Quantification of pore size distribution using diffusion NMR: Experimental design and physical insights

SPHERIOUSLY? The challenges of estimating sphere radius non-invasively in the human brain from diffusion MRI

The Soma and Neurite Density Imaging (SANDI) three-compartment model was recently proposed to disentangle cylindrical and spherical geometries, attributed to neurite and soma compartments, respectively, in brain tissue. There are some recent advances in diffusion-weighted MRI signal encoding and analysis (including the use of multiple so-called 'b-tensor' encodings and analysing the signal in the frequency-domain) that have not yet been applied in the context of SANDI. In this work, using: (i) ultra-strong gradients; (ii) a combination of linear, planar, and spherical b-tensor encodings; and (iii) analysing the signal in the frequency domain, three main challenges to robust estimation of sphere size were identified: First, the Rician noise floor in magnitude-reconstructed data biases estimates of sphere properties in a non-uniform fashion. It may cause overestimation or underestimation of the spherical compartment size and density. This can be partly ameliorated by accounting for the noise floor in the estimation routine. Second, even when using the strongest diffusion-encoding gradient strengths available for human MRI, there is an empirical lower bound on the spherical signal fraction and radius that can be detected and estimated robustly. For the experimental setup used here, the lower bound on the sphere signal fraction was approximately 10%. We employed two different ways of establishing the lower bound for spherical radius estimates in white matter. The first, examining power-law relationships between the DW-signal and diffusion weighting in empirical data, yielded a lower bound of 7μm, while the second, pure Monte Carlo simulations, yielded a lower limit of 3μm and in this low radii domain, there is little differentiation in signal attenuation. Third, if there is sensitivity to the transverse intra-cellular diffusivity in cylindrical structures, e.g., axons and cellular projections, then trying to disentangle two diffusion-time-dependencies using one experimental parameter (i.e., change in frequency-content of the encoding waveform) makes spherical radii estimates particularly challenging. We conclude that due to the aforementioned challenges spherical radii estimates may be biased when the corresponding sphere signal fraction is low, which must be considered.

Optimising magnetic resonance sampling patterns for parametric characterisation

Sampling strategies are often central to experimental design. Choosing efficiently which data to acquire can improve the estimation of parameters and reduce the acquisition time. This work is focused on designing optimal sampling patterns for Nuclear Magnetic Resonance (NMR) applications, illustrated with respect to the best estimate of the parameters characterising a lognormal distribution. Lognormal distributions are commonly used as fitting models for distributions of spin-lattice relaxation time constants, spin-spin relaxation time constants and diffusion coefficients. A method for optimising the choice of points to be sampled is presented which is based on the Cramér-Rao Lower Bound (CRLB) theory. The method's capabilities are demonstrated experimentally by applying it to the problem of estimating the emulsion droplet size distribution from a pulsed field gradient (PFG) NMR diffusion experiment. A difference of <5% is observed between the predictions of CRLB theory and the PFG NMR experimental results. It is shown that CLRB theory is stable down to signal-to-noise ratios of ∼10. A sensitivity analysis for the CRLB theory is also performed. The method of optimizing sampling patterns is easily adapted to distributions other than lognormal and to other aspects of experimental design; case studies of optimising the sampling scheme for a fixed acquisition time and determining the potential for reduction in acquisition time for a fixed parameter estimation accuracy are presented. The experimental acquisition time is typically reduced by a factor of 3 using the proposed method compared to a constant gradient increment approach that would usually be used.

Resolution limit of cylinder diameter estimation by diffusion MRI: The impact of gradient waveform and orientation dispersion

Diffusion MRI has been proposed as a non-invasive technique for axonal diameter mapping. However, accurate estimation of small diameters requires strong gradients, which is a challenge for the transition of the technique from preclinical to clinical MRI scanners, since these have weaker gradients. In this work, we develop a framework to estimate the lower bound for accurate diameter estimation, which we refer to as the resolution limit. We analyse only the contribution from the intra-axonal space and assume that axons can be represented by impermeable cylinders. To address the growing interest in using techniques for diffusion encoding that go beyond the conventional single diffusion encoding (SDE) sequence, we present a generalised analysis capable of predicting the resolution limit regardless of the gradient waveform. Using this framework, waveforms were optimised to minimise the resolution limit. The results show that, for parallel cylinders, the SDE experiment is optimal in terms of yielding the lowest possible resolution limit. In the presence of orientation dispersion, diffusion encoding sequences with square-wave oscillating gradients were optimal. The resolution limit for standard clinical MRI scanners (maximum gradient strength 60-80 mT/m) was found to be between 4 and 8 μm, depending on the noise levels and the level of orientation dispersion. For scanners with a maximum gradient strength of 300 mT/m, the limit was reduced to between 2 and 5 μm.

