5551:
737:, to indicate that the diffusion process is not free in tissues, but hindered and modulated by many mechanisms (restriction in closed spaces, tortuosity around obstacles, etc.) and that other sources of IntraVoxel Incoherent Motion (IVIM) such as blood flow in small vessels or cerebrospinal fluid in ventricles also contribute to the signal attenuation. At the end, images are "weighted" by the diffusion process: In those diffusion-weighted images (DWI) the signal is more attenuated the faster the diffusion and the larger the b factor is. However, those diffusion-weighted images are still also sensitive to T1 and T2 relaxivity contrast, which can sometimes be confusing. It is possible to calculate "pure" diffusion maps (or more exactly ADC maps where the ADC is the sole source of contrast) by collecting images with at least 2 different values,
5627:
2626:
diffusion leads to movement of water molecules along trajectories that proceed along multiple directions over time, leading to a complex projection onto the
Cartesian axes. This pattern is reproducible if the same conditions and forces are applied to the same tissue in the same way. If there is an internal anisotropic organization of the tissue that constrains diffusion, then this fact will be reflected in the pattern of diffusion. The relationship between the properties of driving force that generate diffusion of the water molecules and the resulting pattern of their movement in the tissue can be described by a tensor. The collection of molecular displacements of this physical property can be described with nine componentsâeach one associated with a pair of axes
2921:) 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 componentsâthe 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 ellipsoidâif 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.
2393:. 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 do not 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.
963:
this condition, the ADC seems to depend on the choice of b values (the ADC seems to decrease when using larger b values), as the plot of ln(S/So) is not linear with the b factor, as expected from the above equations. This deviation from a free diffusion behavior is what makes diffusion MRI so successful, as the ADC is very sensitive to changes in tissue microstructure. On the other hand, modeling diffusion in tissues is becoming very complex. Among most popular models are the biexponential model, which assumes the presence of 2 water pools in slow or intermediate exchange and the cumulant-expansion (also called
Kurtosis) model, which does not necessarily require the presence of 2 pools.
604:
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 simplification, Le Bihan suggested gathering all the gradient terms in a "b factor" (which depends only on the acquisition parameters) so that the signal attenuation simply becomes:
5536:
2360:
six or more different diffusion weighted acquisitions, each obtained with a different orientation of the diffusion sensitizing gradients. In some methods, hundreds of measurementsâeach making up a complete imageâare 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 brainâa process called
5566:
5655:. 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.
314:
is applied in the same magnitude but with opposite direction to refocus or rephase the spins. The refocusing will not be perfect for protons that have moved during the time interval between the pulses, and the signal measured by the MRI machine is reduced. This "field gradient pulse" method was initially devised for NMR by
Stejskal and Tanner who derived the reduction in signal due to the application of the pulse gradient related to the amount of diffusion that is occurring through the following equation:
200:(DBSI) further separates DTI signals into discrete anisotropic diffusion tensors and a spectrum of isotropic diffusion tensors to better differentiate sub-voxel cellular structures. For example, anisotropic diffusion tensors correlate to axonal fibers, while low isotropic diffusion tensors correlate to cells and high isotropic diffusion tensors correlate to larger structures (such as the lumen or brain ventricles). DBSI has been shown to differentiate some types of
2371:-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.
9075:
3099:
5619:
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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 mathematics 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 tensor analyses, or we can do a true "model free" analysis that picks the maxima, and then continue to do the tractography.
5772:
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 appears as noise or unexplained decreased anisotropy in a given voxel.
135:) 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
2320:
4349:
1792:
5351:
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310:-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.
2588:
2932:" the matrix. This has two important meanings in imaging. The idea is that there are two equivalent ellipsoidsâof 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
1507:
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to add more evenly spaced apices to the original icosahedron (20 faces)âan idea that also had its precedents in paleomagnetism research several decades earlier. We want to know which direction lines turn up the maximum anisotropic diffusion measures. If there is a single tract, there will be only two
2355:
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
2346:
leads to diffusion restriction, and the difference between images with various DWI weighting will therefore be minor, leading to an ADC image with low signal in the infarcted area. A decreased ADC may be detected minutes after a cerebral infarction. The high signal of infarcted tissue on conventional
255:
can also be investigated using MR spectroscopic approaches). MRI can be made sensitive to the motion of molecules. Regular MRI acquisition utilizes the behavior of protons in water to generate contrast between clinically relevant features of a particular subject. The versatile nature of MRI is due to
5824:
it is usually necessary to know only the magnitude and orientation of the primary axis or vector. This primary axisâthe one with the greatest lengthâis the largest eigenvalue and its orientation is encoded in its matched eigenvector. Only one axis is needed as it is assumed the largest eigenvalue is
3039:
are "off diagonal". It then becomes possible to do a vector processing step in which we rewrite our matrix and replace it with a new matrix multiplied by three different vectors of unit length (length=1.0). The matrix is diagonalized because the off-diagonal components are all now zero. The rotation
2625:
A tensor is often a physical or biophysical property that determines the relationship between two vectors. When a force is applied to an object, movement can result. If the movement is in a single direction, the transformation can be described using a vectorâa tensor of rank 1. However, in a tissue,
2339:
that is otherwise inherent to conventional DWI. ADC imaging does so by acquiring multiple conventional DWI images with different amounts of DWI weighting, and the change in signal is proportional to the rate of diffusion. Contrary to DWI images, the standard grayscale of ADC images is to represent a
313:
To sensitize MRI images to diffusion, the magnetic field strength (B1) 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
5587:
tumors, and also to monitor tumor response to treatment over time. DWI can also be collected to visualize the whole body using a technique called 'diffusion-weighted whole-body imaging with background body signal suppression' (DWIBS). Some more specialized diffusion MRI techniques such as diffusion
3114:
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
3056:" 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.
2665:
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
2359:
In DTI, each voxel has one or more pairs of parameters: a rate of diffusion and a preferred direction of diffusionâdescribed in terms of three-dimensional spaceâfor which that parameter is valid. The properties of each voxel of a single DTI image are usually calculated by vector or tensor math from
603:
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"
5771:
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 voxelâas if all of the axons traveling through a voxel traveled in exactly the same direction.
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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
4120:
193:
of some crystals. Water will then diffuse more rapidly in the direction aligned with the internal structure (axial diffusion), and more slowly as it moves perpendicular to the preferred direction (radial diffusion). This also means that the measured rate of diffusion will differ depending on the
4077:
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
2673:
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
5804:
Note, there is ongoing debate about the best way to preprocess DW-MRI. Several in-vivo studies have shown that the choice of software and functions applied (directed at correcting artefacts arising from e.g. motion and eddy-currents) have a meaningful impact on the DTI parameter estimates from
5792:
The Q-Ball method of tractography is an implementation in which David Tuch provides a mathematical alternative to the tensor model. Instead of forcing the diffusion anisotropy data into a group of tensors, the mathematics used deploys both probability distributions and some classic geometric
174:
and major brain nuclei, or in the bodyâwhere 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
2602:
The idea of a tensor in physical science evolved from attempts to describe the quantity of physical properties. The first properties they were applied to were those that can be described by a single number, such as temperature. Properties that can be described this way are called
5779:
provides a mathematical basis for passing a large number of evenly spaced gradient trajectories through the voxelâeach coinciding with one of the apices of the icosahedron. We can then look into the voxel from a large number of different directions (typically 40 or more). We use
4006:
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
1556:
3156:
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
5154:
4025:). 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 = (
2607:; these can be considered tensors of rank 0, or 0th-order tensors. Tensors can also be used to describe quantities that have directionality, such as mechanical force. These quantities require specification of both magnitude and direction, and are often represented with a
2593:
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.
5705:). Tractography is a useful tool for measuring deficits in white matter, such as in aging. Its estimation of fiber orientation and strength is increasingly accurate, and it has widespread potential implications in the fields of cognitive neuroscience and neurobiology.
1955:
6185:
Vavasour, Irene M; Sun, Peng; Graf, Carina; Yik, Jackie T; Kolind, Shannon H; Li, David KB; Tam, Roger; Sayao, Ana-Luiza; Schabas, Alice; Devonshire, Virginia; Carruthers, Robert; Traboulsee, Anthony; Moore, GR Wayne; Song, Sheng-Kwei; Laule, Cornelia (March 2022).
