Diffusion MRI is a magnetic resonance imaging (MRI) method that produces in vivo images of biological tissues weighted with the local microstructural characteristics of water diffusion. The field of diffusion MRI can be understood in terms of two distinct classes of application—diffusion weighted MRI and diffusion tensor MRI.
In diffusion weighted imaging (DWI), each image voxel (three dimensional pixel) has an image intensity that reflects a single best measurement of the rate of water diffusion at that location. This measurement is more sensitive to early changes after a stroke than more traditional MRI measurements such as T1 or T2 relaxation rates. DWI is most applicable when the tissue of interest is dominated by isotropic water movement e.g. grey matter in the cerebral cortex and major brain nuclei—where the diffusion rate appears to be the same when measured along any axis.
Diffusion tensor imaging (DTI) is important when a tissue—such as the neural axons of white matter in the brain or muscle fibers in the heart—has an internal fibrous structure analogous to the anisotropy of some crystals. Water will then diffuse more rapidly in the direction aligned with the internal structure, and more slowly as it moves perpendicular to the preferred direction. This also means that the measured rate of diffusion will differ depending on the direction from which an observer is looking. In DTI, each voxel therefore 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 is 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.^{[1]}^{[2]}
A more precise statement of the image acquisition process is that, the imageintensities at each position are attenuated, depending on the strength (bvalue) and direction of the socalled 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.
Traditionally, in diffusionweighted imaging (DWI), three gradientdirections are applied, sufficient to estimate the trace of the diffusion tensor or 'average diffusivity', a putative measure of edema. Clinically, traceweighted images have proven to be very useful to diagnose vascular strokes in the brain, by early detection (within a couple of minutes) of the hypoxic edema.
More extended diffusion tensor imaging (DTI) scans derive neural tract directional information from the data using 3D or multidimensional vector algorithms based on three, six, or more gradient directions, sufficient to compute the diffusion tensor. The diffusion model is a rather simple model of the diffusion process, assuming homogeneity and linearity of the diffusion within each image voxel. From the diffusion tensor, diffusion anisotropy measures such as the fractional anisotropy (FA), can be computed. Moreover, the principal direction of the diffusion tensor can be used to infer the whitematter connectivity of the brain (i.e. tractography; trying to see which part of the brain is connected to which other part).
Recently, more advanced models of the diffusion process have been proposed that aim to overcome the weaknesses of the diffusion tensor model. Amongst others, these include qspace imaging and generalized diffusion tensor imaging.
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In 1956, H.C. Torrey mathematically showed how the Bloch equations for magnetization would change with the addition of diffusion.^{[3]} Torrey modified Bloch's original description of transverse magnetization to include diffusion terms and the application of a spatially varying gradient. The BlochTorrey equation neglecting relaxation is:
For the simplest case where the diffusion is isotropic the diffusion tensor is
which means that the Bloch–Torrey equation will have the solution
This demonstrates a cubic dependence of transverse magnetization on time. Anisotropic diffusion will have a similar solution method, but with a more complex diffusion tensor.
Diffusionweighted imaging is an MRI method that produces in vivo magnetic resonance images of biological tissues weighted with the local characteristics of water diffusion.
DWI is a modification of regular MRI techniques, and is an approach which utilizes the measurement of Brownian motion of molecules. Regular MRI acquisition utilizes the behaviour of protons in water to generate contrast between clinically relevant features of a particular subject. The versatile nature of MRI is due to this capability of producing contrast, called weighting. In a typical T_{1}weighted image, water molecules in a sample are excited with the imposition of a strong magnetic field. This causes many of the protons in water molecules to precess simultaneously, producing signals in MRI. In T_{2}weighted images, contrast is produced by measuring the loss of coherence or synchrony between the water protons. When water is in an environment where it can freely tumble, relaxation tends to take longer. In certain clinical situations, this can generate contrast between an area of pathology and the surrounding healthy tissue.
