DTI Color Map
Diffusion-weighted magnetic resonance imaging (DWI or DW-MRI) is an imaging method that uses the diffusion of water molecules to generate contrast in MR images. 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. A special kind of DWI, diffusion tensor imaging (DTI), has been used extensively to map white matter tractography in the brain.
- 1 Introduction
- 2 Diffusion model
- 3 Magnetization dynamics
- 4 Diffusion imaging
- 5 Diffusion tensor imaging
- 6 Mathematical foundation—tensors
- 7 HARDI: High-angular-resolution diffusion imaging and Q-ball vector analysis
- 8 See also
- 9 Notes
- 10 References
- 11 External links
The first diffusion MRI images of the normal and diseased brain were made public in 1985. 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 DWI 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.
In diffusion weighted imaging (DWI), the intensity of each image element (voxel) reflects the best estimate of the rate of water diffusion at that location. Because the mobility of water is driven by thermal agitation and highly dependent on its cellular environment, the hypothesis behind DWI is that findings may indicate (early) pathologic change. For instance, DWI is more sensitive to early changes after a stroke than more traditional MRI measurements such as T1 or T2 relaxation rates. A variant of diffusion weighted imaging, diffusion spectrum imaging (DSI), was used in deriving the Connectome data sets; DSI is a variant of diffusion-weighted imaging that is sensitive to intra-voxel heterogeneities in diffusion directions caused by crossing fiber tracts and thus allows more accurate mapping of axonal trajectories than other diffusion imaging approaches.
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, or in the body—where the diffusion rate appears to be the same when measured along any axis. However, DWI also remains sensitive to T1 and T2 relaxation. To entangle diffusion and relaxation effects on image contrast, one may obtain quantitative images of the diffusion coefficient, or more exactly the apparent diffusion coefficient (ADC). The ADC concept was introduced to take into account the fact that the diffusion process is complex in biological tissues and reflects several different mechanisms.
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.
Traditionally, in diffusion-weighted imaging (DWI), three gradient-directions are applied, sufficient to estimate the trace of the diffusion tensor or 'average diffusivity', a putative measure of edema. Clinically, trace-weighted 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 DTI scans derive neural tract directional information from the data using 3D or multidimensional vector algorithms based on 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 white-matter 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 q-space imaging  and generalized diffusion tensor imaging.
Putting the two together, we get the diffusion equation:
which has terms for precession, T2 relaxation, and T1 relaxation.
In 1956, H.C. Torrey mathematically showed how the Bloch equations for magnetization would change with the addition of diffusion. Torrey modified Bloch's original description of transverse magnetization to include diffusion terms and the application of a spatially varying gradient. Since the magnetization is a vector, there are 3 diffusion equations, one for each dimension. The Bloch–Torrey equation is:
where is now the diffusion tensor.
For the simplest case where the diffusion is isotropic the diffusion tensor is a multiple of the identity:
then the Bloch–Torrey equation will have the solution
The exponential term will be referred to as the attenuation . 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:
where the terms incorporate the gradient fields , , and .
Diffusion imaging is an MRI method that produces in vivo magnetic resonance images of biological tissues sensitized with the local characteristics of molecular diffusion, generally water (but other moieties can also be investigated using MR spectroscopic approaches). MRI can be made sensitive to the 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 related to the structure of tissues at microscopic level. In a typical -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 -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.
To sensitize MRI images to diffusion, 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 magnitude but with opposite direction 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 "field gradient pulse" method was initially devised for NMR by Stejskal and Tanner  who derived the reduction in signal due to the application of the pulse gradient related to the amount of diffusion that is occurring through the following equation:
where is the signal intensity without the diffusion weighting, is the signal with the gradient, is the gyromagnetic ratio, is the strength of the gradient pulse, is the duration of the pulse, is the time between the two pulses, and finally, is the diffusion-coefficient.
