Diffusion MRI

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Diffusion MRI
Illus dti.gif
DTI Color Map
MeSH D038524

Diffusion MRI (or dMRI) is a magnetic resonance imaging (MRI) method which came into existence in the mid-1980s.[1][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, such as 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 get images 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 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 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.[13]




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 [14] and generalized diffusion tensor imaging.


Given the concentration  \rho and flux  J , 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:

 \frac{\partial\rho(x,t)}{\partial t}=-\nabla\cdot J(x,t)

Putting the two together, we get the diffusion equation:

 \frac{\partial\rho(x,t)}{\partial t}=D\nabla^2\rho(x,t)
D is diffusion coefficient

Bloch–Torrey equation[edit]

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.[15] Torrey modified Bloch's original description of transverse magnetization to include diffusion terms and the application of a spatially varying gradient. Since the magnetization  M is a vector, there are 3 diffusion equations, one for each dimension. The Bloch-Torrey equation is:

  \frac{d{M}}{dt}=\gamma{M}\times{B}-\frac{M_x\vec{i}+M_y\vec{j}}{T_2}-\frac{(M_z-M_0)\vec{k}}{T_1}+\nabla\cdot D\nabla{M}

where  D is now the diffusion tensor. For the simplest case where the diffusion is isotropic the diffusion tensor is a multiple of the identity:

{D} = D \cdot \vec {I} = D \cdot \begin{bmatrix}
1 & 0 & 0 \\
0 & 1 & 0 \\
0 & 0 & 1 \end{bmatrix},

then the Bloch–Torrey equation will have the solution

  {M}={M}_{\text{bloch}}e^{-\frac13\gamma^2G^2t^3}\sim e^{-bD_0}

The exponential term will be referred to as the attenuation  A . 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:

  A=e^{ -\sum_{i,j}b_{ij}D_{ij} }

where the  b_{ij} terms incorporate the gradient fields  G_x, ,  G_y , and  G_z .

Diffusion imaging[edit]

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).[16] 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 T_1-weighted image, water molecules in a sample are excited with the imposition of a strong magnetic field. This causes many of the protons in water molecules to precess simultaneously, producing signals in MRI. In T_2-weighted images, contrast is produced by measuring the loss of coherence or synchrony between the water protons. When water is in an environment where it can freely tumble, relaxation tends to take longer. In certain clinical situations, this can generate contrast between an area of pathology and the surrounding healthy tissue.

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 [17] 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:

\frac{S(TE)}{S_0} =  \exp \left[ -\gamma^2 G^2\delta^2 \left( \Delta-\frac{\delta}{3}\right)D \right]

where  S_0 is the signal intensity without the diffusion weighting, S is the signal with the gradient,  \gamma is the gyromagnetic ratio, G is the strength of the gradient pulse,  \delta is the duration of the pulse,  \Delta is the time between the two pulses, and finally,  D 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:[1]

\frac{S(TE)}{S_0} =  \exp (-b\cdot ADC)

Also, the diffusion coefficient,  D, is replaced by an Apparent Diffusion Coefficient,  ADC, 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, b_1 and b_2, of the b factor according to:

\mathrm{ADC}(x,y,z)= \ln [S_2(x,y,z)/S_1(x,y,z)]/(b_1-b_2)

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 [18][19] and the cumulant-expansion (also called Kurtosis) model [20][21][22] which does not necessarily require the presence of 2 pools.




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 [1] 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.[23][24]

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).[13] 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,[25] 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.[26]

Diffusion tensor imaging[edit]

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.[27]




In 1990, Michael Moseley reported that water diffusion in white matter was anisotropic—the effect of diffusion on proton relaxation varied depending on the orientation of tracts relative to the orientation of the diffusion gradient applied by the imaging scanner. He also pointed out that this should best be described by a tensor.[30] 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.).[31] 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, 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.[32] 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][33][34] Filler and colleagues identified both vector and tensor based methods in the patents in July 1992,[34] 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:

 (\text{vector length})^2 = B_X^2 + B_Y^2 + B_Z^2 \,
 \text{diffusion vector angle between }B_X\text{ and }B_Y = \arctan \frac{B_Y}{B_X}
 \text{diffusion vector angle between }B_X\text{ and }B_Z = \arctan \frac{B_Z}{B_X}
 \text{diffusion vector angle between }B_Y\text{ and }B_Z = \arctan \frac{B_Y}{B_Z}

