Diffusion MRI - Wikipedia

7/29/2019 Diffusion MRI - Wikipedia

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Diffusion MRI 1

Diffusion MRI

Diffusion MRI

Diagnostics

DTI Color Map

MeSHD038524

[1]

Diffusion MRI (or dMRI) is a magnetic resonance imaging (MRI) method which came into existence in the

mid-1980s.[][2][3]

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, suchas macromolecules, fibers, membranes, etc. Water molecule diffusion patterns can therefore reveal microscopic

details about tissue architecture, either normal or in a diseased state.

The first diffusion MRI images of the normal and diseased brain were made public in 1985.[4][5]

Since then,

diffusion MRI, also referred to as diffusion tensor imaging or DTI (see section below) has been extraordinarily

successful. Its main clinical application has been in the study and treatment of neurological disorders, especially for

the management of patients with acute stroke. Because it can reveal abnormalities in white matter fiber structure and

provide models of brain connectivity, it is rapidly becoming a standard for white matter disorders.[6]

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 so-called Human Brain Connectome project.[7]

More recently, a

new field has emerged, diffusion functional MRI (DfMRI) as it was suggested that with dMRI one could also getimages of neuronal activation in the brain.

[8]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).[9]

IVIM dMRI is rapidly becoming a major method to obtain images of perfusion in the body,

especially for cancer detection and monitoring.[10]

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),[11]

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
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mapping of axonal trajectories than other diffusion imaging approaches.[12]

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 bodywhere 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 thediffusion process is complex in biological tissues and reflects several different mechanisms.

[]

Diffusion tensor imaging (DTI) is important when a tissuesuch as the neural axons of white matter in the brain or

muscle fibers in the hearthas 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[13]

and generalized diffusion tensor

imaging.

Diffusion

Given the concentration and flux , Fick's first law gives a relationship between the flux and the concentration

gradient:

where D is the diffusion coefficient. Then, given conservation of mass, the continuity equation relates the time

derivative of the concentration with the divergence of the flux:

Putting the two together, we get the diffusion equation:
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BlochTorrey equation

The classical Bloch equation is

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.[14]

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 BlochTorrey 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

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).[15]

MRI can be made sensitive to the 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

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. Thisfield gradient pulse

method was initially devised for NMR by Stejskal and Tanner

[16]

who derivedthe reduction in signal due to the application of the pulse gradient related to the amount of diffusion that is occurring

through the following equation:
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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 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 all the more

attenuated that diffusion is fast and the b factor is large. 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[17][18]

and the

cumulant-expansion (also called Kurtosis) model [19][][20] 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
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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).

History

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, Bihan) 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 Gnrale de Radiologie), a French company located in Buc near

Versailles in France (now GEMS European Headquarters) which patented diffusion and IVIM MRI.[21][22]

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).[]

It worked beautifully and that move was a great achievement.

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,[23]

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 Turners 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.[24]
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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 muscleincluding heart muscleas well as other tissues such as the prostate.[25]

In DTI, each voxel has one or more pairs of parameters: a rate of diffusion and a preferred direction of

diffusiondescribed in terms of three dimensional spacefor 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 measurementseach making up a complete imageare 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 braina 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.

History

In 1990, Michael Moseley reported that water diffusion in white matter was anisotropicthe 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.[26]

Although the

exact mechanism for the anisotropy has remained not completely understood, it became apparent in the early 1990sthat 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 approximatively known.

Only the DTI method, which was introduced shortly after, gave access to all the components of the diffusion tensor

(e.g., Dxy). In this seminal article, the authors also demonstrate that water diffusion is not really restricted, but

merely hindered, even perpendicularly to the fibers, as the diffusion distance kept increasing with the diffusion time.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.[27]

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.[][28][29]

Filler and colleagues identified both vector and tensor based methods in the patents in

July 1992,[29]

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:
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Diffusion MRI 7

The first color maps of white

matter fiber orientation using

diffusion MRI (Douek et al.

1991)[]

The first vector

calculated image using

diffusion anisotropy to

show neural tracts

curving through the

brain in Macaca

fascicularis (Filler et al.

