Diffusion Tensor Imaging Tractography

Adrian Maries

CS2002

Prof. Liz Marai

Diffusion Tensor Imaging Tractography

Introduction

As a result of the advances in Magnetic Resonance Imaging over the last decades, many imaging techniques have been recently developed. One of them is Diffusion Tensor MRI, which is an extension of conventional MRI, with an additionalcapacity to measurethe motion of water molecules, usually in living tissue. The body part most often used by applications of DT-MRI is the brain and that is due to the capacity of the white matter in the brain to restrict the diffusion of the water molecules detected by the MRI scanner.

Applications

The areas in which the Diffusion Tensor MRI has been used are improving the understanding of various body parts, especially understanding of brain development, brain injuries and the ischemic heart, diagnosis of some conditions such as epilepsy and multiple sclerosis and in treatment, for example in tumor resection. [14]The majority of applications include,in one form or another, the visualization of fiber tracts in brain white matter. [24], [16],[19],[4],[10],[23]One of the few applications which are not restricted to the study of living tissue has been developed by Slavin et al. and it uses the DT-MRI to study the behavior of molecules in nematic liquid crystals, especially around topological defects in the structure of the liquid. [17]

Tensors

At the heart of the Diffusion Tensor MRI visualization technique lies the concept of tensor. Basic knowledge of tensors is necessary for fully understanding it. In order to understand how the notion of tensor came about, we have to take a look at the concept of material stress. In physics, stress is the amount of force exerted on a unit of surface. If stress is a scalar, then the stress at any point inside an object can be represented as a number. The fact that scalars don’t have directions creates a problem, as we know that there are two types of stress, tensile (in which the normal force acts perpendicularly on the surface) and tangential (where the tangential force acts in the plane of the surface). Since a vector is not sufficient to describe the properties of the material stress concept, the concept of a dyad was proposed. The dyad product of vectors and is defined the following way:

The dyad product of vectorsU and V is a rank 2 tensor. Similarly, a rank 0 tensoris a scalar, a rank 1 tensor is a vector, a rank 3 tensor is a triad (UVW), and so on.[8]

What is important to remember is that although all rank 0 tensors are scalars, not all scalars are tensors. This is true of rank 1 tensors and scalars, as well as tensors of all other ranks. Intrinsic to the definition of a tensor is the idea that tensors are coordinate independent, which means that they stay the same even if the coordinates of the system in which they are measured change. A simple example of a physical quantity that can be represented as a rank 0 tensor is temperature. If two people measure the temperature at the same point at the same time they will both record the same value, regardless of the position from which they measure it or whether they are moving or stationary. Light frequency, on the other hand, cannot be described as a tensor. It’s a well known fact that the frequency of light emitted from a certain point changes if it is measured while standing still or while moving. As a consequence, it is not coordinate independent, which means that it can be represented as a scalar, but not as a rank 0 tensor.[8] For more on Tensors, read [8] and [18].

Magnetic Resonance Imaging

The MRI is based on a fundamental property of the elementary particle called spin. Depending on the number of number of elementary particles a particle is made up of, it can have paired spin, a whole number, or unpaired spin, fractions such as ½, 1½, 2½, and so on. The particles that have unpaired spin behave like magnets. In usual conditions, the directions of the magnetic moment generated by these particles are random. However, when a group of particles is placed in a magnetic field, the spins of the particles align parallel or anti-parallel with the direction of the magnetic field.The transition between two spin states is caused by the absorption of a photon. The frequency at which it happens is called the Larmour or resonance frequency.[4]

At a macroscopic level, the spins of almost all the particles in the field point in the direction of the field and form the magnetization vector. The MRI works by applying a radio frequency pulse, perpendicular to the direction of the magnetic field, to the area of the body that has to be examined. The pulse makes the particles in the field transition from lower to higher energy states, which, in turn, makes the magnetization vector slide proportionally to the frequency of the pulse.[4]

After the radio frequency pulse is removed, the particles return to their initial state losing energy to the surrounding particles in the process. This energy loss, along with the time it takes the magnetization vector to return to the direction of the field (T1 relaxation) and the time it takes the particles to lose phase coherence (T2 relaxation) are all measured by the scanner. Each of these measurements gives a different piece of information about the area of the body that was scanned. The energy loss is used to calculate the amount of water in the tissue, the T1 relaxation tells us about the chemical surrounding of the water while the T2 relaxation gives information about the surroundings of each individual atom. Each of the two types of relaxations gives a different contrast. Different tissues have different characteristics in T1 and T2 relaxation, which is what enables us to distinguish them. [4]