In diffusionweighted images, instead of a homogeneous magnetic field, the homogeneity is varied linearly by a pulsed field gradient. Since precession is proportional to the magnet strength, the protons begin to precess at different rates, resulting in dispersion of the phase and signal loss. Another gradient pulse is applied in the same direction but with opposite magnitude to refocus or rephase the spins. The refocusing will not be perfect for protons that have moved during the time interval between the pulses, and the signal measured by the MRI machine is reduced. This reduction in signal due to the application of the pulse gradient can be related to the amount of diffusion that is occurring through the following equation:
where S_{0} is the signal intensity without the diffusion weighting, S is the signal with the gradient, γ is the gyromagnetic ratio, G is the strength of the gradient pulse, δ is the duration of the pulse, Δ is the time between the two pulses, and finally, D is the diffusioncoefficient.
By rearranging the formula to isolate the diffusioncoefficient, it is possible to obtain an idea of the properties of diffusion occurring within a particular voxel (volume picture element). These values, called apparent diffusion coefficient (ADC) can then be mapped as an image, using diffusion as the contrast.
The first successful clinical application of DWI was in imaging the brain following stroke in adults. Areas which were injured during a stroke showed up "darker" on an ADC map compared to healthy tissue. At about the same time as it became evident to researchers that DWI could be used to assess the severity of injury in adult stroke patients, they also noticed that ADC values varied depending on which way the pulse gradient was applied. This orientationdependent contrast is generated by diffusion anisotropy, meaning that the diffusion in parts of the brain has directionality. This may be useful for determining structures in the brain which could restrict the flow of water in one direction, such as the myelinated axons of nerve cells (which is affected by multiple sclerosis). However, in imaging the brain following a stroke, it may actually prevent the injury from being seen. To compensate for this, it is necessary to apply a mathematical operator, called a tensor, to fully characterize the motion of water in all directions.
Diffusionweighted images are very useful to diagnose vascular strokes in the brain. It is also used more and more in the staging of non small cell lung cancer, where it is a serious candidate to replace positron emission tomography as the 'gold standard' for this type of disease. Diffusion tensor imaging is being developed for studying the diseases of the white matter of the brain as well as for studies of other body tissues (see below).
Diffusion tensor imaging (DTI) is a magnetic resonance imaging (MRI) 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.^{[4]}
In 1990, Michael Moseley reported that water diffusion in white matter was anisotropic—the effect of diffusion on proton relaxation varied depending on the orientation of tracts relative to the orientation of the diffusion gradient applied by the imaging scanner. He also pointed out that this should best be described by a tensor.^{[5]} Aaron Filler and colleagues reported in 1991 on the use of MRI for tract tracing in the brain using a contrast agent method but pointed out that Moseley's report on polarized water diffusion along nerves would affect the development of tract tracing.^{[6]} A few months after submitting that report, in 1991, the first successful use of diffusion anisotropy data to carry out the tracing of neural tracts curving through the brain without contrast agents was accomplished.^{[1]}^{[7]}^{[8]} Filler and colleagues identified both vector and tensor based methods in the patents in July 1992,^{[8]} before any other group, but the data for these initial images was obtained using the following sets of vector formulas that provide Euler angles and magnitude for the principal axis of diffusion in a voxel, accurately modeling the axonal directions that cause the restrictions to the direction of diffusion:
The first vector calculated image using diffusion anisotropy to show neural tracts curving through the brain in Macaca fascicularis (Filler et al. 1992)^{[8]} 
Aaron Filler loading the 4.7 tesla, 70 millitesla per meter experimental system where experiments leading to the diffusion anisotropy imaging patent were carried out. 