In order to localize this signal attenuation to get images of diffusion one has to combine the pulsed magnetic field gradient pulses used for MRI (aimed at localization of the signal, but those gradient pulses are too weak to produce a diffusion related attenuation) with additional "motion-probing" gradient pulses, according to the Stejskal and Tanner method. This combination is not trivial, as cross-terms arise between all gradient pulses. The equation set by Stejskal and Tanner then becomes inaccurate and the signal attenuation must be calculated, either analytically or numerically, integrating all gradient pulses present in the MRI sequence and their interactions. The result quickly becomes very complex given the many pulses present in the MRI sequence and, as a simplication, Le Bihan suggested to gather all the gradient terms in a "b factor" (which depends only on the acquisition parameters), so that the signal attenuation simply becomes:
Also, the diffusion coefficient, , is replaced by an apparent diffusion coefficient, , to indicate that the diffusion process is not free in tissues, but hindered and modulated by many mechanisms (restriction in closed spaces, tortuosity around obstacles, etc.) and that other sources of IntraVoxel Incoherent Motion (IVIM) such as blood flow in small vessels or cerebrospinal fluid in ventricles also contribute to the signal attenuation. At the end, images are "weighted" by the diffusion process: In those diffusion-weighted images (DWI) the signal is more attenuated the faster the diffusion and the larger the b factor is. However, those diffusion-weighted images are still also sensitive to T1 and T2 relaxivity contrast, which can sometimes be confusing. It is possible to calculate "pure" diffusion maps (or more exactly ADC maps where the ADC is the sole source of contrast) by collecting images with at least 2 different values, and , of the b factor according to:
Although this ADC concept has been extremely successful, especially for clinical applications, it has been challenged recently, as new, more comprehensive models of diffusion in biological tissues have been introduced. Those models have been made necessary, as diffusion in tissues is not free. In this condition, the ADC seems to depend on the choice of b values (the ADC seems to decrease when using larger b values), as the plot of ln(S/So) is not linear with the b factor, as expected from the above equations. This deviation from a free diffusion behavior is what makes diffusion MRI so successful, as the ADC is very sensitive to changes in tissue microstructure. On the other hand, modeling diffusion in tissues is becoming very complex. Among most popular models are the biexponential model, which assumes the presence of 2 water pools in slow or intermediate exchange  and the cumulant-expansion (also called Kurtosis) model  which does not necessarily require the presence of 2 pools.
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 orientation-dependent 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 use a mathematical construct, called a tensor, to fully characterize the motion of water in all directions.
Diffusion-weighted 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).
The main clinical application of diffusion-weighted images has been neurological disorders, especially for the management of acute stroke patients. However, diffusion MRI was originally developed to image the liver. In 1984, Denis Le Bihan, then a medical resident and doctoral student in physics, was asked whether MRI could possibly differentiate liver tumors from angiomas. At that time there were no clinically available MRI contrast media. Le Bihan hypothesized that a molecular diffusion measurement would result in low values for solid tumors, because of some kind of molecular movement 'restriction', while the same measure would be somewhat enhanced in flowing blood. Based on the pioneering work of Stejskal and Tanner in the 1960s he suspected that diffusion encoding could be accomplished using specific magnetic gradient pulses. However this required mixing of such pulses with those used in the MRI sequence for spatial encoding. Thus the diffusion coefficients had to be localized, or mapped on to the tissues. This had never been done before, especially in vivo, with any technique. In the first diffusion MRI paper  he introduced the 'b factor' (from his name, "B"ihan) to take into account the existence of cross-terms between applied diffusion-sensitizing and imaging gradient pulses, and the 'apparent diffusion coefficient' (acronym ADC) concept, as "diffusion" measured by MRI in tissues is modulated by several mechanisms (restriction, hindrance, etc.) and other IntraVoxel Incoherent Motions (IVIM), such as blood microcirculation, etc., all the ingredients necessary to make diffusion MRI successfully working. The first images were obtained on an almost 'home-made' 0.5T scanner called 'Magniscan' by then CGR (Companie Générale de Radiologie), a French company located in Buc near Versailles in France (now GEMS European Headquarters) which patented diffusion and IVIM MRI.
Indeed, the first trials in the liver were very disappointing, and he quickly switched to the brain. He scanned his own brain and that of some of his colleagues before investigating patients (Fig.1). The world first diffusion images of the normal brain were made public in 1985 in London at the international SMRM meeting and the first diffusion images of the brain of patients were shown at the RSNA meeting in Chicago the same year (then published in Radiology).