Diffusion Tensor MRI

A central part of the DT-MRI is the Brownian motion of water molecules. The Brownian motion was first observed in the nineteenth century by Robert Brown and then fully explained by Albert Einstein in the early part of the twentieth century. Despite the fact that water seems stationary, at microscopic level, water molecules are inconstant random motion. This motion, at macroscopic level, produces the phenomenon of water diffusion. [19]

When the diffusion is unimpeded and the diffusion rate is the same in all directions, it is said to be isotropic. In living tissue, diffusion is diminished in some directions by cell membranes, which is then called anisotropic diffusion. Regions like skeletal muscles and especially brain white matter exhibit large diffusion anisotropy due to the large number of fibers pointing in the same direction, which makes them prime candidates for study using the Diffusion Tensor MRI. [4] Polymers and nematic liquid crystals exhibit the same behavior in non-biologic tissue. [19]

The direct measurement of molecular diffusion was pioneered by Stejskal and Tanner, who developed the methodology of the pulse gradient spin echo experiment and the theory behind it.Later on, Lauterbur realized that, he could pinpoint the location where a certain spin echo signal is generated when a magnetic field gradient is superimposed on a static magnetic field. This is possible due to the fact that different types of molecules in the sample have different Larmour frequencies. He then used this information to produce an image of the scanned object. Finally, Basser proposed the diffusion tensor model, which describes anisotropic diffusion as a tensor rather than a scalar. Basser’s contribution was especially important as he identified the major eigenvector of the tensor to indicate the main direction of the fiber, thus enabling us to map it. [19] For more on the history of the DT-MRI, read [19]

The diffusion tensorD can be extracted from the Stejskal-Tanner equations, which relate it to the Si signal values measured by the scanner:

where S0 is the signal intensity without the diffusion sensitization, b is an acquisition-specific constant and gi are the direction of the magnetic field gradient. [3]

D is a symmetricmatrix:

and because of its symmetry, there are only 6 unique components, rather than 9. As a consequence of this, at least 7 images (in practice there usually are 8 or more) are needed for each slice (with different diffusion weightings and field gradients) to solve for the unknowns in the Stejskal-Tanner equations, the 6 independent components of D and S0. Each voxel on each slice will have at least 7 equations and a diffusion tensor associated with it. [4]

The diffusion tensor still does not reveal the main direction of the diffusion. To find out this information the eigenvalues,,and(ordered by size in decreasing order), and eigenvectors,,and, of the tensor must be calculated by solving the equations [4]:

, with

In order to determine whether the diffusion is linear, planar or spherical, Westin et al. developed the following metrics:

where cl measures linear anisotropy, cp measures planar anisotropy and, finally, cs quantifies spherical isotropy. In the case of linear anisotropy, is much larger than and, which are approximately equal and this makes cl large. Similarly, when the anisotropy is planar,andare much larger than, which makes cp large. Lastly, in isotropic diffusion, the 3 eigenvalues are approximately equal and this makes cs large. The reasons why these particular metrics were chosen is, firstly, because the range all three measure is between 0 and 1 and, secondly, due to the fact that their sum is 1. [22]

Several methods have been developed for the visualization of DT-MRI data sets. The most basic method involves the above-mentioned metrics for linear and planar anisotropy and spherical isotropy as well as another metric for fractional anisotropy defined as follows:

where . [2]

Another method for visualization uses eigenvector colormaps. It assigns an (R,G,B) color according to the (X,Y,Z) components of the principal eigenvector and a saturation level that depends on the magnitude of the anisotropy metric.

A third approach utilizes glyphs, which are icons whose shapes, colors, textures and locations correspond in some way to the properties of the data.There are many ways these mappings can be made, but, generally, the shape indicates the directions of the eigenvectors at particular voxels, while colors can point to the value of the anisotropy at those voxels. [25] Several types of glyphs have been developed over the years. Laidlaw et al. normalize glyph size, losing information about mean diffusivity in the process. The advantage of this method is that it increases the number of glyphs displayed in one slice,thereby allowing the researchers to gain more insight into anatomical structures the glyphs represent. [9] Westin et al. felt that it’s often hard to distinguish between the different shapes glyphs can take, so they decided to add color as another indicator, coloring glyphs that represent linear, planar anisotropy and spherical isotropy blue, yellow and, respectively, red. [21] Barr takes a method often used in surface modeling, namely superquadrics, and adapts it for tensor visualization. He uses spheres to represent isotropy, sphere slices for planar anisotropy, cylinders for linear anisotropy and rectangular boxes whose size depend on diffusion values for anything in between. [1]