5169:
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5638:
variations in the MRI magnet) are applied that can determine at least 3 directional vectors (use of 6 different gradients is the minimum and additional gradients improve the accuracy for "off-diagonal" information), it is possible to calculate, for each
4445:
The fractional anisotropy can also be separated into linear, planar, and spherical measures depending on the "shape" of the diffusion ellipsoid. For example, a "cigar" shaped prolate ellipsoid indicates a strongly linear anisotropy, a "flying saucer" or
2402:
3336:. In the setting of the DTI tensor ellipsoid, we can consider each of these as a measure of the diffusivity along each of the three primary axes of the ellipsoid. This is a little different from the ADC since that was a projection on the axis, while
147:(DSI), was used in deriving the Connectome data sets; DSI is a variant of diffusion-weighted imaging that is sensitive to intra-voxel heterogeneities in diffusion directions caused by crossing fiber tracts and thus allows more accurate mapping of
5449:
4836:
3437:
This quantity is an assessment of the degree of restriction due to membranes and other effects and proves to be a sensitive measure of degenerative pathology in some neurological conditions. It can also be called the perpendicular diffusivity
3296:
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
1523:
for magnetization would change with the addition of diffusion. Torrey modified Bloch's original description of transverse magnetization to include diffusion terms and the application of a spatially varying gradient. Since the magnetization
6284:
Ye, Zezhong; Price, Richard L.; Liu, Xiran; Lin, Joshua; Yang, Qingsong; Sun, Peng; Wu, Anthony T.; Wang, Liang; Han, Rowland H.; Song, Chunyu; Yang, Ruimeng; Gary, Sam E.; Mao, Diane D.; Wallendorf, Michael; Campian, Jian L. (2020-10-15).
4667:
4067:). We use the square root of the sum of squares to obtain a sort of weighted averageâdominated 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
2077:
1307:
5550:
175:
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.
2666:
applied measures the movement along the direction of that gradient. Six or more gradients are summed to get all the measurements needed to fill in the matrix, assuming it is symmetric above and below the diagonal (red subscripts).
238:
Recently, more advanced models of the diffusion process have been proposed that aim to overcome the weaknesses of the diffusion tensor model. Amongst others, these include q-space imaging and generalized diffusion tensor imaging.
4986:
4344:{\displaystyle \mathrm {FA} ={\frac {\sqrt {3((\lambda _{1}-\operatorname {E} )^{2}+(\lambda _{2}-\operatorname {E} )^{2}+(\lambda _{3}-\operatorname {E} )^{2})}}{\sqrt {2(\lambda _{1}^{2}+\lambda _{2}^{2}+\lambda _{3}^{2})}}}}
227:. The diffusion tensor 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
1284:
3546:
1185:
4441:
2936:
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
3738:
7606:
Makris N, Worth AJ, Sorensen AG, Papadimitriou GM, Wu O, Reese TG, et al. (December 1997). "Morphometry of in vivo human white matter association pathways with diffusion-weighted magnetic resonance imaging".
3122:, water molecules are moving due to diffusion and they move at equal rates in all directions. By knowing the detailed effects of diffusion gradients we can generate a formula that allows us to convert the signal
283:-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
4507:
5708:
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
4896:
6583:
Niendorf T, Dijkhuizen RM, Norris DG, van
Lookeren Campagne M, Nicolay K (December 1996). "Biexponential diffusion attenuation in various states of brain tissue: implications for diffusion-weighted imaging".
3432:
1787:{\displaystyle {\frac {d{\vec {M}}}{dt}}=\gamma {\vec {M}}\times {\vec {B}}-{\frac {M_{x}{\vec {i}}+M_{y}{\vec {j}}}{T_{2}}}-{\frac {(M_{z}-M_{0}){\vec {k}}}{T_{1}}}+\nabla \cdot {\vec {D}}\nabla {\vec {M}}}
222:
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
4727:
4567:
6101:
Le Bihan D, Breton E, Lallemand D, Grenier P, Cabanis E, Laval-Jeantet M (November 1986). "MR imaging of intravoxel incoherent motions: application to diffusion and perfusion in neurologic disorders".
5001:
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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 matrixâwherein the result is a new matrix in which three diagonal (
686:
957:
8084:
Wade, Ryckie G.; Tam, Winnie; Perumal, Antonia; Pepple, Sophanit; Griffiths, Timothy T.; Flather, Robert; Haroon, Hamied A.; Shelley, David; Plein, Sven; Bourke, Grainne; Teh, Irvin (2023-10-13).
1835:
211:
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
3761:
3284:
5565:
2883:
5346:{\displaystyle \mathrm {RA} ={\frac {\sqrt {(\lambda _{1}-\operatorname {E} )^{2}+(\lambda _{2}-\operatorname {E} )^{2}+(\lambda _{3}-\operatorname {E} )^{2}}}{{\sqrt {3}}\operatorname {E} }}}
444:
5583:. Tumors are in many instances highly cellular, giving restricted diffusion of water, and therefore appear with a relatively high signal intensity in DWI. DWI is commonly used to detect and
3956:{\displaystyle {\begin{aligned}\mathrm {tr} (\Lambda )/3&=\mathrm {tr} (V^{-1}V\Lambda )/3\\&=\mathrm {tr} (V\Lambda V^{-1})/3\\&=\mathrm {tr} (D)/3\\&=ADC_{i}\end{aligned}}}
2185:
1085:
3361:. Historically, this is closest to what Richards originally measured with the vector length in 1991. The diffusivities in the two minor axes are often averaged to produce a measure of
6334:
Ye, Zezhong; Srinivasa, Komal; Meyer, Ashely; Sun, Peng; Lin, Joshua; Viox, Jeffrey D.; Song, Chunyu; Wu, Anthony T.; Song, Sheng-Kwei; Dahiya, Sonika; Rubin, Joshua B. (2021-02-26).
3106:
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
2752:
2583:{\displaystyle {\bar {D}}={\begin{vmatrix}D_{\color {red}xx}&D_{xy}&D_{xz}\\D_{xy}&D_{\color {red}yy}&D_{yz}\\D_{xz}&D_{yz}&D_{\color {red}zz}\end{vmatrix}}}
6660:
Liu C, Bammer R, Moseley ME (2003). "Generalized
Diffusion Tensor Imaging (GDTI): A Method for Characterizing and Imaging Diffusion Anisotropy Caused by Non-Gaussian Diffusion".
5775:
David Tuch was among the first to describe a solution to this problem. The idea is best understood by conceptually placing a kind of geodesic dome around each image voxel. This
3463:
5744:
40:
7078:
de Sénarmont HH (1848). "Mémoire sur la conductibilité des substances cristalisées pour la chaleur" [Memoir on the conductivity of crystallized substances for heat].
6058:
Wedeen VJ, Wang RP, Schmahmann JD, Benner T, Tseng WY, Dai G, et al. (July 2008). "Diffusion spectrum magnetic resonance imaging (DSI) tractography of crossing fibers".
4450:
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
4108:
3650:
3623:
3596:
5894:
Le Bihan, Denis; Breton, E. (1985). "Imagerie de diffusion in-vivo par résonance magnétique nucléaire" [In-vivo diffusion imaging by nuclear magnetic resonance].
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251:
method that produces in vivo magnetic resonance images of biological tissues sensitized with the local characteristics of molecular diffusion, generally water (but other
5535:
1824:
166:
of the brain as well as for studies of other body tissues (see below). DWI is most applicable when the tissue of interest is dominated by isotropic water movement e.g.
4575:
3569:
1502:{\displaystyle {\frac {d{\vec {M}}}{dt}}=\gamma {\vec {M}}\times {\vec {B}}-{\frac {M_{x}{\vec {i}}+M_{y}{\vec {j}}}{T_{2}}}-{\frac {(M_{z}-M_{0}){\vec {k}}}{T_{1}}}}
578:
558:
514:
3025:
2995:
2965:
2218:
1966:
104:. Water molecule diffusion patterns can therefore reveal microscopic details about tissue architecture, either normal or in a diseased state. A special kind of DWI,
6287:"Diffusion Histology Imaging Combining Diffusion Basis Spectrum Imaging (DBSI) and Machine Learning Improves Detection and Classification of Glioblastoma Pathology"
6235:
Ye, Zezhong; George, Ajit; Wu, Anthony T.; Niu, Xuan; Lin, Joshua; Adusumilli, Gautam; Naismith, Robert T.; Cross, Anne H.; Sun, Peng; Song, Sheng-Kwei (May 2020).
2299:
2272:
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989:
789:
762:
474:
308:
281:
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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
7651:
735:
6961:
Manenti G, Carlani M, Mancino S, Colangelo V, Di Roma M, Squillaci E, Simonetti G (June 2007). "Diffusion tensor magnetic resonance imaging of prostate cancer".
4004:
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2103:
1542:
1009:
709:
598:
538:
494:
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4904:
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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 ADC
1200:
231:(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.
3478:
2367:
A more precise statement of the image acquisition process is that the image-intensities at each position are attenuated, depending on the strength (
1108:
7714:
Griffiths, Timothy T.; Flather, Robert; Teh, Irvin; Haroon, Hamied A.; Shelley, David; Plein, Sven; Bourke, Grainne; Wade, Ryckie G. (2021-07-22).
4357:
3149:. 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
6887:"Signal evolution and infarction risk for apparent diffusion coefficient lesions in acute ischemic stroke are both time- and perfusion-dependent"
208:
with higher specificity and sensitivity than conventional DTI. DBSI has also been useful in determining microstructure properties of the brain.
