Two-dimensional diffusion time correlation experiment using a single direction gradient

Multidimensional diffusion MRI with spectrally modulated gradients reveals unprecedented microstructural detail

Characterization of porous media is essential in a wide range of biomedical and industrial applications. Microstructural features can be probed non-invasively by diffusion magnetic resonance imaging (dMRI). However, diffusion encoding in conventional dMRI may yield similar signatures for very different microstructures, which represents a significant limitation for disentangling individual microstructural features in heterogeneous materials. To solve this problem, we propose an augmented multidimensional diffusion encoding (MDE) framework, which unlocks a novel encoding dimension to assess time-dependent diffusion specific to structures with different microscopic anisotropies. Our approach relies on spectral analysis of complex but experimentally efficient MDE waveforms. Two independent contrasts to differentiate features such as cell shape and size can be generated directly by signal subtraction from only three types of measurements. Analytical calculations and simulations support our experimental observations. Proof-of-concept experiments were applied on samples with known and distinctly different microstructures. We further demonstrate substantially different contrasts in different tissue types of a post mortem brain. Our simultaneous assessment of restriction size and shape may be instrumental in studies of a wide range of porous materials, enable new insights into the microstructure of biological tissues or be of great value in diagnostics.

In vivo microscopic diffusional kurtosis imaging with symmetrized double diffusion encoding EPI

PURPOSE

Diffusional kurtosis imaging (DKI) measures the deviation of the displacement probability from a normal distribution, complementing the data commonly acquired by diffusion MRI. It is important to elucidate the sources of kurtosis contrast, particularly in biological tissues where microscopic kurtosis (intrinsic kurtosis) and diffusional heterogeneity may co-exist.

METHODS

We have developed a technique for microscopic kurtosis MRI, dubbed microscopic diffusional kurtosis imaging (µDKI), using a symmetrized double diffusion encoding (s-DDE) EPI sequence. We compared this newly developed µDKI to conventional DKI methods in both a triple compartment phantom and in vivo.

RESULTS

Our results showed that whereas conventional DKI and µDKI provided similar measurements in a compartment of monosphere beads, kurtosis measured by µDKI was significantly less than that measured by conventional DKI in a compartment of mixed Gaussian pools. For in vivo brain imaging, µDKI showed small yet significantly lower kurtosis measurement in regions of the cortex, CSF, and internal capsule compared to the conventional DKI approach.

CONCLUSIONS

Our study showed that µDKI is less susceptible than conventional DKI to sub-voxel diffusional heterogeneity. Our study also provided important preliminary demonstration of our technique in vivo, warranting future studies to investigate its diagnostic use in examining neurological disorders.

Computational approach to integrate 3D X-ray microtomography and NMR data

Nowadays, most of the efforts in NMR applied to porous media are dedicated to studying the molecular fluid dynamics within and among the pores. These analyses have a higher complexity due to morphology and chemical composition of rocks, besides dynamic effects as restricted diffusion, diffusional coupling, and exchange processes. Since the translational nuclear spin diffusion in a confined geometry (e.g. pores and fractures) requires specific boundary conditions, the theoretical solutions are restricted to some special problems and, in many cases, computational methods are required. The Random Walk Method is a classic way to simulate self-diffusion along a Digital Porous Medium. Bergman model considers the magnetic relaxation process of the fluid molecules by including a probability rate of magnetization survival under surface interactions. Here we propose a statistical approach to correlate surface magnetic relaxivity with the computational method applied to the NMR relaxation in order to elucidate the relationship between simulated relaxation time and pore size of the Digital Porous Medium. The proposed computational method simulates one- and two-dimensional NMR techniques reproducing, for example, longitudinal and transverse relaxation times (T₁ and T₂, respectively), diffusion coefficients (D), as well as their correlations. For a good approximation between the numerical and experimental results, it is necessary to preserve the complexity of translational diffusion through the microstructures in the digital rocks. Therefore, we use Digital Porous Media obtained by 3D X-ray microtomography. To validate the method, relaxation times of ideal spherical pores were obtained and compared with the previous determinations by the Brownstein-Tarr model, as well as the computational approach proposed by Bergman. Furthermore, simulated and experimental results of synthetic porous media are compared. These results make evident the potential of computational physics in the analysis of the NMR data for complex porous materials.

Retaining both discrete and smooth features in 1D and 2D NMR relaxation and diffusion experiments

A new method of regularization of 1D and 2D NMR relaxation and diffusion experiments is proposed and a robust algorithm for its implementation is introduced. The new form of regularization, termed the Modified Total Generalized Variation (MTGV) regularization, offers a compromise between distinguishing discrete and smooth features in the reconstructed distributions. The method is compared to the conventional method of Tikhonov regularization and the recently proposed method of L₁ regularization, when applied to simulated data of 1D spin-lattice relaxation, T₁, 1D spin-spin relaxation, T₂, and 2D T₁-T₂ NMR experiments. A range of simulated distributions composed of two lognormally distributed peaks were studied. The distributions differed with regard to the variance of the peaks, which were designed to investigate a range of distributions containing only discrete, only smooth or both features in the same distribution. Three different signal-to-noise ratios were studied: 2000, 200 and 20. A new metric is proposed to compare the distributions reconstructed from the different regularization methods with the true distributions. The metric is designed to penalise reconstructed distributions which show artefact peaks. Based on this metric, MTGV regularization performs better than Tikhonov and L₁ regularization in all cases except when the distribution is known to only comprise of discrete peaks, in which case L₁ regularization is slightly more accurate than MTGV regularization.

