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.

This paper details the derivation of rotationally invariant scalar measures from higher-rank diffusion tensors (DTs) and functions defined on a unit sphere. This was accomplished with the use of an expression that generalizes the evaluation of the trace operator to tensors of arbitrary rank, and even to functions whose domains are the unit sphere. It is shown that the mean diffusivity is invariant to the selection of tensor rank for the model used. However, this rank invariance does not apply to the anisotropy measures. Therefore, a variance-based, general anisotropy measure is proposed. Also an information theoretical parametrization of anisotropy is introduced that is frequently more consistent with the meaning attributed to anisotropy. We accomplished this by associating anisotropy with the amount of orientational information present in the data Diffusional attenuation of the magnetic resonance (MR) signal as a result of the mixing of phase incoherent spins has been known since Hahn (1) first introduced spin echoes in the early days of MR. When the diffusion process is non-Fickian (i.e., the molecular flux density is not oriented opposite to the concentration gradient), the diffusivity is better quantified with a symmetric, rank-2, positive definite tensor (2). At typical resolutions for MRI and microscopy, it has been shown that a macroscopic effective diffusion tensor (DT; henceforth referred to as the "traditional" rank-2 DT) can be calculated that is assumed to have properties similar to those of the true DT (3). The traditional DT has six distinct components, implying that six independent numbers are needed to fully describe this tensor. These six numbers can be chosen to be quantities that are more meaningful than the tensor components. As an example, the principal eigenvector, which can be expressed in terms of two numbers (such as the azimuthal and polar angles that specify a direction in three-dimensional space), has been hypothesized to give the local fiber orientation within the tissue (4). Two other numbers, the mean diffusivity and a measure of anisotropy, were found to be useful in quantitative studies in which comparative analyses were performed (5).The inability of traditional DT imaging (DTI) to resolve more than one fiber direction in a voxel has prompted recent interest in formulating more sophisticated techniques. Tuch et al. (6) developed a clinically feasible approach called high-angular-resolution diffusion imaging (HARDI), in which apparent diffusion coefficients are measured along many directions distributed almost isotropically on the surface of a sphere. In a recent publication (7), we expressed the diffusivities in terms of Cartesian tensors of rank higher than 2 that enabled a straightforward generalization of the Bloch-Torrey result (8) and led to the formulation of a generalized Stejskal-Tanner (9) equation:where u is a unit vector that specifies the direction of the diffusion gradients, whose components are given bywhere and are the polar and azimuthal angles, respectively. In ...

In this paper, we present a novel constrained variational principle for simultaneous smoothing and estimation of the diffusion tensor field from complex valued diffusion-weighted images (DWI). The constrained variational principle involves the minimization of a regularization term of L(P) norms, subject to a nonlinear inequality constraint on the data. The data term we employ is the original Stejskal-Tanner equation instead of the linearized version usually employed in literature. The complex valued nonlinear form leads to a more accurate (when compared to the linearized version) estimate of the tensor field. The inequality constraint requires that the nonlinear least squares data term be bounded from above by a known tolerance factor. Finally, in order to accommodate the positive definite constraint on the diffusion tensor, it is expressed in terms of Cholesky factors and estimated. The constrained variational principle is solved using the augmented Lagrangian technique in conjunction with the limited memory quasi-Newton method. Experiments with complex-valued synthetic and real data are shown to depict the performance of our tensor field estimation and smoothing algorithm.