7097:
Le Bihan D, Mangin JF, Poupon C, Clark CA, Pappata S, Molko N, Chabriat H (April 2001). "Diffusion tensor imaging: concepts and applications".
6391:
Samara, Amjad; Li, Zhaolong; Rutlin, Jerrel; Raji, Cyrus A.; Sun, Peng; Song, Sheng-Kwei; Hershey, Tamara; Eisenstein, Sarah A. (August 2021).
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8433:
1516:
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5517:
The most common application of conventional DWI (without DTI) is in acute brain ischemia. DWI directly visualizes the ischemic necrosis in
4453:
8575:
5626:
2356:
image. It also provides useful structural information about muscleâincluding heart muscleâas well as other tissues such as the prostate.
6548:
Stejskal EO, Tanner JE (1 January 1965). "Spin
Diffusion Measurements: Spin Echoes in the Presence of a Time-Dependent Field Gradient".
5753:: metrics derived from DTI (FA and RD) can differentiate asymptomatic adults from those with compression of the ulnar nerve at the elbow
4848:
8903:
3370:
8175:
6850:
Le Bihan D (August 2013). "Apparent diffusion coefficient and beyond: what diffusion MR imaging can tell us about tissue structure".
6188:"Characterization of multiple sclerosis neuroinflammation and neurodegeneration with relaxation and diffusion basis spectrum imaging"
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154:
Diffusion-weighted images are very useful to diagnose vascular strokes in the brain. It is also used more and more in the staging of
7224:
Westin CF, Maier SE, Mamata H, Nabavi A, Jolesz FA, Kikinis R (June 2002). "Processing and visualization for diffusion tensor MRI".
9011:
5149:{\displaystyle C_{a}=C_{l}+C_{p}=1-C_{s}={\frac {\lambda _{1}+\lambda _{2}-2\lambda _{3}}{\lambda _{1}+\lambda _{2}+\lambda _{3}}}}
3145:, because the measurement misses all the local effects and treats the attenuation as if all the movement rates were solely due to
8541:
7462:
Takahara, Taro; Kwee, Thomas C. (2010). "Diffusion-Weighted Whole-Body
Imaging with Background Body Signal Suppression (DWIBS)".
3293:
subscript to signify that this is what the isotropic diffusion coefficient would be with the effects of anisotropy averaged out.
5958:
Taylor DG, Bushell MC (April 1985). "The spatial mapping of translational diffusion coefficients by the NMR imaging technique".
7773:"Meta-analysis of the normal diffusion tensor imaging values of the median nerve and how they change in carpal tunnel syndrome"
7992:
Basser PJ, Pajevic S (2007). "Spectral decomposition of a 4th-order covariance tensor: applications to diffusion tensor MRI".
7557:"Diffusion Tensor Imaging for Diagnosing Root Avulsions in Traumatic Adult Brachial Plexus Injuries: A Proof-of-Concept Study"
7497:"Diffusion kurtosis MRI as a predictive biomarker of response to neoadjuvant chemotherapy in high grade serous ovarian cancer"
1950:{\displaystyle {\vec {D}}=D\cdot {\vec {I}}=D\cdot {\begin{bmatrix}1&0&0\\0&1&0\\0&0&1\end{bmatrix}},}
610:
162:
as the 'gold standard' for this type of disease. Diffusion tensor imaging is being developed for studying the diseases of the
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797:
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7015:
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can differentiate normal nerves (as shown in the tractogram of the spinal cord and brachial plexus and 3D 4k reconstruction
9114:
8387:
7667:"Diffusion tensor imaging of the roots of the brachial plexus: a systematic review and meta-analysis of normative values"
6712:"Diffusional kurtosis imaging: the quantification of non-gaussian water diffusion by means of magnetic resonance imaging"
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96:
and non-invasively. Molecular diffusion in tissues is not random, but reflects interactions with many obstacles, such as
80:
as well as software that generates images from the resulting data that uses the diffusion of water molecules to generate
6470:
King MD, Houseman J, Roussel SA, van
Bruggen N, Williams SR, Gadian DG (December 1994). "q-Space imaging of the brain".
2784:
2105:. Anisotropic diffusion will have a similar solution for the diffusion tensor, except that what will be measured is the
8939:
8818:
8558:
8467:
6443:
Zhang Y, Wang S, Wu L, Huo Y (Jan 2011). "Multi-channel diffusion tensor image registration via adaptive chaotic PSO".
320:
2686:
At this point, it is helpful to consider the mathematics of ellipsoids. An ellipsoid can be described by the formula:
8397:
5805:
tissue. Consequently, this is the topic of a multinational study directed by the diffusion-study group of the ISMRM.
5667:
5504:
17:
8143:
2389:. Only a special case of the general mathematical notion is relevant to imaging, which is based on the concept of a
2115:
8382:
8377:
8260:
5611:
of the water molecules preferentially along their main direction. Such preferentially oriented diffusion is called
5588:
kurtosis imaging (DKI) have also been shown to predict the response of cancer patients to chemotherapy treatment.
8462:
8445:
256:
this capability of producing contrast related to the structure of tissues at the microscopic level. In a typical
7141:
1025:
8823:
8670:
8570:
8553:
7427:
Koh DM, Collins DJ (June 2007). "Diffusion-weighted MRI in the body: applications and challenges in oncology".
5864:
5486:
5482:
8766:
5651:) that describes the 3-dimensional shape of diffusion. The fiber direction is indicated by the tensor's main
3040:
angles required to get to this equivalent position now appear in the three vectors and can be read out as the
8949:
8803:
8402:
8277:
8269:
8168:
5689:
within white matter. Fiber tracking algorithms can be used to track a fiber along its whole length (e.g. the
6935:
5521:
in the form of a cytotoxic edema, appearing as a high DWI signal within minutes of arterial occlusion. With
2689:
9060:
8994:
8926:
8916:
8894:
8665:
8633:
8264:
5571:
DWI showing cortical ribbon-like high signal consistent with diffusion restriction in a patient with known
1829:
For the simplest case where the diffusion is isotropic the diffusion tensor is a multiple of the identity:
159:
6689:
Relevance of the information about the diffusion distribution in invo given by kurtosis in q-space imaging
6393:"Nucleus accumbens microstructure mediates the relationship between obesity and eating behavior in adults"
3110:
causes a group of water molecules to move out from a central point, and gradually reach the surface of an
2670:
2385:
Diffusion MRI relies on the mathematics and physical interpretations of the geometric quantities known as
8724:
8692:
8675:
4011:. The simplest measure is obtained by dividing the longest axis of the ellipsoid by the shortest = (
3158:
2933:
2678:
showed in the 1850s, diffusion exhibits many of the same patterns as those seen in the transfer of heat.
1012:
5634:
The imaging of this property is an extension of diffusion MRI. If a series of diffusion gradients (i.e.
8798:
8533:
8273:
3441:
155:
6237:"Deep learning with diffusion basis spectrum imaging for classification of multiple sclerosis lesions"
9039:
8707:
8697:
8014:
5444:{\displaystyle \mathrm {VR} ={\frac {\lambda _{1}\lambda _{2}\lambda _{3}}{\operatorname {E} ^{3}}}}
4831:{\displaystyle C_{p}={\frac {2(\lambda _{2}-\lambda _{3})}{\lambda _{1}+\lambda _{2}+\lambda _{3}}}}
9078:
9029:
8808:
8687:
8660:
8421:
8206:
8161:
7149:. Proceedings of the Society for Magnetic Resonance in Medicine. Vol. 11. Berlin. p. 412.
5740:
4081:
2335:, is an MRI image that more specifically shows diffusion than conventional DWI, by eliminating the
50:
7160:
Vaillancourt DE, Spraker MB, Prodoehl J, Abraham I, Corcos DM, Zhou XJ, et al. (April 2009).
6513:
Posse S, Cuenod CA, Le Bihan D (September 1993). "Human brain: proton diffusion MR spectroscopy".
3628:
3601:
3574:
2928:, we can apply a useful technique from standard matrix mathematics and linear algebraâthat is to "
2309:
The standard grayscale of DWI images is to represent increased diffusion restriction as brighter.
8977:
8813:
8754:
8739:
8628:
8450:
8426:
5750:
5557:
5475:
4662:{\displaystyle C_{l}={\frac {\lambda _{1}-\lambda _{2}}{\lambda _{1}+\lambda _{2}+\lambda _{3}}}}
3179:. This leads to the idea of describing the average diffusivity in the voxel which will simply be
5871:, and so on, but the generality, which covers arrays of all sizes, may obscure rather than help.