1992)[30]

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.[31][32][33]

Both vector and tensor methods provide a

"rotationally invariant" measurementthe magnitude will be the same no matter how the tract is oriented relative to

the gradient axesand both provide a three dimensional direction in space, however the tensor method is more

efficient and accurate for carrying out tractography.[]

Practically, this class of calculated image places heavy

demands on image registrationall 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" (non-diffusion) 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 non-diffusion 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,[34]

as was the use of the tensor ellipsoid to describe diffusion.[35][36]

The inventive step

of DTI therefore involved two aspects:

1.1. the application of known methods from other fields for the generation of MRI tensor data; and

2.2. the usable introduction of a three dimensional selective neural tract "vector graphic" concept operating at a

macroscopic level above the scale of the image voxel, in a field where two dimensional pixel imaging (bit

mapped graphics) had been the only method used since MRI was originated.

The abstract with the first tractogram appeared at the August 1992 meeting of the Society for Magnetic Resonance in

Medicine,[28]

Widespread research in the field followed a presentation on March 28, 1993 when Michael Moseley

re-presented the tractographic images from the Filler groupdescribing the new range of neuropathology it had

made detectableand 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.[37][38]

Many groups then paid attention to thepossibility of using tensor based diffusion anisotropy imaging for neural tract tracing, beginning to optimize
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Diffusion MRI 8

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,[39]

Pierpaoli,[40]

Lazar,[41]

Conturo,[42]

Poupon,[43]

and many others.

Diffusion tensor imaging became widely used within the MRI community following the work of Basser, Mattliello

and Le Bihan.[44]

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

[45][46]

.

[47]

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).

Measures of anisotropy and diffusivity

Visualization of DTI data with ellipsoids

In present-day clinical neurology, various brain pathologies may be

best detected by looking at particular measures of anisotropy and

diffusivity. The underlying physical process of diffusion (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.[48]

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 wecan generate a formula that allows us to convert the signal attenuation

of an MRI voxel into a numerical measure of diffusionthe 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. The

long one pointing along the axon direction will be 1 and the two small axes will have lengths 2 and 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
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Diffusion MRI 9

three primary axes of the ellipsoid. This is a little different from the ADC since that was a projection on the axis,

while is an actual measurement of the ellipsoid we have calculated.

The diffusivity along the principal axis, 1

is also called the longitudinal diffusivity or the axial diffusivity or even

the parallel diffusivity. Historically, this is closest to what Richards originally measured with the vector length in

1991.[28]

The diffusivities in the two minor axes are often averaged to produce a measure ofradial 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.[49]

It can also be called the

perpendicular diffusivity ( ).

Another commonly used measure that summarizes the total diffusivity is the Tracewhich 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 equalsADCisince

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 = (1/

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 = (1

2)2

+ (1

3)2

+ (2

3)2. We use the square root of the sum of squares to obtain a

sort of weighted averagedominated 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.[50][51]

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 ,
http://en.wikipedia.org/w/index.php?title=Oblate_spheroidhttp://en.wikipedia.org/w/index.php?title=Oblate_spheroidhttp://en.wikipedia.org/w/index.php?title=Fractional_anisotropyhttp://en.wikipedia.org/w/index.php?title=Prolate_spheroid


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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):

Applications

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.

Tractographic reconstruction of neural connections via DTI

The imaging of this property is an extension of diffusion

MRI. If a series of diffusion gradients (i.e. magnetic fieldvariations 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 33 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 informationtransit from the motor cortex to the spinal cord and the peripheral nerves). Tractography is a useful tool for
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Diffusion MRI 11

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. In one study, DTI identified blast injuries to cerebral tissue in

patients who had normal appearing brains on CT and standard MRI - the study validated the imaging method while

also resolving important questions about the mechanisms of diffuse axonal injuries.

[52][53]

The localization of tumorsin 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.

