isotropic the diffusion tensor is a multiple of the identity: d → = d ⋅ i → = d ⋅ [ 1 0 0 0 1 0 0 0 1 ] , =d\cdot =d\cdot 1&0&0\\0&1&0\\0&0&1\end,} then the bloch-torrey equation will have the solution m = m bloch e − 1 3 γ 2 g 2 t 3 d ∼ e − b d 0 =_e^\gamma ^g^t^d}\sim e^ the exponential term will be referred to as the attenuation a . anisotropic diffusion will have a similar solution for the diffusion tensor, except that what will be measured is the apparent diffusion coefficient (adc). in general, the attenuation is: a = e − ∑ i , j b i j d i j b_d_ where the b i j terms incorporate the gradient fields g x , g y , and g z . grayscale the standard grayscale of dwi images is to represent increased diffusion restriction as brighter. adc image adc image of the same case of cerebral infarction as seen on dwi in section above an apparent diffusion coefficient (adc) image, or an adc map, is an mri image that more specifically shows diffusion than conventional dwi, by eliminating the t2 weighting that is otherwise inherent to conventional dwi. adc imaging does so by acquiring multiple conventional dwi images with different amounts of dwi weighting, and the change in signal is proportional to the rate of diffusion. contrary to dwi images, the standard grayscale of adc images is to represent a smaller magnitude of diffusion as darker. cerebral infarction leads to diffusion restriction, and the difference between images with various dwi weighting will therefore be minor, leading to an adc image with low signal in the infarcted area. a decreased adc may be detected minutes after a cerebral infarction. the high signal of infarcted tissue on conventional dwi is a result of its partial t2 weighting. diffusion tensor imaging diffusion tensor imaging (dti) is a magnetic resonance imaging technique that enables the measurement of the restricted diffusion of water in tissue in order to produce neural tract images instead of using this data solely for the purpose of assigning contrast or colors to pixels in a cross-sectional image. it also provides useful structural information about muscle—including heart muscle—as well as other tissues such as the prostate. in dti, each voxel has one or more pairs of parameters: a rate of diffusion and a preferred direction of diffusion—described in terms of three-dimensional space—for which that parameter is valid. the properties of each voxel of a single dti image are usually calculated by vector or tensor math from six or more different diffusion weighted acquisitions, each obtained with a different orientation of the diffusion sensitizing gradients. in some methods, hundreds of measurements—each making up a complete image—are made to generate a single resulting calculated image data set. the higher information content of a dti voxel makes it extremely sensitive to subtle pathology in the brain. in addition the directional information can be exploited at a higher level of structure to select and follow neural tracts through the brain—a process called tractography. a more precise statement of the image acquisition process is that the image-intensities at each position are attenuated, depending on the strength (b-value) and direction of the so-called magnetic diffusion gradient, as well as on the local microstructure in which the water molecules diffuse. the more attenuated the image is at a given position, the greater diffusion there is in the direction of the diffusion gradient. in order to measure the tissue's complete diffusion profile, one needs to repeat the mr scans, applying different directions (and possibly strengths) of the diffusion gradient for each scan. mathematical foundation—tensors diffusion mri relies on the mathematics and physical interpretations of the geometric quantities known as tensors. only a special case of the general mathematical notion is relevant to imaging, which is based on the concept of a symmetric matrix. diffusion itself is tensorial, but in many cases the objective is not really about trying to study brain diffusion per se, but rather just trying to take advantage of diffusion anisotropy in white matter for the purpose of finding the orientation of the axons and the magnitude or degree of anisotropy. tensors have a real, physical existence in a material or tissue so that they do not move when the coordinate system used to describe them is rotated. there are numerous different possible representations of a tensor (of rank 2), but among these, this discussion focuses on the ellipsoid because of its physical relevance to diffusion and because of its historical significance in the development of diffusion anisotropy imaging in mri. the following matrix displays the components of the diffusion tensor: d ¯ = | d x x d x y d x z d x y d y y d y z d x z d y z d z z | =d_xx}&d_&d_\\d_&d_yy}&d_\\d_&d_&d_zz}\end the same matrix of numbers can have a simultaneous second use to describe the shape and orientation of an ellipse and the same matrix of numbers can be used simultaneously in a third way for matrix mathematics to sort out eigenvectors and eigenvalues as explained below. physical tensors the idea of a tensor in physical science evolved from attempts to describe the quantity of physical properties. the first properties they were applied to were those that can be described by a single number, such as temperature. properties that can be described this way are called scalars; these can be considered tensors of rank 0, or 0th-order tensors. tensors can also be used to describe quantities that have directionality, such as mechanical force. these quantities require specification of both magnitude and direction, and are often represented with a vector. a three-dimensional vector can be described with three components: its projection on the x, y, and z axes. vectors of this sort can be considered tensors of rank 1, or 1st-order tensors. a tensor is often a physical or biophysical property that determines the relationship between two vectors. when a force is applied to an object, movement can result. if the movement is in a single direction, the transformation can be