1800:
8848:
8717:
8648:
8455:
8009:
7918:"High angular resolution diffusion imaging reveals intravoxel white matter fiber heterogeneity"
7162:"High-resolution diffusion tensor imaging in the substantia nigra of de novo Parkinson disease"
5760:
5756:
5710:
2380:
2319:
2072:{\displaystyle {M}={M}_{\text{bloch}}e^{-{\frac {1}{3}}\gamma ^{2}G^{2}t^{3}D}\sim e^{-bD_{0}}}
5923:
Merboldt KD, Hanicke W, Frahm J (1985). "Self-diffusion NMR imaging using stimulated echoes".
3554:
189:
in the brain or muscle fibers in the heartâhas an internal fibrous structure analogous to the
9109:
8712:
8643:
8509:
8392:
8311:
8218:
4069:
3127:
3059:
1546:
1091:
563:
543:
499:
228:
3141:
interfere with the free diffusion, we are measuring an "apparent diffusion coefficient", or
3000:
2970:
2940:
2193:
9055:
8744:
8682:
8504:
8001:
7784:
7508:
6768:
6557:
6347:
5967:
5932:
5798:
5648:
2925:
2611:. A three-dimensional vector can be described with three components: its projection on the
2604:
2277:
2250:
2223:
974:
767:
740:
452:
286:
259:
5759:: Metrics derived from DTI (lower FA and MD) differentiate healthy adults from those with
8:
9001:
8879:
8702:
8438:
7965:
Hext GR (1963). "The estimation of second-order tensors with related tests and designs".
5690:
5679:
5542:
5526:
5518:
3119:
2343:
2336:
1095:
714:
101:
8005:
7788:
7555:
Wade RG, Tanner SF, Teh I, Ridgway JP, Shelley D, Chaka B, et al. (16 April 2020).
7512:
6772:
6561:
6351:
6011:"Mapping complex tissue architecture with diffusion spectrum magnetic resonance imaging"
5971:
5936:
5817:, matrix mathematics and vector mathematics to process the analysis of the tensor data.
8989:
8838:
8833:
8734:
8617:
8472:
8223:
8123:
8086:"Comparison of distortion correction preprocessing pipelines for DTI in the upper limb"
8066:
8027:
7947:
7870:
7813:
7772:
7748:
7715:
7691:
7666:
7632:
7583:
7556:
7529:
7496:
7495:
Deen SS, Priest AN, McLean MA, Gill AB, Brodie C, Crawford R, et al. (July 2019).
7404:
7377:
7285:
7260:
7186:
7177:
7161:
7122:
6996:
6974:
6911:
6886:
6741:
6636:
6629:
Principles and Application of Self-Diffusion Measurements by Nuclear Magnetic Resonance
6609:
6495:
6417:
6392:
6368:
6335:
6311:
6286:
6261:
6236:
6212:
6187:
6162:
6137:
6083:
6040:
5991:
3989:
3969:
2088:
1527:
1191:
994:
694:
583:
523:
517:
479:
252:
215:. Clinically, trace-weighted images have proven to be very useful to diagnose vascular
205:
89:
7237:
2622:
axes. Vectors of this sort can be considered tensors of rank 1, or 1st-order tensors.
8828:
8793:
8788:
8729:
8477:
8326:
8127:
8115:
8107:
8070:
7939:
7898:
7862:
7818:
7800:
7753:
7735:
7696:
7665:
Wade RG, Whittam A, Teh I, Andersson G, Yeh FC, Wiberg M, Bourke G (9 October 2020).
7624:
7588:
7534:
7475:
7444:
7409:
7378:"Magnetic resonance diffusion-perfusion mismatch in acute ischemic stroke: An update"
7331:
7290:
7241:
7191:
7114:
7060:
7037:
6988:
6916:
6867:
6733:
6692:
6640:
6601:
6530:
6487:
6422:
6373:
6336:"Diffusion histology imaging differentiates distinct pediatric brain tumor histology"
6316:
6266:
6217:
6167:
6118:
6075:
6071:
6032:
5995:
5983:
5979:
5944:
4981:{\displaystyle C_{s}={\frac {3\lambda _{3}}{\lambda _{1}+\lambda _{2}+\lambda _{3}}}}
81:
7652:"DTI (Quantitative), a new and advanced MRI procedure for evaluation of Concussions"
7636:
7350:
7000:
6745:
6613:
6526:
6499:
6114:
2662:. These can be written as a matrix similar to the one at the start of this section.
219:
in the brain, by early detection (within a couple of minutes) of the hypoxic edema.
9104:
9099:
8655:
8514:
8286:
8238:
8097:
8058:
8031:
8019:
7974:
7951:
7929:
7874:
7852:
7808:
7792:
7743:
7727:
7686:
7678:
7616:
7578:
7568:
7524:
7516:
7467:
7436:
7399:
7389:
7321:
7280:
7272:
7233:
7181:
7173:
7126:
7106:
7027:
6978:
6970:
6906:
6898:
6859:
6776:
6723:
6669:
6632:
6593:
6565:
6522:
6479:
6452:
6412:
6404:
6363:
6355:
6306:
6298:
6256:
6248:
6207:
6199:
6157:
6149:
6110:
6087:
6067:
6044:
6022:
5975:
5940:
4008:
2675:
2608:
2390:
6302:
8864:
8316:
8282:
8255:
8250:
8184:
8023:
7916:
Tuch DS, Reese TG, Wiegell MR, Makris N, Belliveau JW, Wedeen VJ (October 2002).
7471:
6902:
6798:
5736:
4447:
3146:
2323:
ADC image of the same case of cerebral infarction as seen on DWI in section above
1520:
1279:{\displaystyle {\frac {\partial \rho (x,t)}{\partial t}}=D\nabla ^{2}\rho (x,t).}
171:
56:
7978:
7889:
6673:
9006:
7796:
7731:
7682:
7520:
7326:
7309:
7308:
Grand S, Tahon F, Attye A, Lefournier V, Le Bas JF, Krainik A (December 2013).
7276:
6359:
5814:
5718:
5635:
5584:
5572:
1299:
7771:
Rojoa, Djamila; Raheman, Firas; Rassam, Joseph; Wade, Ryckie G. (2021-10-22).
7143:
Vector analysis of diffusion images in experimental allergic encephalomyelitis
6203:
6136:
Wang Y, Wang Q, Haldar JP, Yeh FC, Xie M, Sun P, et al. (December 2011).
5556:
DWI showing restricted diffusion in the medial dorsal thalami consistent with
3541:{\displaystyle \mathrm {tr} (\Lambda )=\lambda _{1}+\lambda _{2}+\lambda _{3}}
1180:{\displaystyle {\frac {\partial \rho (x,t)}{\partial t}}=-\nabla \cdot J(x,t)}
9093:
8638:
8607:
8595:
8348:
8243:
8111:
7804:
7739:
7573:
7064:
7057:
Physical Properties of Crystals: Their Representation by Tensors and Matrices
6582:
5785:
5522:
4436:{\textstyle \operatorname {E} =(\lambda _{1}+\lambda _{2}+\lambda _{3})/3\,.}
3134:
1295:
140:
136:
97:
7902:
6863:
6780:
6696:
6456:
9021:
8874:
8869:
8776:
8600:
8585:
8519:
8119:
7943:
7866:
7822:
7757:
7700:
7592:
7538:
7448:
7413:
7335:
7294:
7245:
7195:
7118:
7041:
6992:
6920:
6871:
6737:
6631:. Advances in Magnetic and Optical Resonance. Vol. 12. pp. 1â89.
6597:
6483:
6426:
6377:
6320:
6270:
6221:
6171:
6153:
6138:"Quantification of increased cellularity during inflammatory demyelination"
6079:
6036:
5844:
5834:
5821:
5725:
5694:
5686:
5675:
5592:
3468:
Another commonly used measure that summarizes the total diffusivity is the
3299:
3053:
2361:
1544:
is a vector, there are 3 diffusion equations, one for each dimension. The
235:; trying to see which part of the brain is connected to which other part).
232:
186:
163:
116:
113:
77:
7628:
7620:
6605:
6534:
6491:
6122:
5987:
5820:
In some cases, the full set of tensor properties is of interest, but for
8944:
8499:
8331:
8321:
8301:
8291:
8233:
8228:
8197:
7394:
7355:
6940:
5776:
5698:
5678:, surgery is aided by knowing the proximity and relative position of the
5652:
3305:
3138:
3123:
3087:
2775:
201:
167:
7440:
6009:
Wedeen VJ, Hagmann P, Tseng WY, Reese TG, Weisskoff RM (December 2005).
3733:{\displaystyle \mathrm {MD} =(\lambda _{1}+\lambda _{2}+\lambda _{3})/3}
8580:
8343:
8338:
8306:
8062:
6252:
5839:
5794:
5596:
5489: in this section. Unsourced material may be challenged and removed.
4991:
Each measure lies between 0 and 1 and they sum to unity. An additional
3098:
1099:
190:
8102:
8085:
7934:
7917:
7857:
7840:
7110:
7014:
Basser PJ, Pajevic S, Pierpaoli C, Duda J, Aldroubi A (October 2000).
