Effective potential for magnetic resonance measurements of restricted diffusion

Spin echo formation in muscle tissue

Determination of the spin echo signal evolution and of transverse relaxation rates is of high importance for microstructural modeling of muscle tissue in magnetic resonance imaging. So far, numerically exact solutions for the NMR signal dynamics in muscle tissue models have been reported only for the gradient echo free induction decay, with spin echo problems usually solved by approximate methods. In this work, we modeled the spin echo signal numerically exact by discretizing the radial dimension of the Bloch-Torrey equation and expanding the angular dependency in terms of even Chebyshev polynomials. This allows us to express the time dependence of the local magnetization as a closed-form matrix expression. Using this method, we were able to accurately capture the spin echo local and total magnetization dynamics. The obtained transverse relaxation rates showed a high concordance with random walker and finite-element simulations. We could demonstrate that in cases of smaller diffusion coefficients, the commonly used strong collision approximation significantly underestimates the true value considerably. Instead, the limiting behavior in this regime is correctly described either by the full solution or by the slow diffusion approximation. Experimentally measured transverse relaxation rates of a mouse limb muscle showed an angular dependence in accordance with the theoretical prediction.

Q-space trajectory imaging with positivity constraints (QTI+)

Q-space trajectory imaging (QTI) enables the estimation of useful scalar measures indicative of the local tissue structure. This is accomplished by employing generalized gradient waveforms for diffusion sensitization alongside a diffusion tensor distribution (DTD) model. The first two moments of the underlying DTD are made available by acquisitions at low diffusion sensitivity (b-values). Here, we show that three independent conditions have to be fulfilled by the mean and covariance tensors associated with distributions of symmetric positive semidefinite tensors. We introduce an estimation framework utilizing semi-definite programming (SDP) to guarantee that these conditions are met. Applying the framework on simulated signal profiles for diffusion tensors distributed according to non-central Wishart distributions demonstrates the improved noise resilience of QTI+ over the commonly employed estimation methods. Our findings on a human brain data set also reveal pronounced improvements, especially so for acquisition protocols featuring few number of volumes. Our method’s robustness to noise is expected to not only improve the accuracy of the estimates, but also enable a meaningful interpretation of contrast in the derived scalar maps. The technique’s performance on shorter acquisitions could make it feasible in routine clinical practice.

Variance Measures for Symmetric Positive (Semi-) Definite Tensors in Two Dimensions

Calculating the variance of a family of tensors, each represented by a symmetric positive semi-definite second order tensor/matrix, involves the formation of a fourth order tensor R_abcd. To form this tensor, the tensor product of each second order tensor with itself is formed, and these products are then summed, giving the tensor R_abcd the same symmetry properties as the elasticity tensor in continuum mechanics. This tensor has been studied with respect to many properties: representations, invariants, decomposition, the equivalence problem et cetera. In this paper we focus on the two-dimensional case where we give a set of invariants which ensures equivalence of two such fourth order tensors R_abcd and R~abcd. In terms of components, such an equivalence means that components R_ijkl of the first tensor will transform into the components R~ijkl of the second tensor for some change of the coordinate system.

The sensitivity of diffusion MRI to microstructural properties and experimental factors

Diffusion MRI is a non-invasive technique to study brain microstructure. Differences in the microstructural properties of tissue, including size and anisotropy, can be represented in the signal if the appropriate method of acquisition is used. However, to depict the underlying properties, special care must be taken when designing the acquisition protocol as any changes in the procedure might impact on quantitative measurements. This work reviews state-of-the-art methods for studying brain microstructure using diffusion MRI and their sensitivity to microstructural differences and various experimental factors. Microstructural properties of the tissue at a micrometer scale can be linked to the diffusion signal at a millimeter-scale using modeling. In this paper, we first give an introduction to diffusion MRI and different encoding schemes. Then, signal representation-based methods and multi-compartment models are explained briefly. The sensitivity of the diffusion MRI signal to the microstructural components and the effects of curvedness of axonal trajectories on the diffusion signal are reviewed. Factors that impact on the quality (accuracy and precision) of derived metrics are then reviewed, including the impact of random noise, and variations in the acquisition parameters (i.e., number of sampled signals, b-value and number of acquisition shells). Finally, yet importantly, typical approaches to deal with experimental factors are depicted, including unbiased measures and harmonization. We conclude the review with some future directions and recommendations on this topic.