A recent study at Barnes-Jewish Hospital and Washington University School of Medicine of healthy persons and

both newly affected and chronically-afflicted 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.[54]

In October 2009 a report appeared documenting a localized increase in fractional anisotropy following training of a

complex visuo-motor skill (juggling). This was claimed to be the first evidence for experience-dependent changes in

white matter microstructure in healthy human adults.[55]

Mathematical foundation

tensorsDiffusion 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.[56]

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.
http://en.wikipedia.org/w/index.php?title=Symmetric_matrixhttp://en.wikipedia.org/w/index.php?title=Tensorhttp://en.wikipedia.org/w/index.php?title=Jugglinghttp://en.wikipedia.org/w/index.php?title=Diffusion_MRI%23Measures_of_anisotropy_and_diffusivityhttp://en.wikipedia.org/w/index.php?title=Optic_nervehttp://en.wikipedia.org/w/index.php?title=Multiple_sclerosishttp://en.wikipedia.org/w/index.php?title=Optic_neuritishttp://en.wikipedia.org/w/index.php?title=Washington_University_School_of_Medicinehttp://en.wikipedia.org/w/index.php?title=Barnes-Jewish_Hospitalhttp://en.wikipedia.org/w/index.php?title=Tendonhttp://en.wikipedia.org/w/index.php?title=Musclehttp://en.wikipedia.org/w/index.php?title=Sports_medicinehttp://en.wikipedia.org/w/index.php?title=Cardiac_musclehttp://en.wikipedia.org/w/index.php?title=Skeletal_musclehttp://en.wikipedia.org/w/index.php?title=In_vivohttp://en.wikipedia.org/w/index.php?title=Neural_networkhttp://en.wikipedia.org/w/index.php?title=Neural_networkhttp://en.wikipedia.org/w/index.php?title=Dementiahttp://en.wikipedia.org/w/index.php?title=Alzheimer%27s_diseasehttp://en.wikipedia.org/w/index.php?title=Corticospinal_tracthttp://en.wikipedia.org/w/index.php?title=Brain_tumorshttp://en.wikipedia.org/w/index.php?title=Tumorhttp://en.wikipedia.org/w/index.php?title=Traumatic_brain_injury%23Focal_vs._diffusedhttp://en.wikipedia.org/w/index.php?title=Lesion


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Diffusion MRI 12

Physical tensors

The idea of a tensor in physical science evolved from attempts to describe the quantity of a given physical property.

The first instances are the properties that can be described by a single number - such as temperature. There is no

directionality in temperature. A property that can be described this way is denoted a scalarit may also be

considered a tensor of rank 0. The next level of complexity concerns quantities that can only be described with

reference to directiona basic example is mechanical forcethese require a description of magnitude and direction.Properties with a simple directional aspect can be described by a vectoroften represented by an arrowthat has

magnitude and direction. A vector can be described by providing its three componentsits projection on thex-axis,

they-axis and thez-axis. 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 a

force is applied to an object, movement can result. If the movement is in a single directionthis transformation

could be described using a tensor of rank 1a vector (reporting magnitude and direction). However, in a tissue, the

driving force of Brownian Motion will lead to movement of water molecules in an expanding pattern that proceeds

along multiple different directions simultaneously, 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 complex 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 componentseach one associated with

a pair of axesxx,yy,zz,xy,yx,xz,zx,yz,zy.[57]

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 summated 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 Snarmont[58]

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 follows many of the same paths and rules as does 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 ofa, 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 componentsthe 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 ellipsoidif 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 algebrathat is to "diagonalize" the matrix. This has two important meanings in imaging.
http://en.wikipedia.org/w/index.php?title=Matrix_%28mathematics%29http://en.wikipedia.org/w/index.php?title=Diagonalizable_matrix%23How_to_diagonalize_a_matrixhttp://en.wikipedia.org/w/index.php?title=Matrix_%28mathematics%29http://en.wikipedia.org/w/index.php?title=Diagonalizable_matrix%23How_to_diagonalize_a_matrixhttp://en.wikipedia.org/w/index.php?title=Matrix_%28mathematics%29http://en.wikipedia.org/w/index.php?title=Diagonalizable_matrix%23How_to_diagonalize_a_matrixhttp://en.wikipedia.org/w/index.php?title=Diagonalizable_matrix%23How_to_diagonalize_a_matrixhttp://en.wikipedia.org/w/index.php?title=Matrix_%28mathematics%29http://en.wikipedia.org/w/index.php?title=Hyperboloidhttp://en.wikipedia.org/w/index.php?title=Ellipsoidhttp://en.wikipedia.org/w/index.php?title=Quadrichttp://en.wikipedia.org/w/index.php?title=Adolf_Eugen_Fickhttp://en.wikipedia.org/w/index.php?title=Adolf_Eugen_Fickhttp://en.wikipedia.org/w/index.php?title=Henri_Hureau_de_S%C3%A9narmonthttp://en.wikipedia.org/w/index.php?title=Brownian_Motionhttp://en.wikipedia.org/w/index.php?title=Euclidean_vectorhttp://en.wikipedia.org/w/index.php?title=Scalar_%28mathematics%29