6983:
6826:
6728:
6711:
6569:
6408:
6027:
6010:
5618:
2396:
The following matrix displays the components of the diffusion tensor:
8907:
8843:
8296:
5630:
DTI of a healthy human brachial plexus. Taken from Wade et al., 2020.
5608:
3111:
3107:
2771:
85:
8046:
7897:. Proceedings of the 7th Annual Meeting of the ISMRM. Philadelphia.
5863:
Several full mathematical treatments of general tensors exist, e.g.
5464:
4502:{\displaystyle \lambda _{1}\geq \lambda _{2}\geq \lambda _{3}\geq 0}
8930:
8047:"Uber eine geometrische Anwendung der Abelschen Integralgleichnung"
7080:
Comptes Rendus Hebdomadaires des Séances de l'Académie des Sciences
7033:
10.1002/1522-2594(200010)44:4<625::AID-MRM17>3.0.CO;2-O
5714:
5580:
3074:) components have numbers in them but the off-diagonal components (
1016:
8153:
7159:
4891:{\displaystyle \lambda _{1}\simeq \lambda _{2}\simeq \lambda _{3}}
1512:
which has terms for precession, T2 relaxation, and T1 relaxation.
8959:
8362:
7211:
Geometrical diffusion measures for MRI from tensor basis analysis
7209:
Westin CF, Peled S, Gudbjartsson H, Kikinis R, Jolesz FA (1997).
5825:
aligned with the main axon direction to accomplish tractography.
5721:
2755:
93:
7605:
6710:
Jensen JH, Helpern JA, Ramani A, Lu H, Kaczynski K (June 2005).
3427:{\displaystyle \lambda _{\perp }=(\lambda _{2}+\lambda _{3})/2.}
2781:
As it turns out, three more components can be added as follows:
8954:
6691:. Proceedings, 12th ISMRM Annual Meeting. Kyoto. p. 1238.
5663:
5644:
5604:
4722:{\displaystyle \lambda _{1}\simeq \lambda _{2}\gg \lambda _{3}}
4562:{\displaystyle \lambda _{1}\gg \lambda _{2}\simeq \lambda _{3}}
3311:
2386:
224:
216:
7208:
6960:
6469:
6100:
5717:. Applications in brain research include the investigation of
3340:
is an actual measurement of the ellipsoid we have calculated.
8981:
8192:
5702:
5671:
5640:
212:
182:
132:
7891:
High angular resolution diffusion imaging of the human brain
7887:
4995:
can used to describe the deviation from the spherical case:
3052:
components of each of them. Those three vectors are called "
1015:
gives a relationship between the flux and the concentration
39:
5600:
4110:
so that FA has a maximum value of 1. The whole formula for
3150:
3027:
are along the diagonal (red subscripts), but the variables
148:
8148:
7139:
7013:
6885:
An H, Ford AL, Vo K, Powers WJ, Lee JM, Lin W (May 2011).
6759:
Torrey HC (1956). "Bloch Equations with Diffusion Terms".
6057:
5797:
and vector mathematics developed nearly 100 years agoâthe
5622:
Tractographic reconstruction of neural connections via DTI
5159:
There are other metrics of anisotropy used, including the
3126:
of an MRI voxel into a numerical measure of diffusionâthe
2909:
still describe ellipsoids, but the additional components (
1098:
relates the time derivative of the concentration with the
7713:
7307:
7096:
5529:, the latter can be quantified by DWI and perfusion MRI.
3315:. The long one pointing along the axon direction will be
3093:
476:
is the signal intensity without the diffusion weighting,
248:
3133:. When various barriers and restricting factors such as
681:{\displaystyle {\frac {S(TE)}{S_{0}}}=\exp(-b\cdot ADC)}
7888:
Tuch DS, Weisskoff RM, Belliveau JW, Wedeen VJ (1999).
7664:
7223:
6008:
5693:, through which the motor information transit from the
5599:
of the tracts can be measured. The architecture of the
952:{\displaystyle \mathrm {ADC} (x,y,z)=\ln/(b_{1}-b_{2})}
27:
Method of utilizing water in magnetic resonance imaging
7915:
7770:
7258:
6627:
KÀrger, Jörg; Pfeifer, Harry; Heink, Wilfried (1988).
5525:
detecting both the infarcted core and the salvageable
4360:
2426:
1960:
then the Bloch-Torrey equation will have the solution
1886:
151:
trajectories than other diffusion imaging approaches.
7716:"Diffusion tensor imaging in cubital tunnel syndrome"
7259:
Alexander AL, Lee JE, Lazar M, Field AS (July 2007).
7140:
Richards TL, Heide AC, Tsuruda JS, Alvord EC (1992).
6709:
5685:
Diffusion tensor imaging data can be used to perform
5369:
5172:
5004:
4907:
4851:
4738:
4682:
4578:
4522:
4456:
4123:
4084:
3992:
3972:
3759:
3668:
3631:
3604:
3577:
3557:
3481:
3444:
3373:
3188:
3003:
2973:
2943:
2787:
2692:
2405:
2280:
2253:
2226:
2196:
2118:
2091:
1969:
1838:
1803:
1559:
1530:
1310:
1203:
1111:
1028:
997:
977:
800:
770:
743:
717:
697:
613:
586:
566:
546:
526:
502:
482:
455:
323:
289:
262:
8083:
7554:
7494:
6686:
5922:
711:, is replaced by an apparent diffusion coefficient,
181:(DTI) is important when a tissueâsuch as the neural
6512:
6333:
3350:is also called the longitudinal diffusivity or the
3279:{\displaystyle (ADC_{x}+ADC_{y}+ADC_{z})/3=ADC_{i}}
6626:
6390:
6184:
5443:
5345:
5148:
4980:
4890:
4830:
4721:
4661:
4561:
4509:, then the measures can be calculated as follows:
4501:
4435:
4343:
4102:
3998:
3978:
3955:
3732:
3644:
3617:
3590:
3563:
3540:
3457:
3426:
3278:
3019:
2989:
2959:
2878:{\displaystyle ax^{2}+by^{2}+cz^{2}+dyz+ezx+fxy=1}
2877:
2746:
2582:
2374:
2293:
2266:
2239:
2212:
2179:
2097:
2071:
1949:
1818:
1786:
1536:
1501:
1278:
1179:
1079:
1003:
983:
951:
783:
756:
729:
703:
680:
592:
572:
552:
532:
508:
488:
468:
438:
302:
275:
7834:
7832:
6135:
3309:. The lengths are symbolized by the Greek letter
1294:With no diffusion present, the change in nuclear
580:is the time between the two pulses, and finally,
439:{\displaystyle {\frac {S(TE)}{S_{0}}}=\exp \left}
9091:
6659:
2082:The exponential term will be referred to as the
143:rates. A variant of diffusion weighted imaging,
6792:
6790:
5666:such as trauma and in defining the severity of
5591:The principal application is in the imaging of
7829:
7348:
7301:
7016:"In vivo fiber tractography using DT-MRI data"
6884:
6547:
6442:
6241:Annals of Clinical and Translational Neurology
5893:
5541:DWI showing necrosis (shown as brighter) in a
2924:When the ellipsoid/tensor is represented by a
2180:{\displaystyle A=e^{-\sum _{i,j}b_{ij}D_{ij}}}
8169:
7991:
6438:
6436:
6283:
6234:
5957:
5889:
5887:
2347:DWI is a result of its partial T2 weighting.
194:direction from which an observer is looking.
158:, where it is a serious candidate to replace
66:Diffusion-weighted magnetic resonance imaging
7461:
7217:
7213:. ISMRM '97. Vancouver Canada. p. 1742.
7133:
7077:
6843:
6787:
5662:Tract-specific localization of white matter
3472:âwhich is the sum of the three eigenvalues,
2681:
131:(DWI), the intensity of each image element (
6680:
3655:If we divide this sum by three we have the
2350:
84:in MR images. It allows the mapping of the
8176:
8162:
7426:
6878:
6433:
6094:
5884:
3343:The diffusivity along the principal axis,
3115:of diffusion in each voxel of the tissue.
3102:Visualization of DTI data with ellipsoids.
2340:smaller magnitude of diffusion as darker.
1080:{\displaystyle J(x,t)=-D\nabla \rho (x,t)}
38:
8101:
8013:
7933:
7856:
7812:
7747:
7690:
7582:
7572:
7550:
7548:
7528:
7403:
7393:
7325:
7284:
7185:
7031:
6982:
6910:
6827:"MRI Physics: Diffusion-Weighted Imaging"
6820:
6818:
6727:
6416:
6367:
6310:
6260:
6211:
6161:
6026:
5896:Comptes Rendus de l'Académie des Sciences
5813:For DTI, it is generally possible to use
5747:) from traumatically injured nerve roots.
