Dxx Dxy Dxz Dyx Dyy Dyz Dzx Dzy Dzz

The diffusion tensor may be diagonalized to calculate the eigenvalues (Ai, A2, A3) and corresponding eigenvectors (ei, e2, e3) of the diffusion tensor, which describe the relative amplitudes of diffusion and the directions of the principle diffusion axes. A common visual representation of the diffusion tensor is an ellipsoid with the principal axes aligned with the eigenvectors and axes lengths a function of the eigenvalues (see Fig. 1). In the case where the diffusion eigenvalues are (roughly) equal (e.g., Ai ~ A2 ~ A3), the diffusion tensor is (nearly) isotropic. When the eigenvalues are significantly different in magnitude (e.g., A4 > A2 > A3), the diffusion tensor is anisotropic. Changes in local tissue microstructure with many types of tissue injury, disease or normal physiological changes (i.e., aging) will cause changes in the eigenvalue magnitudes. Thus, the diffusion tensor is an extremely sensitive probe for characterizing both normal and abnormal tissue microstructure.

More specifically in the CNS, water diffusion is typically anisotropic in white matter regions, and isotropic in both gray matter and cerebrospinal fluid (CSF). The major diffusion eigenvector ^-direction of greatest diffusivity) is assumed to be parallel to the tract orientation in regions of homogenous white matter. This directional relationship is the basis for estimating the trajectories of white matter pathways with tractography algorithms.

The random motion of water molecules in biological tissues may cause the signal intensity to decrease in MRI. The NMR signal attenuation from molecular

Fig. 1. Schematic representations of diffusion displacement distributions for the diffusion tensor. Ellipsoids (right) are used to represent diffusion displacements. The diffusion is highly anisotropic in fibrous tissues such as white matter (left). The direction of greatest diffusivity is generally assumed to be parallel to the local direction of white matter

Fig. 1. Schematic representations of diffusion displacement distributions for the diffusion tensor. Ellipsoids (right) are used to represent diffusion displacements. The diffusion is highly anisotropic in fibrous tissues such as white matter (left). The direction of greatest diffusivity is generally assumed to be parallel to the local direction of white matter diffusion was first observed more than a half century ago by Hahn (1950). Subsequently, Stejskal & Tanner (1965) described the NMR signal attenuation in the presence of field gradients. More recently, field gradient pulses have been used to create diffusion-weighted MR images (Le Bihan 1990).

Typically, the diffusion weighting is performed using two gradient pulses with equal magnitude and duration (Fig. 2). The first gradient pulse dephases the magnetization across the sample (or voxel in imaging); and the second pulse rephases the magnetization. For stationary (non-diffusing) molecules, the phases induced by both gradient pulses will completely cancel, the magnetization will be maximally coherent, and there will be no signal attenuation from diffusion. In the case of coherent flow in the direction of the applied gradient, the bulk motion will cause the signal phase to change by different amounts for each pulse so that there will be a net phase difference, A^ = yvGSA, which is proportional to the velocity, v, the area of the gradient pulses defined by the amplitude, G, and the duration, 5, and the spacing between the pulses, A. The gyromagnetic ratio is y This is also the basis for phase contrast angiography. For the case of diffusion, the water molecules are also moving, but in arbitrary directions and with variable effective velocities. Thus, in the presence of diffusion gradients, each diffusing molecule will accumulate a different amount of phase. The diffusion-weighted signal is created by summing the magnetization from all water molecules in a voxel. The phase dispersion from diffusion will cause destructive interference, which will cause signal attenuation. For simple isotropic Gaussian diffusion, the signal attenuation for the diffusion gradient pulses in Fig. 2 is described by

where S is the diffusion-weighted signal, So is the signal without any diffusion-weighting gradients (but otherwise identical imaging parameters), D is the apparent diffusion coefficient, and b is the diffusion weighting described by the properties of the pulse pair:

Diffusion weighting may be achieved using either a bipolar gradient pulse pair or identical gradient pulses that bracket a 180° refocusing pulse as shown in Fig. 2.

Fig. 2. Spin echo pulse sequence scheme for pulsed-gradient diffusion weighting. A spin-echo refocusing pulse (180°) causes the gradient pulses to be diffusion-weighted

Fig. 2. Spin echo pulse sequence scheme for pulsed-gradient diffusion weighting. A spin-echo refocusing pulse (180°) causes the gradient pulses to be diffusion-weighted

The large gradients make DW MRI extremely sensitive to subject motion. Very small amounts of subject motion may lead to phase inconsistencies in the raw k-space data, causing severe ghosting artifacts in the reconstructed images. Recently, the advances in gradient hardware (maximum gradient amplitude and speed) and the availability of echo planar imaging (EPI) (Mansfield 1984; Turner et al. 1990) on clinical MRI scanners have made routine DW-MRI studies possible. A schematic of a DW-EPI pulse sequence is shown in Fig. 3. With EPI, the image data for a single slice may be collected in 100ms or less, effectively "freezing" any head motion. The fast acquisition speed of EPI makes it highly efficient, which is important for maximizing the image signal-to-noise ratio (SNR) and the accuracy of the diffusion measurements. Thus, single-shot EPI is the most common acquisition method for diffusion-weighted imaging. However, the disadvantages of single shot EPI can also be significant. First, both magnetic field inhomogeneities (Jezzard and Balaban 1995) and eddy currents (Haselgrove and Moore 1996) can warp the image data, thereby compromising the spatial fidelity. Distortions from eddy currents may be either minimized using bipolar diffusion gradient encoding schemes (Alexander et al. 1997; Reese et al. 2003), or corrected retrospectively using image co-registration methods (Haselgrove and Moore 1996; Andersson and Skare 2002; Rohde et al. 2004). Distortions from static field inhomogeneities may be either reduced by using parallel imaging methods such as SENSE (Pruessmann et al. 1999) or retrospectively corrected using maps of the magnetic field (Jezzard and Balaban 1995). Misalignments of k-space data on odd and even lines of k-space will lead to Nyquist or half-field ghosts in the image data. In general, the system should be calibrated to minimize this ghosting although post-processing correction methods have been developed (Zhang and Wehrli 2004). The spatial resolution of 2D EPI pulse sequences also tends

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