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Diffusion MRI 13

The idea is that there are two equivalent ellipsoidsof 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, by

2, and cz

2are along the diagonal (red subscripts), but the variables d, e andfare "off

diagonal". It then becomes possible to do a vector processing step in which we rewrite our matrix and replace it wit h

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 thex,y, andz 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 matrixwherein 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 voxelas 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.[59][60]

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 voxeleach 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.[61]

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,

[62]

or we can do a true "model free" analysis that just picks the maximaand goes on about doing the tractography. We could use very high angular resolution (256 different directions) but i t

is often necessary to do ten or fifteen complete runs to get the information correct and this could mean 2,000 or more

imagesit 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.[63]

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 agothe Funk Radon Transform.[64]
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Diffusion MRI 14

Summary

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 axisthe one with the greatest lengthis

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.

Notes

1. Filler AG, Tsuruda JS, Richards TL, Howe FA: Images, apparatus, algorithms and methods. Patent application

no. GB9216383.1[65]

, UK Patent Office, (1992) - now: Filler AG, Tsuruda JS, Richards TL, Howe FA: Image

Neurography and Diffusion Anisotropy Imaging. US 5,560,360[66]

, United States Patent Office, (1996)

References

[1] http:/ /www.nlm.nih. gov/cgi/mesh/2011/MB_cgi?field=uid& term=D038524

[6] Hagmann et al, "Understanding Diffusion MR Imaging Techniques: From Scalar Diffusion-weighted Imaging to Diffusion Tensor Imaging

and Beyond,"RadioGraphics. Oct 2006. http://radiographics. rsna. org/content/26/suppl_1/S205. full

[7] Dillow, Clay. "The Human Connectome Project Is a First-of-its-Kind Map of the Brain's Circuitry." Popular Science. Sept 2010. http://

www.popsci.com/science/article/2010-09/introducing-human-connectome-project-first-its-kind-map-brains-circuitry

[27][27] Filler AG, Winn HR, Howe FA, Griffiths JR, Bell BA, Deacon TW: Axonal transport of superparamagnetic metal oxide particles: Potential

for magnetic resonance assessments of axoplasmic flow in clinical neurosciece. Presented at Society for Magnetic Resonance in Medicine,

San Francisco, SMRM Proceedings 10:985, 1991 (abstr).

[28][28] Richards TL, Heide AC, Tsuruda JS, Alvord EC: Vector analysis of diffusion images in experimental allergic encephalomyelitis. Presented

at Society for Magnetic Resonance in Medicine, Berlin, SMRM Proceedings 11:412, 1992 (abstr).

[29] Filler AG, Tsuruda JS, Richards TL, Howe FA: Images, apparatus, algorithms and methods. GB9216383.1 (http://www.neurography. com/

images/neurography-GB9216383A. pdf), UK Patent Office, 1992.

[31] Filler AG, Howe FA: Images, apparatus, and methods. GB9210810 (http:/

/

www.

neurography.

com/

images/

neurography-GB9210810.pdf), UK Patent Office, 1992.

[32][32] Basser PJ, LeBihan D: Fiber orientation mapping in an anisotropic medium with NMR diffusion spectroscopy. Presented at Society for

Magnetic Resonance in Medicine, Berlin, SMRM Proceedings 11:1221, 1992 (abstr).