5505:Learn how and when to remove this message
4429:
3322:and the two small axes will have lengths
1289:
9012:Orthogonal polarization spectral imaging
7369:
6936:"Diffusion weighted MRI in acute stroke"
6849:
6687:Chabert S, Mecca CC, Le Bihan D (2004).
5647:(i.e. a symmetric positive definite 3Ă3
5625:
5617:
3153:, and slower perpendicularly across it.
3097:
2318:
1094:. Then, given conservation of mass, the
7466:. Medical Radiology. pp. 227â252.
7261:"Diffusion tensor imaging of the brain"
2109:(ADC). In general, the attenuation is:
540:is the strength of the gradient pulse,
88:process of molecules, mainly water, in
14:
9092:
7545:
7429:AJR. American Journal of Roentgenology
7375:
7349:Weerakkody Y, Gaillard F, et al.
6815:
6796:
6758:
5579:Another application area of DWI is in
3571:is a diagonal matrix with eigenvalues
3094:Measures of anisotropy and diffusivity
2770:determine if the quadric describes an
2747:{\displaystyle ax^{2}+by^{2}+cz^{2}=1}
2220:terms incorporate the gradient fields
8157:
7314:Diagnostic and Interventional Imaging
7099:Journal of Magnetic Resonance Imaging
5595:where the location, orientation, and
1190:Putting the two together, we get the
8044:
7964:
7838:
7649:
7310:"Perfusion imaging in brain disease"
5850:
5732:Applications for peripheral nerves:
5487:adding citations to reliable sources
5458:
3086:) are 0. The second matrix provides
1298:over time is given by the classical
112:), has been used extensively to map
8183:
7054:
2597:
24:
8819:Sestamibi parathyroid scintigraphy
7671:Clinical and Translational Imaging
7178:10.1212/01.wnl.0000340982.01727.6e
6975:10.1097/01.rli.0000264059.46444.bf
6824:
6637:10.1016/b978-0-12-025512-2.50004-x
5416:
5374:
5371:
5325:
5291:
5247:
5203:
5177:
5174:
4361:
4248:
4204:
4160:
4128:
4125:
3986:is the matrix of eigenvectors and
3902:
3899:
3861:
3851:
3848:
3823:
3800:
3797:
3775:
3768:
3765:
3673:
3670:
3558:
3493:
3486:
3483:
3303:and the measures of their lengths
1769:
1751:
1246:
1230:
1207:
1153:
1138:
1115:
1056:
966:
808:
805:
802:
567:
407:
25:
9126:
8137:
6933:
3458:{\displaystyle \lambda _{\perp }}
3354:or even the parallel diffusivity
2562:
2498:
2434:
691:Also, the diffusion coefficient,
496:is the signal with the gradient,
9074:
9073:
6072:10.1016/j.neuroimage.2008.03.036
5564:
5549:
5534:
5463:
2758:surface. The relative values of
791:, of the b factor according to:
198:Diffusion Basis Spectrum Imaging
8576:Cholangiopancreatography (MRCP)
8077:
8038:
7985:
7958:
7909:
7881:
7764:
7707:
7658:
7643:
7599:
7488:
7455:
7420:
7342:
7252:
7202:
7153:
7090:
7071:
7048:
7007:
6954:
6927:
6752:
6703:
6653:
6620:
6576:
6550:The Journal of Chemical Physics
6541:
6527:10.1148/radiology.188.3.8351339
6506:
6463:
6384:
6327:
6277:
6115:10.1148/radiology.161.2.3763909
5960:Physics in Medicine and Biology
5857:
5603:in parallel bundles, and their
5474:needs additional citations for
5454:
3118:In an isotropic medium such as
2375:Mathematical foundationâtensors
122:
8824:Radioactive iodine uptake test
8090:Magnetic Resonance in Medicine
7922:Magnetic Resonance in Medicine
7845:Magnetic Resonance in Medicine
7020:Magnetic Resonance in Medicine
6716:Magnetic Resonance in Medicine
6586:Magnetic Resonance in Medicine
6472:Magnetic Resonance in Medicine
6228:
6178:
6129:
6051:
6015:Magnetic Resonance in Medicine
6002:
5951:
5916:
5668:diffuse traumatic brain injury
5429:
5422:
5337:
5331:
5307:
5303:
5297:
5275:
5263:
5259:
5253:
5231:
5219:
5215:
5209:
5187:
4784:
4758:
4418:
4379:
4373:
4367:
4335:
4281:
4273:
4264:
4260:
4254:
4232:
4220:
4216:
4210:
4188:
4176:
4172:
4166:
4144:
4141:
3912:
3906:
3877:
3855:
3826:
3804:
3778:
3772:
3719:
3680:
3496:
3490:
3413:
3387:
3246:
3189:
2412:
2329:apparent diffusion coefficient
2107:apparent diffusion coefficient
1866:
1845:
1810:
1778:
1763:
1729:
1720:
1694:
1669:
1644:
1616:
1601:
1572:
1519:mathematically showed how the
1480:
1471:
1445:
1420:
1395:
1367:
1352:
1323:
1270:
1258:
1225:
1213:
1174:
1162:
1133:
1121:
1074:
1062:
1044:
1032:
946:
920:
912:
909:
891:
873:
855:
842:
830:
812:
675:
654:
629:
620:
600:is the diffusion-coefficient.
560:is the duration of the pulse,
339:
330:
13:
1:
8804:Radionuclide ventriculography
8278:Lower gastrointestinal series
8270:Upper gastrointestinal series
7464:Diffusion-Weighted MR Imaging
7238:10.1016/s1361-8415(02)00053-1
6303:10.1158/1078-0432.ccr-20-0736
5925:Journal of Magnetic Resonance
5878:
4103:{\displaystyle 1/{\sqrt {2}}}
1826:is now the diffusion tensor.
8995:Optical coherence tomography
8917:Myocardial perfusion imaging
8505:Dental panoramic radiography
8024:10.1016/j.sigpro.2006.02.050
7472:10.1007/978-3-540-78576-7_14
7376:Chen F, Ni YC (March 2012).
7055:Nye, John Frederick (1957).
6903:10.1161/STROKEAHA.110.610501
5945:10.1016/0022-2364(85)90111-8
3645:{\displaystyle \lambda _{3}}
3618:{\displaystyle \lambda _{2}}
3591:{\displaystyle \lambda _{1}}
2754:. This equation describes a
2314:
2304:
242:
160:positron emission tomography
7:
6674:10.1560/HB5H-6XBR-1AW1-LNX9
6662:Israel Journal of Chemistry
5828:
5766:
5658:Applications in the brain:
10:
9131:
9115:Magnetic resonance imaging
8799:Ventilation/perfusion scan
8274:Small-bowel follow-through
7797:10.1038/s41598-021-00353-z
7732:10.1038/s41598-021-94211-7
7683:10.1007/s40336-020-00393-x
7521:10.1038/s41598-019-47195-4
7382:World Journal of Radiology
7327:10.1016/j.diii.2013.06.009
7277:10.1016/j.nurt.2007.05.011
6360:10.1038/s41598-021-84252-3
6192:Multiple Sclerosis Journal
5980:10.1088/0031-9155/30/4/009
5808:
2378:
1819:{\displaystyle {\vec {D}}}
156:non-small-cell lung cancer
145:diffusion spectrum imaging
129:diffusion weighted imaging
9069:
9048:
9040:Dynamic angiothermography
9020:
8976:
8925:
8902:
8892:
8857:
8784:
8774:
8765:
8708:Abdominal ultrasonography
8616:
8532:
8492:
8411:
8370:
8361:
8214:
8205:
8191:
8144:PNRC: About Diffusion MRI
7979:10.1093/biomet/50.3-4.353
7839:Tuch DS (December 2004).
