Thomas H. Mareci

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

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 Reson Med 50:955–965, 2003.

Resolution of complex tissue microarchitecture using the diffusion orientation transform (DOT)

This article describes an accurate and fast method for fiber orientation mapping using multidirectional diffusion-weighted magnetic resonance (MR) data. This novel approach utilizes the Fourier transform relationship between the water displacement probabilities and diffusion-attenuated MR signal expressed in spherical coordinates. The radial part of the Fourier integral is evaluated analytically under the assumption that MR signal attenuates exponentially. The values of the resulting functions are evaluated at a fixed distance away from the origin. The spherical harmonic transform of these functions yields the Laplace series coefficients of the probabilities on a sphere of fixed radius. Alternatively, probability values can be computed nonparametrically using Legendre polynomials. Orientation maps calculated from excised rat nervous tissue data demonstrate this techniques ability to accurately resolve crossing fibers in anatomical regions such as the optic chiasm. This proposed methodology has a trivial extension to multiexponential diffusion-weighted signal decay. The developed methods will improve the reliability of tractography schemes and may make it possible to correctly identify the neural connections between functionally connected regions of the nervous system.

A constrained variational principle for direct estimation and smoothing of the diffusion tensor field from complex DWI

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/sup 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.

A novel tensor distribution model for the diffusion-weighted MR signal☆

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.

Mapping proton-proton coupling via double-quantum coherence

Several recent carbon13 structural studies have made use of a two-dimensional Fourier transform (I, 2) version of the double-quantum coherence experiment “INADEQUATE” (3-6). This technique establishes the connectivity of the carbon skeleton of a molecule unambiguously by displaying a clean spectrum of the satellites from directly coupled pairs of 13C nuclei, suppressing the singlet signal from isolated 13C spins (which cannot generate double-quantum coherence). The coupled t3C resonances (A and X) are identified by measuring the doublequantum frequency (6, + 6x) in a two-dimensional NMR experiment where the doublequantum coherence is allowed to evolve for a variable period t, (4, 5). The timing of the prep aration phase of this experiment ensures that only single-bond “direct” coupling is effective. The pulse sequence can be written

Generalized scalar measures for diffusion MRI using trace, variance, and entropy

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, regardless of the imaging technique used. Using a simplified model of fibrous tissue, we simulated anisotropy values with varying orientational complexity and tensor models. Simulations suggested that a lower‐rank tensor model may produce artificially low anisotropy values in voxels with complex structure. This was confirmed with a spin‐echo experiment performed on an excised rat brain. Magn Reson Med 53:866–876, 2005.

Visualization of neural tissue water compartments using biexponential diffusion tensor MRI

The apparent diffusion tensor (ADT) imaging method was extended to account for multiple diffusion components. A biexponential ADT imaging experiment was used to obtain separate images of rapidly and slowly diffusing water fractions in excised rat spinal cord. The fast and slow component tensors were compared and found to exhibit similar gross features, such as fractional anisotropy, in both white and gray matter. However, there were also some important differences, which are consistent with the different structures occupying intracellular and extracellular spaces. Evidence supporting the assignment of the two tensor components to extracellular and intracellular water fractions is provided by an NMR spectroscopic investigation of homogeneous samples of brain tissue. Magn Reson Med 45:580–587, 2001.

Experimental study of optimal selective 180° radiofrequency pulses

Abstract By solving an optimal control problem with the conjugate gradient algorithm, we previously obtained a set of optimal selective 180° inversion RF pulses. In this article we demonstrate by computer simulation that the magnetization responses to a selective 180° pulse are the same whether the pulse is used for refocusing or inversion. An optimal selective 180° inversion RF pulse can be used as a selective refocusing RF pulse; this is verified experimentally by a spin-echo pulse sequence. We also demonstrate by both simulation and experiment that there exists a minimum ratio of the width of a slice edge to the thickness of the selected slice at a certain slice thickness. Thus, “optimal” has a dual meaning here: a desired slice profile is optimized and the corresponding RF pulse is optimized. To obtain a slice with a sharper edge at a certain pulse length, an optimal selective pulse with a higher peak amplitude is required. The Fourier coefficients of four selective RF pulses are supplied in the Appendix.

Fiber tract mapping from diffusion tensor MRI

To understand evolving pathology in the central nervous system (CNS) and develop effective treatments, it is essential to correlate the nerve fiber connectivity with the visualization of function. Diffusion tensor imaging (DTI) can provide the fundamental information required for viewing structural connectivity. We present a novel algorithm for automatic fiber tract mapping in the CNS specifically, the spinal cord. The automatic fiber tract mapping problem is solved in two phases, namely a data smoothing phase and a fiber tract mapping phase. In the former, smoothing is achieved via a new weighted total variation (TV)-norm minimization (for vector-valued data) which strives to smooth while retaining all relevant detail. For the fiber tract mapping, a smooth 3D vector field indicating the dominant anisotropic direction at each spatial location is computed from the smoothed data. Fiber tracts are then determined as the smooth integral curves of this vector field in a variational framework.

Patterns of Gene Expression Reveal a Temporally Orchestrated Wound Healing Response in the Injured Spinal Cord

Spinal cord injury (SCI) induces a progressive pathophysiology affecting cell survival and neurological integrity via complex and evolving molecular cascades whose interrelationships are not fully understood. The present experiments were designed to: (1) determine potential functional interactions within transcriptional expression profiles obtained after a clinically relevant SCI and (2) test the consistency of transcript expression after SCI in two genetically and immunologically diverse rat strains characterized by differences in T cell competence and associated inflammatory responses. By interrogating Affymetrix U34A rat genome GeneChip microarrays, we defined the transcriptional expression patterns in midcervical contusion lesion sites between 1 and 90 d postinjury of athymic nude (AN) and Sprague Dawley (SD) strains. Stringent statistical analyses detected significant changes in 3638 probe sets, with 80 genes differing between the AN and SD groups. Subsequent detailed functional categorization of these transcripts unveiled an overall tissue remodeling response that was common to both strains. The functionally organized gene profiles were temporally distinct and correlated with repair indices observed microscopically and by magnetic resonance microimaging. Our molecular and anatomical observations have identified a novel, longitudinal perspective of the post-SCI response, namely, that of a highly orchestrated tissue repair and remodeling repertoire with a prominent cutaneous wound healing signature that is conserved between two widely differing rat strains. These results have significant bearing on the continuing development of cellular and pharmacological therapeutics directed at tissue rescue and neuronal regeneration in the injured spinal cord.