This work describes a new diffusion MR framework for imaging and modeling of microstructure that we call q-space trajectory imaging (QTI). The QTI framework consists of two parts: encoding and modeling. First we propose q-space trajectory encoding, which uses time-varying gradients to probe a trajectory in q-space, in contrast to traditional pulsed field gradient sequences that attempt to probe a point in q-space. Then we propose a microstructure model, the diffusion tensor distribution (DTD) model, which takes advantage of additional information provided by QTI to estimate a distributional model over diffusion tensors. We show that the QTI framework enables microstructure modeling that is not possible with the traditional pulsed gradient encoding as introduced by Stejskal and Tanner. In our analysis of QTI, we find that the well-known scalar b-value naturally extends to a tensor-valued entity, i.e., a diffusion measurement tensor, which we call the b-tensor. We show that b-tensors of rank 2 or 3 enable estimation of the mean and covariance of the DTD model in terms of a second order tensor (the diffusion tensor) and a fourth order tensor. The QTI framework has been designed to improve discrimination of the sizes, shapes, and orientations of diffusion microenvironments within tissue. We derive rotationally invariant scalar quantities describing intuitive microstructural features including size, shape, and orientation coherence measures. To demonstrate the feasibility of QTI on a clinical scanner, we performed a small pilot study comparing a group of five healthy controls with five patients with schizophrenia. The parameter maps derived from QTI were compared between the groups, and 9 out of the 14 parameters investigated showed differences between groups. The ability to measure and model the distribution of diffusion tensors, rather than a quantity that has already been averaged within a voxel, has the potential to provide a powerful paradigm for the study of complex tissue architecture.

Diffusion-weighted magnetic resonance (MR) signals reflect information about underlying tissue microstructure and cytoarchitecture. We propose a quantitative, efficient, and robust mathematical and physical framework for representing diffusion-weighted MR imaging (MRI) data obtained in “q-space,” and the corresponding “mean apparent propagator (MAP)” describing molecular displacements in “r-space.” We also define and map novel quantitative descriptors of diffusion that can be computed robustly using this MAP-MRI framework. We describe efficient analytical representation of the three-dimensional q-space MR signal in a series expansion of basis functions that accurately describes diffusion in many complex geometries. The lowest order term in this expansion contains a diffusion tensor that characterizes the Gaussian displacement distribution, equivalent to diffusion tensor MRI (DTI). Inclusion of higher order terms enables the reconstruction of the true average propagator whose projection onto the unit “displacement” sphere provides an orientational distribution function (ODF) that contains only the orientational dependence of the diffusion process. The representation characterizes novel features of diffusion anisotropy and the non-Gaussian character of the three-dimensional diffusion process. Other important measures this representation provides include the return-to-the-origin probability (RTOP), and its variants for diffusion in one- and two-dimensions—the return-to-the-plane probability (RTPP), and the return-to-the-axis probability (RTAP), respectively. These zero net displacement probabilities measure the mean compartment (pore) volume and cross-sectional area in distributions of isolated pores irrespective of the pore shape. MAP-MRI represents a new comprehensive framework to model the three-dimensional q-space signal and transform it into diffusion propagators. Experiments on an excised marmoset brain specimen demonstrate that MAP-MRI provides several novel, quantifiable parameters that capture previously obscured intrinsic features of nervous tissue microstructure. This should prove helpful for investigating the functional organization of normal and pathologic nervous tissue.

A new method for mapping diffusivity profiles in tissue is presented. The Bloch-Torrey equation is modified to include a diffusion term with an arbitrary rank Cartesian tensor. This equation is solved to give the expression for the generalized Stejskal-Tanner formula quantifying diffusive attenuation in complicated geometries. This makes it possible to calculate the components of higher-rank tensors without using the computationally-difficult spherical harmonic transform. General theoretical relations between the diffusion tensor (DT) components measured by traditional (rank-2) DT imaging (DTI) and 3D distribution of diffusivities, as measured by high angular resolution diffusion imaging (HARDI) methods, are derived. Also, the spherical tensor components from HARDI are related to the rank-2 DT. The relationships between higher-and lower-rank Cartesian DTs are also presented. The inadequacy of the traditional rank-2 tensor model is demonstrated with simulations, and the method is applied to excised rat brain data collected in a spin-echo HARDI experiment.Magn Key words: diffusion tensor magnetic resonance imaging; high angular resolution; diffusion imaging; spherical harmonics; higher-rank tensor MRI of water diffusion in tissue has been used to infer anatomical structure and to aid in the diagnosis of many pathologies (e.g., Refs. 1 and 2). This is partly due to the fact that such diffusion images are sensitive to restrictions of water molecular motion resulting from tissue structures that may be much smaller than the resolution of the images. In addition, diffusion images can be made sensitive to the direction of diffusion by employing magnetic field gradients oriented along different directions. This has proven to be very useful for the study of fibrous tissues, such as white matter (3) and muscle (4), in which the measured characteristics of water diffusion have directional dependence. Thus it is possible to calculate diffusion constants along different directions to quantify the level of anisotropy as well as the local orientation of fibers within a voxel.Up to this time, quantification of diffusion anisotropy in biological tissues has mainly involved the use of a rank-2 Cartesian tensor model (5) that was originally developed to measure the diffusional motion of molecules that have organizational anisotropy, such as in liquid crystals (6). However, unlike liquid crystals, anisotropy in tissue is primarily caused by geometric restrictions to water translational motion. The diffusion processes inside tissue are extremely complex, which makes it difficult to achieve accurate mathematical modeling (7-9) and necessitates very demanding experiments to test these models (10,11). Even though diffusion tensor imaging (DTI) applications employ a relatively simple rank-2 tensor model of diffusion, very informative maps of anisotropy, as well as fiber directions, have been generated that make fiber tract mapping possible in highly structured tissues (12-15).Despite the early success of diffusion imaging, it has been sh...