6204:10.1177/13524585211023345
2682:Mathematics of ellipsoids
2671:Henri Hureau de SĂ©narmont
76:) is the use of specific
49:
37:
32:
9030:Non-contact thermography
8809:Radionuclide angiography
8661:Doppler echocardiography
7574:10.3389/fsurg.2020.00019
6291:Clinical Cancer Research
5607:sheaths, facilitate the
3564:{\displaystyle \Lambda }
2351:Diffusion tensor imaging
971:Given the concentration
247:Diffusion imaging is an
179:Diffusion tensor imaging
106:diffusion tensor imaging
8814:Radioisotope renography
6963:Investigative Radiology
6864:10.1148/radiol.13130420
6781:10.1103/PhysRev.104.563
6457:10.4304/jcp.6.4.825-829
5751:Cubital Tunnel Syndrome
5558:Wernicke encephalopathy
2885:. Many combinations of
573:{\displaystyle \Delta }
553:{\displaystyle \delta }
509:{\displaystyle \gamma }
8849:Gastric emptying study
7226:Medical Image Analysis
6799:"Restricted Diffusion"
6598:10.1002/mrm.1910360607
6484:10.1002/mrm.1910320605
5761:carpal tunnel syndrome
5757:Carpal Tunnel Syndrome
5670:. The localization of
5631:
5623:
5445:
5347:
5150:
4982:
4892:
4832:
4723:
4663:
4563:
4503:
4437:
4345:
4104:
4000:
3980:
3957:
3734:
3646:
3619:
3592:
3565:
3542:
3459:
3428:
3280:
3103:
3021:
3020:{\displaystyle cz^{2}}
2991:
2990:{\displaystyle by^{2}}
2961:
2960:{\displaystyle ax^{2}}
2879:
2748:
2584:
2381:Principal axis theorem
2324:
2295:
2268:
2241:
2214:
2213:{\displaystyle b_{ij}}
2181:
2099:
2073:
1951:
1820:
1788:
1538:
1503:
1290:Magnetization dynamics
1280:
1181:
1081:
1005:
985:
953:
785:
758:
731:
705:
682:
594:
574:
554:
534:
510:
490:
470:
440:
304:
277:
8510:X-ray motion analysis
8393:X-ray microtomography
8312:Hysterosalpingography
8219:Pneumoencephalography
7621:10.1002/ana.410420617
5629:
5621:
5613:anisotropic diffusion
5446:
5348:
5151:
4983:
4893:
4833:
4724:
4664:
4564:
4504:
4438:
4346:
4105:
4070:fractional anisotropy
4001:
3981:
3958:
3735:
3647:
3620:
3593:
3566:
3543:
3460:
3429:
3281:
3128:diffusion coefficient
3101:
3022:
2992:
2962:
2880:
2749:
2585:
2322:
2296:
2294:{\displaystyle G_{z}}
2269:
2267:{\displaystyle G_{y}}
2242:
2240:{\displaystyle G_{x}}
2215:
2182:
2100:
2074:
1952:
1821:
1789:
1547:Bloch-Torrey equation
1539:
1504:
1281:
1182:
1092:diffusion coefficient
1082:
1006:
986:
984:{\displaystyle \rho }
954:
786:
784:{\displaystyle b_{2}}
759:
757:{\displaystyle b_{1}}
732:
706:
683:
595:
575:
555:
535:
511:
491:
471:
469:{\displaystyle S_{0}}
441:
305:
303:{\displaystyle T_{2}}
278:
276:{\displaystyle T_{1}}
229:fractional anisotropy
9035:Contact thermography
8745:Emergency ultrasound
8683:Transcranial Doppler
8434:Abdominal and pelvis
7561:Frontiers in Surgery
7395:10.4329/wjr.v4.i3.63
6445:Journal of Computers
6154:10.1093/brain/awr307
6148:(Pt 12): 3590â3601.
5799:Funk Radon Transform
5713:from other types of
5483:improve this article
5367:
5170:
5002:
4905:
4849:
4736:
4680:
4576:
4520:
4454:
4358:
4121:
4082:
3990:
3970:
3757:
3666:
3629:
3602:
3575:
3555:
3479:
3442:
3371:
3186:
3001:
2971:
2941:
2785:
2690:
2403:
2278:
2251:
2224:
2194:
2116:
2089:
1967:
1836:
1801:
1557:
1528:
1308:
1201:
1109:
1026:
995:
975:
798:
768:
741:
715:
695:
611:
584:
564:
544:
524:
500:
480:
453:
321:
287:
260:
9002:Confocal microscopy
8880:Indium-111 WBC scan
8703:Echoencephalography
8439:Virtual colonoscopy
8006:2007SigPr..87..220B
7789:2021NatSR..1120935R
7609:Annals of Neurology
7513:2019NatSR...910742D
7441:10.2214/AJR.06.1403
7059:. Clarendon Press.
6773:1956PhRv..104..563T
6562:1965JChPh..42..288S
6352:2021NatSR..11.4749Y
5972:1985PMB....30..345T
5937:1985JMagR..64..479M
5711:Alzheimer's disease
5701:and the peripheral
5691:corticospinal tract
5680:corticospinal tract
5543:cerebral infarction
5519:cerebral infarction
5161:relative anisotropy
4334:
4316:
4298:
3120:cerebrospinal fluid
2344:Cerebral infarction
2331:(ADC) image, or an
1096:continuity equation
730:{\displaystyle ADC}
8990:Optical tomography
8839:Dacryoscintigraphy
8834:Immunoscintigraphy
8473:Whole body imaging
8224:Dental radiography
8149:White Matter Atlas
8063:10.1007/BF01456824
7777:Scientific Reports
7720:Scientific Reports
7501:Scientific Reports
7351:"Ischaemic stroke"
6797:Elster AD (2021).
6340:Scientific Reports
6253:10.1002/acn3.51037
5632:
5624:
5441:
5343:
5146:
4993:anisotropy measure
4978:
4888:
4828:
4719:
4659:
4559:
4499:
4433:
4341:
4320:
4302:
4284:
4100:
3996:
3976:
3953:
3951:
3730:
3642:
3615:
3588:
3561:
3538:
3455:
3424:
3363:radial diffusivity
3276:
3104:
3017:
2987:
2957:
2875:
2744:
2580:
2574:
2569:
2505:
2441:
2325:
2291:
2264:
2237:
2210:
2177:
2148:
2095:
2069:
1947:
1938:
1816:
1784:
1534:
1499:
1276:
1192:diffusion equation
1177:
1077:
1001:
981:
949:
781:
754:
727:
701:
678:
590:
570:
550:
530:
518:gyromagnetic ratio
506:
486:
466:
436:
300:
273:
206:multiple sclerosis
90:biological tissues
9087:
9086:
9049:Target conditions
8972:
8971:
8968:
8967:
8888:
8887:
8829:Bone scintigraphy
8794:Scintimammography
8789:Cholescintigraphy
8634:contrast-enhanced
8528:
8527:
8488:
8487:
8478:Full-body CT scan
8378:General operation
8357:
8356:
8327:Angiocardiography
8103:10.1002/mrm.29881
7994:Signal Processing
7935:10.1002/mrm.10268
7858:10.1002/mrm.20279
7481:978-3-540-78575-0
7320:(12): 1241â1257.
7265:Neurotherapeutics
7172:(16): 1378â1384.
7111:10.1002/jmri.1076
6803:mriquestions.com/
6729:10.1002/mrm.20508
6646:978-0-12-025512-2
6570:10.1063/1.1695690
6409:10.1002/oby.23201
6297:(20): 5388â5399.
6028:10.1002/mrm.20642
5902:(15): 1109â1112.
5851:Explanatory notes
5515:
5514:
5507:
5439:
5341:
5323:
5316:
5144:
4976:
4826:
4657:
4339:
4338:
4276:
4114:looks like this:
4098:
3999:{\displaystyle D}
3979:{\displaystyle V}
3652:on its diagonal.
3352:axial diffusivity
2415:
2133:
2098:{\displaystyle A}
2006:
1987:
1869:
1848:
1813:
1781:
1766:
1746:
1732:
1686:
1672:
1647:
1619:
1604:
1587:
1575:
1537:{\displaystyle M}
1497:
1483:
1437:
1423:
1398:
1370:
1355:
1338:
1326:
1237:
1145:
1004:{\displaystyle J}
704:{\displaystyle D}
643:
593:{\displaystyle D}
533:{\displaystyle G}
489:{\displaystyle S}
421:
353:
63:
62:
18:Diffusion imaging
16:(Redirected from
9122:
9077:
9076:
8900:
8899:
8782:
8781:
8772:
8771:
8656:Echocardiography
8515:Hounsfield scale
8368:
8367:
8287:Cholecystography
8212:
8211:
8203:
8202:
8178:
8171:
8164:
8155:
8154:
8132:
8131:
8105:
8081:
8075:
8074:
8042:
8036:
8035:
8017:
7989:
7983:
7982:
7973:(3â4): 353â373.
7962:
7956:
7955:
7937:
7913:
7907:
7906:
7896:
7885:
7879:
7878:
7860:
7851:(6): 1358â1372.
7841:"Q-ball imaging"
7836:
7827:
7826:
7816:
7768:
7762:
7761:
7751:
7711:
7705:
7704:
7694:
7662:
7656:
7655:
7647:
7641:
7640:
7603:
7597:
7596:
7586:
7576:
7552:
7543:
7542:
7532:
7492:
7486:
7485:
7459:
7453:
7452:
7435:(6): 1622â1635.
7424:
7418:
7417:
7407:
7397:
7373:
7367:
7366:
7364:
7363:
7346:
7340:
7339:
7329:
7305:
7299:
7298:
7288:
7256:
7250:
7249:
7221:
7215:
7214:
7206:
7200:
7199:
7189:
7157:
7151:
7150:
7148:
7137:
7131:
7130:
7094:
7088:
7087:
7075:
7069:
7068:
7052:
7046:
7045:
7035:
7011:
7005:
7004:
6986:
6958:
6952:
6951:
6949:
6948:
6931:
6925:
6924:
6914:
6897:(5): 1276â1281.