Peter J. Basser, James Mattiello and Denis Le Bihan showed how the classical ellipsoid tensor formalism could be deployed to analyze diffusion MR data 
The use of mixed contributions from gradients in the three primary orthogonal axes in order to generate an infinite number of differently oriented gradients for tensor analysis was also identified in 1992 as the basis for accomplishing tensor descriptions of water diffusion in MRI voxels.^{[9]}^{[10]}^{[11]} Both vector and tensor methods provide a "rotationally invariant" measurement—the magnitude will be the same no matter how the tract is oriented relative to the gradient axes—and both provide a three dimensional direction in space, however the tensor method is more efficient and accurate for carrying out tractography.^{[1]} Practically, this class of calculated image places heavy demands on image registration—all of the images collected should ideally be identically shaped and positioned so that the calculated composite image will be correct. In the original FORTRAN program written on a MacIntosh computer by Todd Richards in late 1991, all of the tasks of image registration, and normalized anisotropy assessment (stated as a fraction of 1 and corrected for a "B0" (nondiffusion) basis), as well as calculation of the Euler angles, image generation and tract tracing were simplified by initial development with vectors (three diffusion images plus one nondiffusion image) as opposed to six or more required for a full 2nd rank tensor analysis.
The use of electromagnetic data acquisitions from six or more directions to construct a tensor ellipsoid was known from other fields at the time,^{[12]} as was the use of the tensor ellipsoid to describe diffusion.^{[13]}^{[14]} The inventive step of DTI therefore involved two aspects:
The abstract with the first tractogram appeared at the August 1992 meeting of the Society for Magnetic Resonance in Medicine,^{[7]} Widespread research in the field followed a presentation on March 28, 1993 when Michael Moseley represented the tractographic images from the Filler group—describing the new range of neuropathology it had made detectable—and drew attention to this new direction in MRI at a plenary session of Society for Magnetic Resonance Imaging in front of an audience of 700 MRI scientists.^{[15]}^{[16]} Many groups then paid attention to the possibility of using tensor based diffusion anisotropy imaging for neural tract tracing, beginning to optimize tractography. There is now an annual "Fibre Cup" in which various groups compete to provide the most effective new tractographic algorithm. Further advances in the development of tractography can be attributed to Mori,^{[17]} Pierpaoli,^{[18]} Lazar,^{[19]} Conturo ^{[20]} and many others.
Diffusion Tensor Imaging became widely used within the MRI community following the work of Basser, Mattliello and LeBihan.^{[21]} Working at the National Institutes of Health, Peter Basser and his coworkers published a series of highly influential papers in the 1990s, establishing diffusion tensor imaging as a viable imaging method^{[22]}^{[23]} .^{[24]} For this body of work, Basser was awarded the 2008 International Society for Magnetic Resonance in Medicine Gold Medal for "his pioneering and innovative scientific contributions in the development of Diffusion Tensor Imaging (DTI)."
Early in the development of DTI based tractography, a number of researchers pointed out a flaw in the diffusion tensor model. The tensor analysis assumes that there is a single ellipsoid in each imaging voxel—as if all of the axons traveling through a voxel traveled in exactly the same direction. This is often true, but it can be estimated that in more than 30% of the voxels in a standard resolution brain image, there are at least two different neural tracts traveling in different directions that pass through each other. In the classic diffusion ellipsoid tensor model, the information from the crossing tract just appears as noise or unexplained decreased anisotropy in a given voxel. David Tuch was among the first to describe a working solution to this problem.^{[25]}^{[26]}
The idea is best understood by conceptually placing a kind of geodesic dome around each image voxel. This an icosahedron that provides a mathematical basis for passing a large number of evenly spaced gradient trajectories through the voxel—each coinciding with one of the apices of the icosahedron. Basically, we are now going to look into the voxel from a large number of different directions (typically 40 or more). We use "ntuple" tessellations to add more evenly spaced apices to the original icosahedron (20 faces)—an idea that also had its precedents in paleomagnetism research several decades earlier.^{[27]} We just want to know which direction lines turn up the maximum anisotropic diffusion measures. If there is a single tract, there will be just two maxima pointing in opposite directions. If two tracts cross in the voxel, there will be two pairs of maxima, and so on. We can still use tensor math to use the maxima to select groups of gradients to package into several different tensor ellipsoids in the same voxel, or use more complex higher rank tensors analyses,^{[28]} or we can do a true "model free" analysis that just picks the maxima and goes on about doing the tractography. We could use very high angular resolution (256 different directions) but it is often necessary to do ten or fifteen complete runs to get the information correct and this could mean 2,000 or more images—it gets to be over an hour to do the image and so becomes impossible. At forty angles, we can do 10 repetitions and get done in ten minutes. Also, in order to make this work, the gradient strengths have to be considerably higher than for standard DTI. This is because we can reduce the apparent noise (nondiffusion contributions to signal) at higher b values (a combination of gradient strength and pulse duration) and improve the spatial resolution.