At that time diffusion MRI was a very slow method, very sensitive to motion artifacts. It was not until the availability of echo-planar imaging (EPI) on clinical MRI scanners that diffusion and IVIM MRI (and soon later DTI) could really take off in the early 1990s, as results became much more reliable and free of motion artifacts. This move into the clinical field was the result of an intense and fruitful collaboration between Denis Le Bihan and Robert Turner, who was also at NIH. With Turner's unique expertise in gradient hardware and EPI gained during the years he spent with Peter Mansfield, they were able to obtain the first IVIM-EPI images also with the help of colleagues from General Electric Medical Systems (Joe Maier, Bob Vavrek, and James MacFall). With EPI IVIM and diffusion, images could be obtained in a matter of seconds and motion artifacts became history (of course, new types of artifacts came along later). Interestingly, thanks to EPI, diffusion and IVIM MRI could be extended outside the brain, and the very first hypothesis set by Denis Le Bihan to distinguish tumors from angiomas in the liver was confirmed.
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 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.
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.
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. Although the exact mechanism for the anisotropy has remained not completely understood, it became apparent in the early 1990s that this anisotropy effect could be exploited to map out the orientation in space of the white matter tracks in the brain, assuming that the direction of the fastest diffusion would indicate the overall orientation of the fibres, as first shown by D. Le Bihan (Douek et al.). While the diffusion tensor concept was introduced in this article the authors used a simple approach in 2 dimensions (within the imaging plane) to obtain color maps of fiber orientation from the ratio between diffusion coefficients measured in the X and Y direction (Dyy/Dxx). This ratio (which is the tangent of the angle between the diffusion vector in the XY plane and the X axis) was displayed with a color scale (blue to green to red). The limitation of this "vector" approach was that Dxx and Dyy were only approximately known. Only the DTI method, which was introduced shortly after by Peter Besser, gave access to all the components of the diffusion tensor (e.g., Dxy).
Diffusion tensor imaging became widely used within the MRI community following the work of Basser, Mattliello and Le Bihan. 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 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)." (D. Le Bihan and M. Moseley were awarded the Gold Medal of the International Society for Magnetic Resonance in 2001 for their pioneering work on the diffusion MRI method and its applications). Many groups then paid attention to the possibility of using tensor based diffusion anisotropy imaging for tractography. There was once a "Fibre Cup" competition in which various groups competed to provide the most effective new tractographic algorithm. Further advances in the development of tractography can be attributed to Mori, Pierpaoli, Lazar, Conturo, Poupon, and many others.
Measures of anisotropy and diffusivity
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 cerebro-spinal 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 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 ADCx, ADCy, ADCz. 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. The long one pointing along the axon direction will be λ1 and the two small axes will have lengths λ2 and λ3. 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, λ1 is also called the longitudinal diffusivity or the axial diffusivity or even the parallel diffusivity λ∥. Historically, this is closest to what Richards originally measured with the vector length in 1991. 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. It can also be called the perpendicular diffusivity ().
Another commonly used measure that summarizes the total diffusivity is the Trace—which is the sum of the three eigenvalues,
where is a diagonal matrix with eigenvalues , and on its diagonal.
If we divide this sum by three we have the mean diffusivity,
which equals ADCi since
where is the matrix of eigenvectors and is the diffusion tensor. 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 = (λ1/λ3). 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 = (λ1 − λ2)2 + (λ1 − λ3)2 + (λ2 − λ3)2. We use the square root of the sum of squares to obtain 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 so that FA has a maximum value of 1. 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. 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:
There are other metrics of anisotropy used, including the relative anisotropy (RA):
and the volume ratio (VR):
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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 "off-diagonal" information), it is possible to calculate, for each voxel, a tensor (i.e. a symmetric positive definite 3×3 matrix) that describes the 3-dimensional shape of diffusion. The fiber direction is indicated by the tensor's main eigenvector. This vector can be color-coded, yielding a cartography of the tracts' position and direction (red for left-right, blue for superior-inferior, and green for anterior-posterior). 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 tract-specific 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.
DTI also has applications in the characterization of skeletal and cardiac muscle. The sensitivity to fiber orientation also appears to be helpful in the area of sports medicine where it greatly aids imaging of structure and injury in muscles and tendons.
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.[notes 1] 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 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.
HARDI: High-angular-resolution diffusion imaging and Q-ball vector analysis
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.
The idea is best understood by conceptually placing a kind of geodesic dome around each image voxel. This icosahedron 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 "n-tuple" 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. 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, 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 (non-diffusion contributions to signal) at higher b values (a combination of gradient strength and pulse duration) and improve the spatial resolution.
The Q-Ball method of tractography is an implementation of the HARDI approach in which David Tuch provides a mathematical alternative to the tensor model. 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.
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.
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