Generalized diffusion tensor imaging and analytical relationships between diffusion tensor imaging and high angular resolution diffusion imaging

Abstract

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...

Diffusion MR imaging characteristics of the developing primate brain

“…The best way to convey these statistical features in terms of tissue characteristics such as cell density, type, or shape is currently an area of active investigation (Alexander et al, 2002;Clark et al, 2002;Frank, 2002;Liu et al, 2004;Maier et al, 2004a,b;Ozarslan and Mareci, 2003;Tuch et al, 2002Tuch et al, , 2003Zhan et al, 2003). An optimal diffusion model retains the full complement of structural information present in the data, but contains a minimal number of adjustable parameters.…”

“…The most commonly adopted mathematical formalism used to express diffusion anisotropy in brain is the diffusion tensor imaging (DTI) model (Basser et al, 1994). Other expressions, such as a sum of diffusion tensors (Clark et al, 2002; Maier et al., 2004a,b), a spherical harmonic expansion of the diffusion profile (Alexander et al, 2002;Frank, 2002;Zhan et al, 2003), a function possessing higher moments than the DTI model function (e.g., kurtosis) (Liu et al, 2004;Ozarslan and Mareci, 2003), and methods for expressing diffusive displacement without reference to simplifying models (Tuch et al, 2002(Tuch et al, , 2003 have been proposed. We have found that water diffusion within fixed baboon brain may be modeled with high fidelity by selecting a single model from a series of expressions possessing varying degrees of radial symmetry on a voxel-by-voxel basis, provided that a constant term is added to the expression for the MR signal (see Appendix A).…”

Hybrid diffusion imaging

“…The methods vary in their acquisition sampling and analysis approaches. High angular resolution diffusion imaging (HARDI) methods (including QBI: q-ball imaging; and GDTI: generalized DTI) typically obtain images with a constant moderate to high level of diffusion weighting (b > 1500 s/mm 2 ) over a large number (> 30) of non-collinear encoding directions (Liu et al, 2004;Özarslan and Mareci, 2003). The HARDI approaches may be used to detect and characterize regions of crossing white matter tracts; however, they are insufficient for characterizing aspects of non-monoexponential diffusion.…”

Efficient and robust computation of PDF features from diffusion MR signal

“…Indeed, whereas HARDI focuses on having a lot of samples on a single sphere, HYDI focuses on a better distribution of samples on the qspace. Very few methods have been proposed so far to take advantage of this kind of sampling: the generalized DTI (GDTI) method [28,29] is based on the Fick's diffusion law, and the DOT method [26,30] has been extended to multi-exponential radial decay. Nonetheless these methods use a larger set of data and are still based on prior models of the input signal radial behavior.…”