The QBall method of tractography is an implementation of the HARDI approach in which David Tuch provides a mathematical alternative to the tensor model.^{[29]} Instead of forcing the diffusion anisotropy data into a group of tensors, the mathematics used deploys both probability distributions and a classic bit of geometric tomography and vector math developed nearly 100 years ago—the Funk Radon Transform.^{[30]}
In presentday clinical neurology, various brain pathologies may be best detected by looking at particular measures of anisotropy and diffusivity. The underlying physical process of diffusion (by Brownian motion) 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 generate a formula that allows us to convert the signal attenuation of an MRI voxel into a numerical measure of diffusion—the diffusion coefficient D. When various barriers and restricting factors such as cell membranes and microtubules interfere with the free diffusion, we are measuring an "apparent diffusion coefficient" or ADC because the measurement misses all the local effects and treats it as if all the movement rates were solely due to Brownian motion. The ADC in anisotropic tissue varies depending on the direction in which it is measured. Diffusion is fast along the length of (parallel to) an axon, and slower perpendicularly across it.
Once we have measured the voxel from six or more directions and corrected for attenuations due to T2 and T1 effects, we can use information from our calculated ellipsoid tensor to describe what is happening in the voxel. If you consider an ellipsoid sitting at an angle in a Cartesian grid (x, y, z axes) then you can consider the projection of that ellipse onto the three axes. The three projections can give you the ADC along each of the three axes ADC_{x}, ADC_{y}, ADC_{z}. This leads to the idea of describing the average diffusivity in the voxel which will simply be
We use the i subscript to signify that this is what the isotropic diffusion coefficient would be with the effects of anisotropy averaged out.
The ellipsoid itself has a principal long axis and then two more small axes that describe its width and depth. All three of these are perpendicular to each other and cross at the center point of the ellipsoid. We call the axes in this setting eigenvectors and the measures of their lengths eigenvalues. The lengths are symbolized by the Greek letter λ (lambda). The long one pointing along the axon direction will be λ_{1} and the two small axes will have lengths λ_{2} and λ_{3}. In the setting of the DTI tensor ellipsoid, we can consider each of these as a measure of the diffusivity along each of the three primary axes of the ellipsoid. This is a little different from the ADC since that was a projection on the axis, while λ is an actual measurement of the ellipsoid we have calculated.
The diffusivity along the principal axis, λ_{1} is also called the longitudinal diffusivity or the axial diffusivity or even the parallel diffusivity λ_{ǁ}. Historically, this is closest to what Richards originally measured with the vector length in 1991.^{[7]} The diffusivities in the two minor axes are often averaged to produce a measure of radial diffusivity
This quantity is an assessment of the degree of restriction due to membranes and other effects and proves to be a sensitive measure of degenerative pathology in some neurological conditions.^{[31]} It can also be called the perpendicular diffusivity (λ).