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.[35][36][37] Both vector and tensor methods provide a "rotationally invariant" measurement—the magnitude will be the same no matter how the tract is oriented relative to the gradient axes—and both provide a three-dimensional direction in space, however the tensor method is more efficient and accurate for carrying out tractography.[28] Practically, this class of calculated image places heavy demands on image registration—all of the images collected should ideally be identically shaped and positioned so that the calculated composite image will be correct. In the original FORTRAN program written on a Macintosh computer by Todd Richards in late 1991, all of the tasks of image registration, and normalized anisotropy assessment (stated as a fraction of 1 and corrected for a "B0" (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,[38] as was the use of the tensor ellipsoid to describe diffusion.[39][40] The inventive step of DTI therefore involved two aspects:

  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,[33] Widespread research in the field followed a presentation on March 28, 1993 when Michael Moseley re-presented the tractographic images from the Filler group—describing the new range of neuropathology it had made detectable—and drew attention to this new direction in MRI at a plenary session of Society for Magnetic Resonance Imaging in front of an audience of 700 MRI scientists.[41][42] Many groups then paid attention to the possibility of using tensor based diffusion anisotropy imaging for neural tract tracing, beginning to optimize tractography. There is now an annual "Fibre Cup" in which various groups compete to provide the most effective new tractographic algorithm. Further advances in the development of tractography can be attributed to Mori,[43] Pierpaoli,[44] Lazar,[45] Conturo,[46] Poupon,[47] and many others.

Diffusion tensor imaging became widely used within the MRI community following the work of Basser, Mattliello and Le Bihan.[48] 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[49][50] .[51] 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[edit]

Visualization of DTI data with ellipsoids




(ADC_x + ADC_y + ADC_z)/3 = ADC_i



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.[33] The diffusivities in the two minor axes are often averaged to produce a measure of radial diffusivity

\lambda_{\perp} = (\lambda_2 + \lambda_3)/2 .

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.[53] It can also be called the perpendicular diffusivity (


\mathrm{tr}(\Lambda) = \lambda_1 + \lambda_2 + \lambda_3

where \Lambda is a diagonal matrix with eigenvalues \lambda_1, \lambda_2 and \lambda_3 on its diagonal.

If we divide this sum by three we have the mean diffusivity,

MD = (\lambda_1 + \lambda_2 + \lambda_3) /3

which equals ADCi since

\mathrm{tr}(\Lambda)/3 &= \mathrm{tr}(V V^{-1} \Lambda)/3 \\
&= \mathrm{tr}(V \Lambda V^{-1})/3 \\ 
&= \mathrm{tr}(D)/3 \\
&= ADC_i

where V is the matrix of eigenvectors and D 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 1/\sqrt{2} so that FA has a maximum value of 1. The whole formula for FA looks like this:

  FA=\frac{\sqrt{3( (\lambda_1-\mathbb E[\lambda])^2+(\lambda_2-\mathbb E[\lambda])^2+(\lambda_3-\mathbb E[\lambda])^2 )}}{\sqrt{2( \lambda_1^2+\lambda_2^2+\lambda_3^2 )}}










There are other metrics of anisotropy used, including the relative anisotropy (RA):

  RA=\frac{\sqrt{(\lambda_1-\mathbb E[\lambda])^2+(\lambda_2-\mathbb E[\lambda])^2+(\lambda_3-\mathbb E[\lambda])^2}}{\sqrt{3\mathbb E[\lambda]}}

and the volume ratio (VR):

  VR=\frac{\lambda_1\lambda_2\lambda_3}{\mathbb E[\lambda]^3}



Tractographic reconstruction of neural connections via DTI



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.[56][57] 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.


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.


In 2013, researchers from the University of Pennsylvania reported in the Proceedings of the National Academy of Sciences a study to map differences in male and female brains.[59] The report demonstrated that in male brains, intralobe neural connections dominate, while in female brains interlobe connections dominate. It was also demonstrated that this distinction is acquired rather that inborn.[60][61]

Mathematical foundation—tensors[edit]


The following matrix displays the components of the diffusion tensor:

 \bar{D} = \begin{vmatrix}


Physical tensors[edit]

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 scalar—it may also be considered a tensor of rank 0. The next level of complexity concerns quantities that can only be described with reference to direction—a basic example is mechanical force—these require a description of magnitude and direction. Properties with a simple directional aspect can be described by a vector—often represented by an arrow—that has magnitude and direction. A vector can be described by providing its three components—its projection on the x-axis, the y-axis and the z-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 direction—this transformation could be described using a tensor of rank 1—a 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 components—each one associated with a pair of axes xx, yy, zz, xy, yx, xz, zx, yz, zy.[63] 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 Sénarmont[64] 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[edit]

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.



HARDI: High-angular-resolution diffusion imaging and Q-ball vector analysis[edit]



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.[69] 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.[70]




See also[edit]




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