Brain White-Matter Degeneration Due to Aging and Parkinson Disease as Revealed by Double Diffusion Encoding

Microstructure imaging by means of multidimensional diffusion encoding is increasingly applied in clinical research, with expectations that it yields a parameter that better correlates with clinical disability than current methods based on single diffusion encoding. Under the assumption that diffusion within a voxel can be well described by a collection of diffusion tensors, several parameters of this diffusion tensor distribution can be derived, including mean size, variance of sizes, orientational dispersion, and microscopic anisotropy. The information provided by multidimensional diffusion encoding also enables us to decompose the sources of the conventional fractional anisotropy and mean kurtosis. In this study, we explored the utility of the diffusion tensor distribution approach for characterizing white-matter degeneration in aging and in Parkinson disease by using double diffusion encoding. Data from 23 healthy older subjects and 27 patients with Parkinson disease were analyzed. Advanced age was associated with greater mean size and size variances, as well as smaller microscopic anisotropy. By analyzing the parameters underlying diffusion kurtosis, we found that the reductions of kurtosis in aging and Parkinson disease reported in the literature are likely driven by the reduction in microscopic anisotropy. Furthermore, microscopic anisotropy correlated with the severity of motor impairment in the patients with Parkinson disease. The present results support the use of multidimensional diffusion encoding in clinical studies and are encouraging for its future clinical implementation.

Multimodal integration of diffusion MRI for better characterization of tissue biology

The contrast in diffusion-weighted MR images is due to variations of diffusion properties within the examined specimen. Certain microstructural information on the underlying tissues can be inferred through quantitative analyses of the diffusion-sensitized MR signals. In the first part of the paper, we review two types of approach for characterizing diffusion MRI signals: Bloch's equations with diffusion terms, and statistical descriptions. Specifically, we discuss expansions in terms of cumulants and orthogonal basis functions, the confinement tensor formalism and tensor distribution models. Further insights into the tissue properties may be obtained by integrating diffusion MRI with other techniques, which is the subject of the second part of the paper. We review examples involving magnetic susceptibility, structural tensors, internal field gradients, transverse relaxation and functional MRI. Integrating information provided by other imaging modalities (MR based or otherwise) could be a key to improve our understanding of how diffusion MRI relates to physiology and biology.

Orientationally-averaged diffusion-attenuated magnetic resonance signal for locally-anisotropic diffusion

Diffusion-attenuated MR signal for heterogeneous media has been represented as a sum of signals from anisotropic Gaussian sub-domains to the extent that this approximation is permissible. Any effect of macroscopic (global or ensemble) anisotropy in the signal can be removed by averaging the signal values obtained by differently oriented experimental schemes. The resulting average signal is identical to what one would get if the micro-domains are isotropically (e.g., randomly) distributed with respect to orientation, which is the case for “powdered” specimens. We provide exact expressions for the orientationally-averaged signal obtained via general gradient waveforms when the microdomains are characterized by a general diffusion tensor possibly featuring three distinct eigenvalues. This extends earlier results which covered only axisymmetric diffusion as well as measurement tensors. Our results are expected to be useful in not only multidimensional diffusion MR but also solid-state NMR spectroscopy due to the mathematical similarities in the two fields.

Influence of the size and curvedness of neural projections on the orientationally averaged diffusion MR signal

Neuronal and glial projections can be envisioned to be tubes of infinitesimal diameter as far as diffusion magnetic resonance (MR) measurements via clinical scanners are concerned. Recent experimental studies indicate that the decay of the orientationally-averaged signal in white-matter may be characterized by the power-law, Ē(q) ∝ q^-1, where q is the wavenumber determined by the parameters of the pulsed field gradient measurements. One particular study by McKinnon et al. [1] reports a distinctively faster decay in gray-matter. Here, we assess the role of the size and curvature of the neurites and glial arborizations in these experimental findings. To this end, we studied the signal decay for diffusion along general curves at all three temporal regimes of the traditional pulsed field gradient measurements. We show that for curvy projections, employment of longer pulse durations leads to a disappearance of the q^-1 decay, while such decay is robust when narrow gradient pulses are used. Thus, in clinical acquisitions, the lack of such a decay for a fibrous specimen can be seen as indicative of fibers that are curved. We note that the above discussion is valid for an intermediate range of q-values as the true asymptotic behavior of the signal decay is Ē(q) ∝ q^-4 for narrow pulses (through Debye-Porod law) or steeper for longer pulses. This study is expected to provide insights for interpreting the diffusion-weighted images of the central nervous system and aid in the design of acquisition strategies.