DT_MRI data sets can also be visualized using a technique called tractography or fiber tracking. Tractography is a method of visualizing individual fibers and fiber bundles in brain white matter. This method is currently the only one which can non-invasively illustrate these neural pathways. Glyphs are not a good option because they are not continuous. Instead, Zhang et al. use streamtubes and streamsurfaces. [24]

In his survey of the field, Bjornemo has identified four algorithms and several variants that are used for fiber tracking[4]:

• Principal diffusion direction tracking is one of the basic ones that simply follows the direction of the principal eigenvector. [5], [6]A variant of this method, rather than calculating the eigenvector directly, uses the previous direction and it transforms it using the current tensor.[20]One problem with this method is that it doesn’t handle fibers that cross each other or fibers that split into multiple fibers very well.
• A second algorithm finds the direction of the fiber by solving the diffusion equation. The diffusion equation is a differential equation in the water concentration with respect to time and the 3 possible directions of diffusion. (Gembris 2001)
• Fast marching tractography is a method that uses Level Set theory. Level Set theory is usually used for modeling evolving fronts or interfaces. Some of its numerical methods can be applied to tractography to find, for example the shortest path between two points. One advantage of Fast marching tractography is that that it can use Euclidian metric in calculations.[13]
• The final algorithm is called Markov random field regularization. This algorithm uses a separate direction map, rather than the main eigenvectors of the tensors. It uses a Markov random field to model a regularized direction field. The property that makes it a good algorithm is its resistance to noise and partial volume effects. [15]

ClusteringAlgorithms used in fiber tracking

There are several problems in tractography. One of them is that there often are too many fibers cluttering the screen at the same time.This makes it very hard to extract any information from it, so there has to be a way to cull some of them. Another common problem is that it is almost always useful to have some visual indication of which fibers from a bundle. This is usually achieved by coloring all the fibers in one bundle with one color, and this is often not an easy task. Many algorithms have been proposed to achieve these two goals. Some of them are detailed below.

Zhang et al. developed a method called Hierarchical clustering which works as follows. It starts with n clusters, each made up of one fiber and it keeps merging the nearest clusters until the distance between the fibers reaches a predefined threshold. Before applying this algorithm, the fibers are culled using 3 metrics: length of trajectory (shorter fibers are removed), average linear anisotropy (tracts with lower average linear anisotropy are culled) and similarity to an already selected group (fibers that were very similar to a group that was already selected are eliminated). The similarity measure used is defined as the mean of the smallest distances between curves Q and R that are larger than t:

where Q and R are two curves, t is the minimum threshold between the two curves, a and b are points on the curves and is the Euclidian distance between a and b. [23]

The Shared nearest neighbor clustering algorithm proposed by Moberts et al. is based on the idea that two data members that share a lot of neighbors are probably in the same cluster. It starts with a k-nearest neighbor graph, in which there is an edge between a fiber and its closest k neighbors. Then, the algorithm gives each fiber a weight that depends on the number and ordering of shared neighbors. Finally, the edges whose weights are below a predefined threshold are removed. The authors used several similarity measures, but the best results were obtained using the mean distance between fibers.[10]

Shimony et al. use an established clustering algorithm, namely the fuzzy c-means algorithm, that had not been used for fiber clustering before. The algorithm input is the number of desired clusters and a distance metric to be used in deciding which fibers form a cluster and which do not. The algorithm works by minimizing the weighted sum of squared differences within one cluster. The output is the probability that each fiber is a member of the cluster to which it has initially been assigned. Among the used metrics, the best-performing were found to be the average distance between the fibers and the dot product between corresponding tangents to the tracks.[16]

Tensor Visualization

Tensor visualization is alsoan important scientific tool in non-medical fields like computational geomechanics and solid mechanics. It can be used to illustrate stresses that occur in the soil or in bridges during earthquakes or to model physical properties such as solid stiffness.

Jeremic at al. demonstrate how it can be used in the visualization and study of the stresses that occur in earthquakes. The three methods they use are hedgehogs, which display the lines associated with the 3 tensor eigenvalues, hyperstreamlines, which are very similar to the streamtubes used by Zhang et al., and hyperstreamsurfaces, which are obtained by connecting multiple hyperstreamlines. [7]

Stresses that can occur in earthquakes have also been studied by Neeman et al. They model shear stresses in the soil and bridges using plane-in-a-box glyphs, which are scaled, planar glyphs that lie in a plane. The glyphs are scaled in order to minimize occlusion and are planar due to the specific properties of theshear forces they model, which have two major eigenvalues and a third minor one that is largely irrelevant and implied by the normal to the plane. [12]