Another commonly used measure that summarizes the total diffusivity is the Trace—which is the sum of the three eigenvalues,
If we divide this sum by three we have the mean diffusivity
Aside from describing the amount of diffusion, it is often important to describe the relative degree of anisotropy in a voxel. At one extreme would be the sphere of isotropic diffusion and at the other extreme would be a cigar or pencil shaped very thin prolate spheroid. The simplest measure is obtained by dividing the longest axis of the ellipsoid by the shortest = (λ_{1}/λ_{3}). However this proves to be very susceptible to measurement noise, so increasingly complex measures were developed to capture the measure while minimizing the noise. An important element of these calculations is the sum of squares of the diffusivity differences = (λ_{1} − λ_{2})^{ 2} + (λ_{1} − λ_{ 3})^{2} + (λ_{2} − λ_{3})^{2}. We use the square root of the sum of squares to obtained a sort of weighted average—dominated by the largest component. One objective is to keep the number near 0 if the voxel is spherical but near 1 if it is elongate. This leads to the fractional Anisotropy or FA which is the square root of the sum of squares (SRSS) of the diffusivity differences, divided by the SRSS of the diffusivities. When the second and third axes are small relative to the principal axis, the number in the numerator is almost equal the number in the denominator. We also multiply by the square root of 1/2 = 0.707 to help get the resulting number to be less than one. The whole formula for FA looks like this:
The fractional anisotropy can also be separated into linear, planar, and spherical measures depending on the "shape" of the diffusion ellipsoid.^{[32]} For example, a "cigar" shaped prolate ellipsoid indicates a strongly linear anisotropy, a "flying saucer" or oblate spheroid represents diffusion in a plane, and a sphere is indicative of isotropic diffusion, equal in all directions. If the eigenvalues of the diffusion vector are sorted such that , then the measures can be calculated as follows:
For the linear case, where ,
For the planar case, where ,
For the spherical case, where ,
Each measure lies between 0 and 1 and they sum to unity. An additional anisotropy measure can used to describe the deviation from the spherical case:
The principal application is in the imaging of white matter where the location, orientation, and anisotropy of the tracts can be measured. The architecture of the axons in parallel bundles, and their myelin sheaths, facilitate the diffusion of the water molecules preferentially along their main direction. Such preferentially oriented diffusion is called anisotropic diffusion.
The imaging of this property is an extension of diffusion MRI. If a series of diffusion gradients (i.e. magnetic field variations in the MRI magnet) are applied that can determine at least 3 directional vectors (use of 6 different gradients is the minimum and additional gradients improve the accuracy for "offdiagonal" information), it is possible to calculate, for each voxel, a tensor (i.e. a symmetric positive definite 3×3 matrix) that describes the 3dimensional shape of diffusion. The fiber direction is indicated by the tensor's main eigenvector. This vector can be colorcoded, yielding a cartography of the tracts' position and direction (red for leftright, blue for superiorinferior, and green for anteriorposterior). The brightness is weighted by the fractional anisotropy which is a scalar measure of the degree of anisotropy in a given voxel. Mean diffusivity (MD) or trace is a scalar measure of the total diffusion within a voxel. These measures are commonly used clinically to localize white matter lesions that do not show up on other forms of clinical MRI.
Diffusion tensor imaging data can be used to perform tractography within white matter. Fiber tracking algorithms can be used to track a fiber along its whole length (e.g. the corticospinal tract, through which the motor information transit from the motor cortex to the spinal cord and the peripheral nerves). Tractography is a useful tool for measuring deficits in white matter, such as in aging. Its estimation of fiber orientation and strength is increasingly accurate, and it has widespread potential implications in the fields of cognitive neuroscience and neurobiology.
Some clinical applications of DTI are in the tractspecific localization of white matter lesions such as trauma and in defining the severity of diffuse traumatic brain injury. The localization of tumors in relation to the white matter tracts (infiltration, deflection), has been one of the most important initial applications. In surgical planning for some types of brain tumors, surgery is aided by knowing the proximity and relative position of the corticospinal tract and a tumor.
The use of DTI for the assessment of white matter in development, pathology and degeneration has been the focus of over 2,500 research publications since 2005. It promises to be very helpful in distinguishing Alzheimer's disease from other types of dementia. Applications in brain research cover e.g. connectionistic investigation of neural networks in vivo.^{[33]}
DTI also has applications in the characterization of skeletal and cardiac muscle. The sensitivity to fiber orientation also appears to be helpful in the arena of sports medicine where it greatly aids imaging of structure and injury in muscles and tendons.