[33][33] Basser PJ, Mattiello J, LeBihan D: Diagonal and off-diagonal components of the self-diffusion tensor: their relation to and estimation from

the NMR spin-echo signal. Presented at Society for Magnetic Resonance in Medicine, Berlin, SMRM Proceedings 11:1222, 1992 (abstr).

[35][35] Jost, W. Diffusion in Solids, Liquids and Gases. Academic Press, New York, 1952

[50][50] Westin CF, Peled S, Gudbjartsson H, Kikinis R, Jolesz FA. Geometrical diffusion measures for MRI from tensor basis analysis. In ISMRM

'97. Vancouver Canada, 1997;1742.

[51][51] Westin CF, Maier SE, Mamata H, Nabavi A, Jolesz FA, Kikinis R. Processing and visualization of diffusion tensor MRI. Medical Image

Analysis 2002;6(2):93-108.

[56] Several full mathematical treatments of general tensors exist, e.g. classical, component free, intermediate, but the generality, which covers

arrays of all sizes, may obscure rather than help.

[65] http://www.neurography. com/images/neurography-GB9216383A. pdf

[66] http://www.neurography. com/images/neurography-DiffusionImaging-US05560360. pdf
http://www.neurography.com/images/neurography-DiffusionImaging-US05560360.pdfhttp://www.neurography.com/images/neurography-GB9216383A.pdfhttp://en.wikipedia.org/w/index.php?title=Intermediate_treatment_of_tensorshttp://en.wikipedia.org/w/index.php?title=Component-free_treatment_of_tensorshttp://en.wikipedia.org/w/index.php?title=Classical_treatment_of_tensorshttp://www.neurography.com/images/neurography-GB9210810.pdfhttp://www.neurography.com/images/neurography-GB9210810.pdfhttp://www.neurography.com/images/neurography-GB9216383A.pdfhttp://www.neurography.com/images/neurography-GB9216383A.pdfhttp://www.popsci.com/science/article/2010-09/introducing-human-connectome-project-first-its-kind-map-brains-circuitryhttp://www.popsci.com/science/article/2010-09/introducing-human-connectome-project-first-its-kind-map-brains-circuitryhttp://radiographics.rsna.org/content/26/suppl_1/S205.fullhttp://www.nlm.nih.gov/cgi/mesh/2011/MB_cgi?field=uid&term=D038524http://www.neurography.com/images/neurography-DiffusionImaging-US05560360.pdfhttp://www.neurography.com/images/neurography-GB9216383A.pdfhttp://en.wikipedia.org/w/index.php?title=Tractographyhttp://en.wikipedia.org/w/index.php?title=Linear_algebra


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Diffusion MRI 15

External links

MITK Diffusion: Free software for the processing of diffusion-weighted MR data (http://www.mitk.org/

DiffusionImaging)

DTITool: Software to visualize Diffusion MRI data (http://bmia.bmt.tue.nl/Software/DTITool/)

PNRC: About Diffusion MRI (http://pnrc.cchmc.org/research/dti.php)

White Matter Atlas (http://www.dtiatlas.org)

Thesis on DTI (http://www.rsierra.com/DA/thesis.html)

PhD Thesis on diffusion tensor MRI (2006): Modeling and Processing of Diffusion Tensor Magnetic Resonance

Images for Improved Analysis of Brain Connectivity (http://www.visielab.ua.ac.be/publications/

modeling-and-processing-diffusion-tensor-magnetic-resonance-images-improved-analysis)

PhD Thesis on diffusion tensor MRI (2009): Coregistration, Atlas Construction, and Voxel Based Analysis (http:/

/www.visielab.ua.ac.be/publications/

improved-processing-diffusion-tensor-magnetic-resonance-images-coregistration-atlas)

PhD Thesis on diffusion MRI (2012): Improved analysis of brain connectivity using high angular resolution

diffusion imaging (http://www.visielab.ua.ac.be/publications/

improved-analysis-brain-connectivity-using-high-angular-resolution-diffusion-imaging)