described using a vector—a tensor of rank 1. however, in a tissue, diffusion leads to movement of water molecules along trajectories that proceed along multiple directions over time, leading to a complex projection onto the cartesian axes. this pattern is reproducible if the same conditions and forces are applied to the same tissue in the same way. if there is an internal anisotropic organization of the tissue that constrains diffusion, then this fact will be reflected in the pattern of diffusion. the relationship between the properties of driving force that generate diffusion of the water molecules and the resulting pattern of their movement in the tissue can be described by a tensor. the collection of molecular displacements of this physical property can be described with nine components—each one associated with a pair of axes xx, yy, zz, xy, yx, xz, zx, yz, zy. these can be written as a matrix similar to the one at the start of this section. diffusion from a point source in the anisotropic medium of white matter behaves in a similar fashion. the first pulse of the stejskal tanner diffusion gradient effectively labels some water molecules and the second pulse effectively shows their displacement due to diffusion. each gradient direction applied measures the movement along the direction of that gradient. six or more gradients are summed to get all the measurements needed to fill in the matrix, assuming it is symmetric above and below the diagonal (red subscripts). in 1848, henri hureau de sénarmont applied a heated point to a polished crystal surface that had been coated with wax. in some materials that had "isotropic" structure, a ring of melt would spread across the surface in a circle. in anisotropic crystals the spread took the form of an ellipse. in three dimensions this spread is an ellipsoid. as adolf fick showed in the 1850s, diffusion exhibits many of the same patterns as those seen in the transfer of heat. mathematics of ellipsoids at this point, it is helpful to consider the mathematics of ellipsoids. an ellipsoid can be described by the formula: ax2 + by2 + cz2 = 1. this equation describes a quadric surface. the relative values of a, b, and c determine if the quadric describes an ellipsoid or a hyperboloid. as it turns out, three more components can be added as follows: ax2 + by2 + cz2 + dyz + ezx + fxy = 1. many combinations of a, b, c, d, e, and f still describe ellipsoids, but the additional components (d, e, f) describe the rotation of the ellipsoid relative to the orthogonal axes of the cartesian coordinate system. these six variables can be represented by a matrix similar to the tensor matrix defined at the start of this section (since diffusion is symmetric, then we only need six instead of nine components—the components below the diagonal elements of the matrix are the same as the components above the diagonal). this is what is meant when it is stated that the components of a matrix of a second order tensor can be represented by an ellipsoid—if the diffusion values of the six terms of the quadric ellipsoid are placed into the matrix, this generates an ellipsoid angled off the orthogonal grid. its shape will be more elongated if the relative anisotropy is high. when the ellipsoid/tensor is represented by a matrix, we can apply a useful technique from standard matrix mathematics and linear algebra—that is to "diagonalize" the matrix. this has two important meanings in imaging. the idea is that there are two equivalent ellipsoids—of identical shape but with different size and orientation. the first one is the measured diffusion ellipsoid sitting at an angle determined by the axons, and the second one is perfectly aligned with the three cartesian axes. the term "diagonalize" refers to the three components of the matrix along a diagonal from upper left to lower right (the components with red subscripts in the matrix at the start of this section). the variables ax2, by2, and cz2 are along the diagonal (red subscripts), but the variables d, e and f are "off diagonal". it then becomes possible to do a vector processing step in which we rewrite our matrix and replace it with a new matrix multiplied by three different vectors of unit length (length=1.0). the matrix is diagonalized because the off-diagonal components are all now zero. the rotation angles required to get to this equivalent position now appear in the three vectors and can be read out as the x, y, and z components of each of them. those three vectors are called "eigenvectors" or characteristic vectors. they contain the orientation information of the original ellipsoid. the three axes of the ellipsoid are now directly along the main orthogonal axes of the coordinate system so we can easily infer their lengths. these lengths are the eigenvalues or characteristic values. diagonalization of a matrix is done by finding a second matrix that it can be multiplied with followed by multiplication by the inverse of the second matrix—wherein the result is a new matrix in which three diagonal (xx, yy, zz) components have numbers in them but the off-diagonal components (xy, yz, zx) are 0. the second matrix provides eigenvector information. measures of anisotropy and diffusivity visualization of dti data with ellipsoids. in present-day clinical neurology, various brain pathologies may be best detected by looking at particular measures of anisotropy and diffusivity. the underlying physical process of diffusion causes a group of water molecules to move out from a central point, and gradually reach the surface of an ellipsoid if the medium is anisotropic (it would be the surface of a sphere for an isotropic medium). the ellipsoid formalism functions also as a mathematical method of organizing tensor data. measurement of an ellipsoid tensor further permits a retrospective analysis, to gather information about the process of diffusion in each voxel of the tissue. in an isotropic medium such as cerebrospinal fluid, water molecules are moving due to diffusion and they move at equal rates in all directions. by knowing the detailed effects of diffusion gradients we can