Obtaining sparse distributions in 2D inverse problems

The mathematics of inverse problems has relevance across numerous estimation problems in science and engineering. L₁ regularization has attracted recent attention in reconstructing the system properties in the case of sparse inverse problems; i.e., when the true property sought is not adequately described by a continuous distribution, in particular in Compressed Sensing image reconstruction. In this work, we focus on the application of L₁ regularization to a class of inverse problems; relaxation-relaxation, T₁-T₂, and diffusion-relaxation, D-T₂, correlation experiments in NMR, which have found widespread applications in a number of areas including probing surface interactions in catalysis and characterizing fluid composition and pore structures in rocks. We introduce a robust algorithm for solving the L₁ regularization problem and provide a guide to implementing it, including the choice of the amount of regularization used and the assignment of error estimates. We then show experimentally that L₁ regularization has significant advantages over both the Non-Negative Least Squares (NNLS) algorithm and Tikhonov regularization. It is shown that the L₁ regularization algorithm stably recovers a distribution at a signal to noise ratio<20 and that it resolves relaxation time constants and diffusion coefficients differing by as little as 10%. The enhanced resolving capability is used to measure the inter and intra particle concentrations of a mixture of hexane and dodecane present within porous silica beads immersed within a bulk liquid phase; neither NNLS nor Tikhonov regularization are able to provide this resolution. This experimental study shows that the approach enables discrimination between different chemical species when direct spectroscopic discrimination is impossible, and hence measurement of chemical composition within porous media, such as catalysts or rocks, is possible while still being stable to high levels of noise.

Imaging Local Diffusive Dynamics Using Diffusion Exchange Spectroscopy MRI

The movement of water between microenvironments presents a central challenge in the physics of soft matter and porous media. Diffusion exchange spectroscopy (DEXSY) is a powerful 2D nuclear magnetic resonance method for measuring such exchange, yet it is rarely used because of its long scan time requirements. Moreover, it has never been combined with magnetic resonance imaging (MRI). Using probability theory, we vastly reduce the required data, making DEXSY MRI feasible for the first time. Experiments are performed on a composite nerve tissue phantom with restricted and free water-exchanging compartments.

Direct correlation of diffusion and pore size distributions with low field NMR

The time-dependent diffusion coefficient (D) is a powerful tool to probe microstructure in porous media, and can be obtained by the NMR method. In a real porous sample, molecular diffusion is very complex. Here we present a new method which directly measures the relationship between effective diffusion coefficients and pore size distributions without knowing surface relaxivity. This method is used to extract structural information and explore the relationship between D and a in porous media having broad pore size distributions. The diffusion information is encoded by the Pulsed Field Gradient (PFG) method and the pore size distributions are acquired by the Decay due to Diffusion in the Internal Field (DDIF) method. Two model samples were measured to verify this method. Restricted diffusion was analyzed, and shows that most fluid molecules experience pore wall. The D(a) curves obtained from correlation maps were fitted to the Padé approximant equation and a good agreement was found between the fitting lines and the measured data. Then a sandstone sample with unknown structure was measured. The state of confined fluids was analyzed and structural information, such as pore size distributions, were extracted. The D - T1 correlation maps were also obtained using the same method, which yielded surface relaxivities for different samples. All the experiments were conducted on 2MHz NMR equipment to obtain accurate diffusion information, where internal gradients can be neglected. This method is expected to have useful applications in the oil industry, particularly for NMR logging in the future.

The robust identification of exchange from T2-T2 time-domain features

Two-dimensional spin-spin relaxation (T2-T2) techniques have been developed to probe coupling between different environments such as diffusive coupling between small and large pores or chemical exchange with clays. In these studies, Numerical Laplace Inversion (NLI) is used to obtain two-dimensional T2-T2 relaxation distribution spectrum from the T2-T2 signal decays, and the off-diagonal peaks characterize coupling. Often, these coupling peaks are both weak and close to the diagonal and combined with the inherently ill-conditioned nature of the inversion, their presence is difficult to differentiate from inversion related artifacts and blurring. This manuscript presents a time domain based analysis to identify the presence of coupling that avoids the ambiguities present in T2-T2 spectra. The approach utilizes the symmetric nature of the two-dimensional time domain data, where the presence of curvature along one of these symmetries gives an unambiguous indicator of coupling. Measurements on porous glass beads are used to verify the technique.

Detecting compartmental non-Gaussian diffusion with symmetrized double-PFG MRI

Diffusion in tissue and porous media is known to be non-Gaussian and has been used for clinical indications of stroke and other tissue pathologies. However, when conventional NMR techniques are applied to biological tissues and other heterogeneous materials, the presence of multiple compartments (pores) with different Gaussian diffusivities will also contribute to the measurement of non-Gaussian behavior. Here we present symmetrized double PFG (sd-PFG), which can separate these two contributions to non-Gaussian signal decay as having distinct angular modulation frequencies. In contrast to prior angular d-PFG methods, sd-PFG can unambiguously extract kurtosis as an oscillation from samples with isotropic or uniformly oriented anisotropic pores, and can generally extract a combination of compartmental anisotropy and kurtosis. The method further fixes its sensitivity with respect to the time dependence of the apparent diffusion coefficient. We experimentally demonstrate the measurement of the fourth cumulant (kurtosis) of diffusion and find it consistent with theoretical predictions. By enabling the unambiguous identification of contributions of compartmental kurtosis to the signal, sd-PFG has the potential to help identify the underlying micro-structural changes corresponding to current kurtosis based diagnostics, and act as a novel source of contrast to better resolve tissue micro-structure.