Information, with image gallery (http://www.sci.utah.edu/research/diff-tensor-imaging.html)

Multimodal Neurosurgery Planning, with DTI information (http://www.irisa.fr/visages/demo/

demo_MultiModality-Planning/MultiModalityPlanning-eng.html)

UK diffusion MRI interest group (http://www.diffusion-mri.org.uk)

Camino (diffusion MRI toolkit) (http://www.camino.org.uk)

Dipy: Diffusion Imaging in Python (http://dipy.org)

HARDI Tools (http://neuroimagen.es/webs/hardi_tools/)
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Article Sources and Contributors 16

Article Sources and ContributorsDiffusion MRI Source: http://en.wikipedia.org/w/index.php?oldid=570783957 Contributors: 1000Faces, Adam Riggall, Aeth909, Afiller, Ahoerstemeier, Anonymous Dissident, Arcadian,

BasicTruth, Benscripps, Biological965, Charles Matthews, Chris Capoccia, ChrisGualtieri, CogitoErgoSum101, CommonsDelinker, Coneslayer, CopperKettle, Darrell Greenwood, Dblandford,

Decltype, Deli nk, Denislb, Dkstiles, Dougher, EdH, Ejcanales, Emersoni, Facts707, Fnielsen, Fram, Gabrieleg, Gcastellanos, Gene Nygaard, Gumby55555, GyroMagician, Heatv4, Irbisgreif,

Jimw338, Jjron, Joel7687, John of Reading, Johnuniq, JordiGH, Joshiansmith, Jstrater, Juliancolton, Justanothervisitor, Kpmiyapuram, Kwamikagami, Lambyte, LaurenJOD, Linas, LizzardKitty,

Lova Falk, MaSt, Mattfiller, Michael Hardy, Mightypile, MisfitToys, Monsterman222, Moritz37, NMRwiki, Neelix, Pcap, Pmg, Prhone, Pwjb, Qwfp, Radagast83, RandowWalk, Rich

Farmbrough, Rjanag, Rjwilmsi, Rod57, Schlegel, Shadowjams, Sjschen, Solleo, Spellwanderer, Sprevrha, Stevenfruitsmaak, Stwalkerster, Sun Creator, Sawomir Biay, Tarinth, Thomas Schultz,

Tmangray, Ugur Basak, Visor, Wouterstomp, Yamaken, Zinzone, , 129 anonymous edits

Image Sources, Licenses and ContributorsFile:Illus dti.gif Source: http://en.wikipedia.org/w/index.php?title=File:Illus_dti.gifLicense: Copyrighted free use Contributors: Lipothymia, 2 anonymous edits

Image:AnisoColor.tif Source: http://en.wikipedia.org/w/index.php?title=File:AnisoColor.tifLicense: Public Domain Contributors: Christian1985, Nevit, Yamaken

Image:AGF ArctanLR.jpg Source: http://en.wikipedia.org/w/index.php?title=File:AGF_ArctanLR.jpg License: Creative Commons Attribution-Sharealike 3.0 Contributors: Original uploader

was Afiller at en.wikipedia

Image:UK_lab_MRI_Aaron_Filler.png Source: http://en.wikipedia.org/w/index.php?title=File:UK_lab_MRI_Aaron_Filler.pngLicense: Creative Commons Attribution-Share Alike

Contributors: Aaron Filler

Image:Ellipsoid_Patent.jpg Source: http://en.wikipedia.org/w/index.php?title=File:Ellipsoid_Patent.jpg License: Public Domain Contributors: Afiller, Monkeybait

File:DTI-axial-ellipsoids.jpg Source: http://en.wikipedia.org/w/index.php?title=File:DTI-axial-ellipsoids.jpg License: Creative Commons Attribution-ShareAlike 3.0 Unported Contributors:

Thomas Schultz

Image:DTI-sagittal-fibers.jpg Source: http://en.wikipedia.org/w/index.php?title=File:DTI-sagittal-fibers.jpg License: Creative Commons Attribution-ShareAlike 3.0 Unported Contributors:

Thomas Schultz

License

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Diffusion MRI - Wikipedia

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