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Diffusion MRI is a non-invasive imaging technique that allows the measurement of water molecule diffusion through tissue in vivo. The directional features of water diffusion allow one to infer the connectivity patterns prevalent in tissue and possibly track changes in this connectivity over time for various clinical applications. In this paper, we present a novel statistical model for diffusion-weighted MR signal attenuation which postulates that the water molecule diffusion can be characterized by a continuous mixture of diffusion tensors. An interesting observation is that this continuous mixture and the MR signal attenuation are related through the Laplace transform of a probability distribution over symmetric positive definite matrices. We then show that when the mixing distribution is a Wishart distribution, the resulting closed form of the Laplace transform leads to a Rigaut-type asymptotic fractal expression, which has been phenomenologically used in the past to explain the MR signal decay but never with a rigorous mathematical justification until now. Our model not only includes the traditional diffusion tensor model as a special instance in the limiting case, but also can be adjusted to describe complex tissue structure involving multiple fiber populations. Using this new model in conjunction with a spherical deconvolution approach, we present an efficient scheme for estimating the water molecule displacement probability functions on a voxel-by-voxel basis. Experimental results on both simulations and real data are presented to demonstrate the robustness and accuracy of the proposed algorithms.

We consider a general double pulsed field gradient experiment with arbitrary experimental parameters and calculate an exact expression for the NMR signal attenuation from restricted geometries, which is valid at long wavelengths, i.e., when the product of the gyromagnetic ratio of the spins, the pulsed gradients' duration, and their magnitude is small compared to the reciprocal of the pore size. It is possible to observe microscopic anisotropy within the pore space induced by the boundaries of the pore, which can be used to differentiate restricted from free or multicompartmental diffusion and to estimate a characteristic pore dimension in the former case. Explicit solutions for diffusion taking place between parallel plates as well as in cylindrical and spherical pores are provided. In coherently packed cylindrical pores, it is possible to measure simultaneously the cylinders' orientation and diameter using small gradient strengths. The presence of orientational heterogeneity of cylinders is addressed, and a scheme for differentiating microscopic from ensemble anisotropy is proposed.

Stejskal and Tanner's ingenious pulsed field gradient design from 1965 has made diffusion NMR and MRI the mainstay of most studies seeking to resolve microstructural information in porous systems in general and biological systems in particular. Methods extending beyond Stejskal and Tanner's design, such as double diffusion encoding (DDE) NMR and MRI, may provide novel quantifiable metrics that are less easily inferred from conventional diffusion acquisitions. Despite the growing interest on the topic, the terminology for the pulse sequences, their parameters, and the metrics that can be derived from them remains inconsistent and disparate among groups active in DDE. Here, we present a consensus of those groups on terminology for DDE sequences and associated concepts. Furthermore, the regimes in which DDE metrics appear to provide microstructural information that cannot be achieved using more conventional counterparts (in a model-free fashion) are elucidated. We highlight in particular DDE's potential for determining microscopic diffusion anisotropy and microscopic fractional anisotropy, which offer metrics of microscopic features independent of orientation dispersion and thus provide information complementary to the standard, macroscopic, fractional anisotropy conventionally obtained by diffusion MR. Finally, we discuss future vistas and perspectives for DDE. Magn Reson Med 75:82-87, 2016. V C 2015 Wiley Periodicals, Inc.

The multiple scattering extensions of the pulsed field gradient (PFG) experiments can be used to characterize restriction-induced anisotropy at different length scales. In double-PFG acquisitions that involve two pairs of diffusion gradient pulses, the dependence of the MR signal attenuation on the angle between the two gradients is a signature of restriction that can be observed even at low gradient strengths. In this article, a comprehensive theoretical treatment of the double-PFG observation of restricted diffusion is presented. In the first part of the article, the problem is treated for arbitrarily shaped pores under idealized experimental conditions, comprising infinitesimally narrow gradient pulses with long separation times and long or vanishing mixing times. New insights are obtained when the treatment is applied to simple pore shapes of spheres, ellipsoids, and capped cylinders. The capped cylinder geometry is considered in the second part of the article where the solution for a double-PFG experiment with arbitrary experimental parameters is introduced. Although compartment shape anisotropy (CSA) is emphasized here, the findings of this article can be used in gleaning the volume, eccentricity, and orientation distribution function associated with ensembles of anisotropic compartments using double-PFG acquisitions with arbitrary experimental parameters.