In a conventional diffusion-weighted MRI experiment, the signal amplitude depends on the squared magnitude of the Fourier transformation of the pore or cell geometry [1]. From such data, the pore or cell geometry cannot be reconstructed because the phase information is missing [1]. Recently, several approaches have been proposed that offer access to the phase and, thus, allow to reconstruct the pore or cell geometry directly [2-4]. Here, the performance of these methods is compared with respect to their applicability in practice, in particular in biological systems. considering the impact of the noise level, mixtures of pore sizes, orientations, and shapes, a freely diffusing compartment, and gradient pulse durations and diffusion times achievable on conventional MRI systems.

The pore imaging methods considered are the method of Laun et al. based on a short and a long diffusion-weighting gradient pulse [2] (”asymmetric” method), the method of Shemesh et al. involving the signal ratio of a conventional and a double diffusion encoding experiment [3] (“ratio” method), and the method of Kuder et al. that estimates the phase information iteratively from a double diffusion encoding experiment [4] (“phase iteration” method). The pulse sequences required for these methods are presented in Fig. 1.

Numerical simulations were performed using an IDL algorithm (version 7.1, ITT Visual Information Solutions, Boulder, Colorado, USA) of a random-walk model without or within confining geometries with diffuse reflection at the boundaries (see, e.g., [5]). Only two dimensions were considered for simplicity but all results can be expected to hold for three dimensions as well. Up to 10⁷ spins were simulated with a temporal resolution of 10 µs using a diffusion coefficient of 2.0×10^‑3 mm² s^‑1. Different pore shapes (square, rectangle, circle), orientations, and sizes (radius and edge length between 5 µm and 20 µm) were considered. Simulations were performed for a radial grid, regridded to a cartesian grid, and Fourier transformed to obtain the pore image. For some simulations, Gaussian noise was added prior to the Fourier transformation to investigate the performance in the presence of a finite signal-to-noise ratio (SNR).

For most simulations ideal timing parameters were used, however, some simulations were performed with timing parameters compatible with state-of-the-art gradient systems for animal and whole-body MR systems (500 mT m^‑1 and 80 mT m^‑1, respectively) or total durations of the diffusion weighting limited to 200 ms. For the “ratio” method, a thresholding approach was used that reduces noise amplification by setting the ratio to 0 for small values of the denominator (zero crossings of the denominator are also zero crossings of the nominator).

Pore images obtained for different noise levels are presented in Fig. 2. While the “phase iteration” method amplifies noise due to the iterative calculation of the phase, the “ratio” method suffers from blurring making the “asymmetric” method the most reliable one.

Results for mixtures of different pore orientations, sizes, and shapes are presented in Fig. 3. Again the “asymmetric” method provides the best performance resolving the orientations, sizes, and shapes while the other methods fail to reconstruct the underlying geometries correctly.

Images obtained for realistic timing parameters and gradient amplitudes are presented in Fig. 4. With finite gradient amplitudes, the pore size is underestimated, in particular for the weaker gradient amplitudes available on whole-body MR systems (about 13-14 µm for a pore size of 20 µm; Fig. 4c). This is because for finite gradient pulses the center-of-mass of the spin trajectory defines the effective encoding position [6] which moves towards the pore’s center for longer gradient pulses. Considering a finite time for the diffusion weighting does not have a significant impact on the “ratio” and “phase iteration” method but significantly affects the “asymmetric” method that yields a featureless image approaching the pore’s autocorrelation function (Fig. 4d and e) as for a conventional experiment. Since in practice the duration of the long gradient pulse is limited by T2 relaxation, improved gradient hardware cannot solve this problem but larger diffusion coefficients, smaller pores, or longer relaxation times are beneficial.

Adding a free compartment (Fig. 5) as can be relevant in biological systems does seem to have a significant impact on either of the methods.

The method of Laun et al. ("asymmetric" method) is the most robust one regarding the noise level and, unlike the other two methods, is able to resolve the pore geometry in mixtures which is important for more complex samples like biological tissues. However, in contrast to the other methods, it faces limitations for the timing parameters that are required in practice.