measuring an average image of the ensemble of pore shapes contained in the considered imaging volume element ^2,3,4.

Recently, further approaches to diffusion pore imaging have been introduced which use double diffusion encodings (DDE) employing short gradient pulses only ^5,6. While conventional q-space measurements allow to determine the magnitude of the Fourier transform of the pore space function only ⁷, the phase information necessary for unambiguous image reconstruction of arbitrarily shaped pores is obtained from DDE measurements ⁶. So far, one of these methods has been demonstrated experimentally using cylindrical pores ⁵. In these point-symmetric pores, the DDE signal is strictly real.

Here, we present initial results demonstrating that the necessary imaginary signal parts can be detected for non-point-symmetric pore shapes, which thus can be imaged without the application of a long gradient pulse.

For the method used here, two measurements are needed ⁶: During q-space imaging two short gradient pulses of duration $$$\delta$$$ and amplitude $$$G$$$ are applied (Fig. 1a), where the case of long diffusion time $$$T$$$ is considered. For the DDE measurement, three short gradient pulses are applied, where the second pulse has twice the amplitude of the other two pulses and is split into two pulses, separated by the mixing time $$$T_M$$$, to realize a spin-echo sequence (Fig. 1b). Both measurements produce the same q-space vector $$${\bf q}=\gamma{\bf G}\delta$$$, where $$$\gamma$$$ is the gyromagnetic ratio.

For the idealized case of short gradients, long diffusion time and zero mixing time ($$$\delta\rightarrow 0$$$, $$$T\rightarrow\infty$$$, $$$T_{M}\rightarrow 0$$$), the complex signal attenuation in q-space resulting from the application of the DDE measurement is given by $$$S_{121}({\bf q})=\rho^\ast({\bf q}/2)^2\rho({\bf q})$$$ ⁸, where $$$\rho({\bf q})$$$ is the form factor and the asterisk indicates the complex conjugate. The form factor is the Fourier transform of the pore space function, which is zero outside the pore and defined as $$$\rho({\bf x})=1/V_P$$$ inside the pore where $$$V_P$$$ is the volume inside the pore.

To retrieve the complex phase information for non-point-symmetric pore shapes from $$$S_{121}({\bf q})$$$, an iterative approach can be applied ⁶: The magnitude of $$$\rho({\bf q})$$$ can be determined by q-space measurements employing $$$|\rho({\bf q})|=\sqrt{S_{11}({\bf q})}$$$. Using appropriate initial conditions for the phase of $$$\rho({\bf q})$$$ at small q-values, one can recursively estimate the necessary phase from $$$S_{121}({\bf q})$$$ for radial acquisitions in q-space and thus reconstruct pore images.

The measurements were performed on a clinical 1.5 T MR scanner (Symphony, Siemens) using a phantom with 170 pores of equilateral triangular shape (3.4 mm edge length). These were filled with hyperpolarized Xe-129 gas, which was generated by spin exchange optical pumping, flowing along the z-direction of the scanner. The gradients were applied orthogonal to the gas flow. The vector $$${\bf G}$$$ in Fig. 2 indicates the gradient direction relative to the triangles. 18 measurement points homogeneously spaced up to a maximal q-value of 8 mm^-1 were sampled with a gradient duration of $$$\delta$$$ = 3.58 ms and a diffusion time of $$$T$$$ = 300 ms. The gradient strength $$$G$$$ increased up to 30 mT/m for the highest q-value. Both q-space and DDE experiment were averaged over 10 acquisitions.

For comparison, the diffusion process was simulated by using a matrix approach to solve the Bloch-Torrey equations ^9,10.

Fig. 2 shows the real and imaginary parts of the measured signals $$$S_{11}(q)$$$ and $$$S_{121}(q)$$$ (dots). Due to the missing point symmetry, a non-zero imaginary part arises in $$$S_{121}(q)$$$ (Fig. 2, purple). This allows to estimate the phase of $$$\rho({q})$$$. The simulations (lines) are in good agreement with the experimental results, except for small deviations of $$$\operatorname{Im}(S_{121}(q))$$$, mainly at small q-values. Fig. 3 shows the inverse Fourier transform of $$$\rho({q})$$$ obtained by means of the recursive reconstruction approach. The projection of the triangular shape on the gradient direction is clearly observable. Simulation and measurement are in good agreement except for the region near the tip of the triangle, which can be attributed to the deviations in the measured $$$\operatorname{Im}(S_{121}(q))$$$.These initial experiments demonstrate that diffusion pore imaging using DDE is feasible for non-point-symmetric pore shapes and that the necessary phase can be estimated. However, the recursive estimation process tends to amplify errors in $$$S_{121}(q)$$$ at small q-values. This will require further improvements in the measurement sequence and reconstruction approach to yield decent phase stability.