Tensor

Tensors are geometric objects that describe linear relations between geometric vectors, scalars, and other tensors. Elementary examples of such relations include the dot product, the cross product, and linear maps. Euclidean vectors, often used in physics and engineering applications, and scalars themselves are also tensors. A more sophisticated example is the Cauchy stress tensor T, which takes a direction v as input and produces the stress T^(v) on the surface normal to this vector for output, thus expressing a relationship between these two vectors, shown in the figure (right).

In terms of a coordinate basis or fixed frame of reference, a tensor can be represented as an organized multidimensional array of numerical values. The order (also degree) of a tensor is the dimensionality of the array needed to represent it, or equivalently, the number of indices needed to label a component of that array. For example, a linear map is represented by a matrix (a 2-dimensional array) in a basis, and therefore is a 2nd-order tensor. A vector is represented as a 1-dimensional array in a basis, and is a 1st-order tensor. Scalars are single numbers and are thus 0th-order tensors. Because they express a relationship between vectors, tensors themselves must be independent of a particular choice of coordinate system. The coordinate independence of a tensor then takes the form of a "covariant" transformation law that relates the array computed in one coordinate system to that computed in another one. The precise form of the transformation law determines the type (or valence) of the tensor. The tensor type is a pair of natural numbers (n, m), where n is the number of contravariant indices and m is the number of covariant indices. The total order of a tensor is the sum of these two numbers.

Diffusion MRI

Diffusion MRI (or dMRI) is a magnetic resonance imaging (MRI) method which came into existence in the mid-1980s. It allows the mapping of the diffusion process of molecules, mainly water, in biological tissues, in vivo and non-invasively. Molecular diffusion in tissues is not free, but reflects interactions with many obstacles, such as macromolecules, fibers, and membranes. Water molecule diffusion patterns can therefore reveal microscopic details about tissue architecture, either normal or in a diseased state.

Introduction

The first diffusion MRI images of the normal and diseased brain were made public in 1985. Since then, diffusion MRI, also referred to as diffusion tensor imaging or DTI (see section below) has been extraordinarily successful. Its main clinical application has been in the study and treatment of neurological disorders, especially for the management of patients with acute stroke. Because it can reveal abnormalities in white matter fiber structure and provide models of brain connectivity, it is rapidly becoming a standard for white matter disorders. The ability to visualize anatomical connections between different parts of the brain, noninvasively and on an individual basis, has emerged as a major breakthrough for neuroscience's Human Connectome Project. More recently, a new field has emerged, diffusion functional MRI (DfMRI) as it was suggested that with dMRI one could also get images of neuronal activation in the brain. Finally, the method of diffusion MRI has also been shown to be sensitive to perfusion, as the movement of water in blood vessels mimics a random process, intravoxel incoherent motion (IVIM). IVIM dMRI is rapidly becoming a major method to obtain images of perfusion in the body, especially for cancer detection and monitoring.