A recent study at BarnesJewish Hospital and Washington University School of Medicine of healthy persons and both newly affected and chronicallyafflicted individuals with optic neuritis caused by multiple sclerosis (MS) showed that DTI can be used to assess the course of the condition's effects on the eye's optic nerve and the vision because it can assess axial diffusivity of water flow in the area.
In October 2009 a report appeared documenting a localized increase in fractional anisotropy following training of a complex visuomotor skill (juggling). This was claimed to be the first evidence for experiencedependent changes in white matter microstructure in healthy human adults.^{[34]}
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 symmetric matrix.^{[35]} 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 don't 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:
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.
The idea of a tensor in physical science evolved from attempts to describe the quantity of a given physical property. The first instance are those properties that can be described by a single number  such as temperature. There is no directionality in temperature. A property that can be described in this way is denoted as a scalar—it may be also considered as a tensor of rank 0. The next level of complexity concerns quantities that can only be described with reference to direction—a basic example is mechanical force—these require a description of magnitude and direction. Properties with a simple directional aspect are described by a vector—often represented by an arrow—that has magnitude and direction. A vector can be described by providing its three components—its projection on the xaxis, the yaxis and the zaxis. Vectors of this sort can be tensors of rank 1.
A tensor is often a physical or biophysical property that determines the relationship between two vectors. When an electric field is applied to a crystal a current can be detected. If the current were in a notional infinitely thin wire, then it would flow purely in a single direction—this transformation could be described using a tensor of rank 1—a vector. However, in a crystal, the applied field will lead to movement of charge in an expanding pattern that proceeds along multiple different directions at the same time, leading to a complex projection onto the Cartesian axes. This pattern will be reproducible—the same each time we apply the same field to the same crystal in the same way—particularly as long as the external field is applied in the same orientation relative to the internal anisotropic organization of the crystal. The relationship between the direction of the applied field and the complex pattern of the charge movement in the crystal is described by a tensor. This collection of charge displacements in this physical property can be described with nine components—each of which is associated with a pair of axes xx, yy, zz, xy, yx, xz, zx, yz, zy.^{[36]} 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 where they have moved to due to diffusion. Each gradient direction applied measures the movement along the direction of that applied gradient. We summate six or more gradients to get all the measurements we need to fill in the matrix —assuming it is symmetric above and below the diagonal (red subscripts).
In 1848, Henri Hureau de Sénarmont^{[37]} 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 melting 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 follows many of the same paths and rules as does heat.
At this point, it's helpful to consider a little of the mathematics of ellipsoids. An ellipsoid can be described by the formula: ax^{2} + by^{2} + cz^{2} = 1. A great way to get a feel for this is to download "Graphing Calculator" nucalc.com. This equation describes a quadric surface. The relative value of a, b, and c determine if the quadric will be an ellipsoid or a hyperboloid.
As it turns out, three more components can be added as follows: ax^{2} + by^{2} + cz^{2} + dyz + ezx + fxy = 1. The results of many mixes of a, b, c, d, e, and f will still be ellipsoids, but the additional components (d, e, f) typically force the ellipsoid to be rotated into an angled position off of the orthogonal axes of the Cartesian coordinate system. These six variables can be written into a matrix which looks a lot like the tensor matrix appearing 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 for our 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, it becomes possible to do something helpful 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 ax^{2}, by^{2}, and cz^{2} 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 from 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 read out the length of each. 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 offdiagonal components (xy, yz, zx) are 0. The second matrix provides eigenvector information.
For DTI, it is generally possible to use linear algebra, matrix mathematics and vector mathematics to process the analysis of the tensor data.
In some cases, the full set of tensor properties is of interest, but for tractography it is usually necessary to know only the magnitude and orientation of the primary axis or vector. This primary axis—the one with the greatest length—is the largest eigenvalue and its orientation is encoded in its matched eigenvector. Only one axis is needed because the interest is in the vectorial property of axon direction to accomplish tractography.