White matter microstructure from nonparametric axon diameter distribution mapping

We report the development of a double diffusion encoding (DDE) MRI method to estimate and map the axon diameter distribution (ADD) within an imaging volume. A variety of biological processes, ranging from development to disease and trauma, may lead to changes in the ADD in the central and peripheral nervous systems. Unlike previously proposed methods, this ADD experimental design and estimation framework employs a more general, nonparametric approach, without a priori assumptions about the underlying form of the ADD, making it suitable to analyze abnormal tissue. In the current study, this framework was used on an ex vivo ferret spinal cord, while emphasizing the way in which the ADD can be weighted by either the number or the volume of the axons. The different weightings, which result in different spatial contrasts, were considered throughout this work. DDE data were analyzed to derive spatially resolved maps of average axon diameter, ADD variance, and extra-axonal volume fraction, along with a novel sub-micron restricted structures map. The morphological information contained in these maps was then used to segment white matter into distinct domains by using a proposed k-means clustering algorithm with spatial contiguity and left-right symmetry constraints, resulting in identifiable white matter tracks. The method was validated by comparing histological measures to the estimated ADDs using a quantitative similarity metric, resulting in good agreement. With further acquisition acceleration and experimental parameters adjustments, this ADD estimation framework could be first used preclinically, and eventually clinically, enabling a wide range of neuroimaging applications for improved understanding of neurodegenerative pathologies and assessing microstructural changes resulting from trauma.

Joint radius-length distribution as a measure of anisotropic pore eccentricity: an experimental and analytical framework

In this work, we present an experimental design and analytical framework to measure the nonparametric joint radius-length (R-L) distribution of an ensemble of parallel, finite cylindrical pores, and more generally, the eccentricity distribution of anisotropic pores. Employing a novel 3D double pulsed-field gradient acquisition scheme, we first obtain both the marginal radius and length distributions of a population of cylindrical pores and then use these to constrain and stabilize the estimate of the joint radius-length distribution. Using the marginal distributions as constraints allows the joint R-L distribution to be reconstructed from an underdetermined system (i.e., more variables than equations), which requires a relatively small and feasible number of MR acquisitions. Three simulated representative joint R-L distribution phantoms corrupted by different noise levels were reconstructed to demonstrate the process, using this new framework. As expected, the broader the peaks in the joint distribution, the less stable and more sensitive to noise the estimation of the marginal distributions. Nevertheless, the reconstruction of the joint distribution is remarkably robust to increases in noise level; we attribute this characteristic to the use of the marginal distributions as constraints. Axons are known to exhibit local compartment eccentricity variations upon injury; the extent of the variations depends on the severity of the injury. Nonparametric estimation of the eccentricity distribution of injured axonal tissue is of particular interest since generally one cannot assume a parametric distribution a priori. Reconstructing the eccentricity distribution may provide vital information about changes resulting from injury or that occurred during development.

Nonparametric pore size distribution using d-PFG: comparison to s-PFG and migration to MRI

Here we present the successful translation of a pore size distribution (PSD) estimation method from NMR to MRI. This approach is validated using a well-characterized MRI phantom consisting of stacked glass capillary arrays (GCA) having different diameters. By employing a double pulsed-field gradient (d-PFG) MRI sequence, this method overcomes several important theoretical and experimental limitations of previous single-PFG (s-PFG) based MRI methods by allowing the relative diffusion gradients' direction to vary. This feature adds an essential second dimension in the parameters space, which can potentially improve the reliability and stability of the PSD estimation. To infer PSDs from the MRI data in each voxel an inverse linear problem is solved in conjunction with the multiple correlation function (MCF) framework, which can account for arbitrary experimental parameters (e.g., long diffusion pulses). This scheme makes no a priori assumptions about the functional form of the underlying PSD. Creative use of region of interest (ROI) analysis allows us to create different underlying PSDs using the same GCA MRI phantom. We show that an s-PFG experiment on the GCA phantom fails to accurately reconstruct the size distribution, thus demonstrating the superiority of the d-PFG experiment. In addition, signal simulations corrupted by different noise levels were used to generate continuous and complex PSDs, which were then successfully reconstructed. Finally, owing to the reduced q- or b- values required to measure microscopic PSDs via d-PFG MRI, this method will be better suited to biomedical and clinical applications, in which gradient strength of scanners is limited.