6882:
6876:
6875:
6847:
6841:
6840:
6838:
6837:
6822:
6813:
6812:
6810:
6809:
6794:
6785:
6784:
6756:
6750:
6749:
6731:
6722:(6): 1432â1440.
6707:
6701:
6700:
6684:
6678:
6677:
6657:
6651:
6650:
6624:
6618:
6617:
6580:
6574:
6573:
6545:
6539:
6538:
6510:
6504:
6503:
6467:
6461:
6460:
6440:
6431:
6430:
6420:
6403:(8): 1328â1337.
6388:
6382:
6381:
6371:
6331:
6325:
6324:
6314:
6281:
6275:
6274:
6264:
6232:
6226:
6225:
6215:
6182:
6176:
6175:
6165:
6133:
6127:
6126:
6098:
6092:
6091:
6066:(4): 1267â1277.
6055:
6049:
6048:
6030:
6021:(6): 1377â1386.
6006:
6000:
5999:
5955:
5949:
5948:
5920:
5914:
5913:
5908:
5891:
5872:
5861:
5724:, as well as in
5568:
5553:
5538:
5510:
5503:
5499:
5496:
5490:
5467:
5459:
5450:
5448:
5447:
5442:
5440:
5438:
5437:
5436:
5414:
5413:
5412:
5403:
5402:
5393:
5392:
5382:
5377:
5352:
5350:
5349:
5344:
5342:
5340:
5324:
5319:
5315:
5314:
5287:
5286:
5271:
5270:
5243:
5242:
5227:
5226:
5199:
5198:
5186:
5185:
5180:
5155:
5153:
5152:
5147:
5145:
5143:
5142:
5141:
5129:
5128:
5116:
5115:
5105:
5104:
5103:
5088:
5087:
5075:
5074:
5064:
5059:
5058:
5040:
5039:
5027:
5026:
5014:
5013:
4987:
4985:
4984:
4979:
4977:
4975:
4974:
4973:
4961:
4960:
4948:
4947:
4937:
4936:
4935:
4922:
4917:
4916:
4897:
4895:
4894:
4889:
4887:
4886:
4874:
4873:
4861:
4860:
4837:
4835:
4834:
4829:
4827:
4825:
4824:
4823:
4811:
4810:
4798:
4797:
4787:
4783:
4782:
4770:
4769:
4753:
4748:
4747:
4728:
4726:
4725:
4720:
4718:
4717:
4705:
4704:
4692:
4691:
4668:
4666:
4665:
4660:
4658:
4656:
4655:
4654:
4642:
4641:
4629:
4628:
4618:
4617:
4616:
4604:
4603:
4593:
4588:
4587:
4568:
4566:
4565:
4560:
4558:
4557:
4545:
4544:
4532:
4531:
4508:
4506:
4505:
4500:
4492:
4491:
4479:
4478:
4466:
4465:
4442:
4440:
4439:
4434:
4425:
4417:
4416:
4404:
4403:
4391:
4390:
4350:
4348:
4347:
4342:
4340:
4333:
4328:
4315:
4310:
4297:
4292:
4277:
4272:
4271:
4244:
4243:
4228:
4227:
4200:
4199:
4184:
4183:
4156:
4155:
4137:
4136:
4131:
4109:
4107:
4106:
4101:
4099:
4094:
4092:
4009:prolate spheroid
4005:
4003:
4002:
3997:
3985:
3983:
3982:
3977:
3962:
3960:
3959:
3954:
3952:
3948:
3947:
3926:
3919:
3905:
3891:
3884:
3876:
3875:
3854:
3840:
3833:
3819:
3818:
3803:
3785:
3771:
3739:
3737:
3736:
3731:
3726:
3718:
3717:
3705:
3704:
3692:
3691:
3676:
3657:mean diffusivity
3651:
3649:
3648:
3643:
3641:
3640:
3624:
3622:
3621:
3616:
3614:
3613:
3597:
3595:
3594:
3589:
3587:
3586:
3570:
3568:
3567:
3562:
3547:
3545:
3544:
3539:
3537:
3536:
3524:
3523:
3511:
3510:
3489:
3464:
3462:
3461:
3456:
3454:
3453:
3433:
3431:
3430:
3425:
3420:
3412:
3411:
3399:
3398:
3383:
3382:
3285:
3283:
3282:
3277:
3275:
3274:
3253:
3245:
3244:
3226:
3225:
3207:
3206:
3026:
3024:
3023:
3018:
3016:
3015:
2996:
2994:
2993:
2988:
2986:
2985:
2966:
2964:
2963:
2958:
2956:
2955:
2884:
2882:
2881:
2876:
2832:
2831:
2816:
2815:
2800:
2799:
2753:
2751:
2750:
2745:
2737:
2736:
2721:
2720:
2705:
2704:
2598:Physical tensors
2589:
2587:
2586:
2581:
2579:
2578:
2571:
2570:
2554:
2553:
2539:
2538:
2522:
2521:
2507:
2506:
2490:
2489:
2473:
2472:
2458:
2457:
2443:
2442:
2417:
2416:
2408:
2391:symmetric matrix
2300:
2298:
2297:
2292:
2290:
2289:
2273:
2271:
2270:
2265:
2263:
2262:
2246:
2244:
2243:
2238:
2236:
2235:
2219:
2217:
2216:
2211:
2209:
2208:
2186:
2184:
2183:
2178:
2176:
2175:
2174:
2173:
2161:
2160:
2147:
2104:
2102:
2101:
2096:
2078:
2076:
2075:
2070:
2068:
2067:
2066:
2065:
2042:
2041:
2037:
2036:
2027:
2026:
2017:
2016:
2007:
1999:
1989:
1988:
1985:
1983:
1974:
1956:
1954:
1953:
1948:
1943:
1942:
1871:
1870:
1862:
1850:
1849:
1841:
1825:
1823:
1822:
1817:
1815:
1814:
1806:
1793:
1791:
1790:
1785:
1783:
1782:
1774:
1768:
1767:
1759:
1747:
1745:
1744:
1735:
1734:
1733:
1725:
1719:
1718:
1706:
1705:
1692:
1687:
1685:
1684:
1675:
1674:
1673:
1665:
1662:
1661:
1649:
1648:
1640:
1637:
1636:
1626:
1621:
1620:
1612:
1606:
1605:
1597:
1588:
1586:
1578:
1577:
1576:
1568:
1561:
1543:
1541:
1540:
1535:
1508:
1506:
1505:
1500:
1498:
1496:
1495:
1486:
1485:
1484:
1476:
1470:
1469:
1457:
1456:
1443:
1438:
1436:
1435:
1426:
1425:
1424:
1416:
1413:
1412:
1400:
1399:
1391:
1388:
1387:
1377:
1372:
1371:
1363:
1357:
1356:
1348:
1339:
1337:
1329:
1328:
1327:
1319:
1312:
1285:
1283:
1282:
1277:
1254:
1253:
1238:
1236:
1228:
1205:
1186:
1184:
1183:
1178:
1146:
1144:
1136:
1113:
1086:
1084:
1083:
1078:
1013:Fick's first law
1010:
1008:
1007:
1002:
990:
988:
987:
982:
958:
956:
955:
950:
945:
944:
932:
931:
919:
890:
889:
880:
854:
853:
811:
790:
788:
787:
782:
780:
779:
763:
761:
760:
755:
753:
752:
736:
734:
733:
728:
710:
708:
707:
702:
687:
685:
684:
679:
644:
642:
641:
632:
615:
599:
597:
596:
591:
579:
577:
576:
571:
559:
557:
556:
551:
539:
537:
536:
531:
515:
513:
512:
507:
495:
493:
492:
487:
475:
473:
472:
467:
465:
464:
445:
443:
442:
437:
435:
431:
427:
423:
422:
414:
401:
400:
391:
390:
381:
380:
354:
352:
351:
342:
325:
309:
307:
306:
301:
299:
298:
282:
280:
279:
274:
272:
271:
59:
42:
30:
29:
21:
9130:
9129:
9125:
9124:
9123:
9121:
9120:
9119:
9090:
9089:
9088:
9083:
9065:
9044:
9016:
8964:
8950:PET mammography
8921:
8884:
8870:Gallium-67 scan
8865:Octreotide scan
8853:
8761:
8612:
8524:
8484:
8407:
8388:High-resolution
8353:
8317:Skeletal survey
8283:Cholangiography
8196:
8187:
8185:Medical imaging
8182:
8140:
8135:
8082:
8078:
8045:Funk P (1919).
8043:
8039:
8015:10.1.1.104.9041
7990:
7986:
7963:
7959:
7914:
7910:
7894:
7886:
7882:
7837:
7830:
7769:
7765:
7712:
7708:
7663:
7659:
7650:Soll E (2015).
7648:
7644:
7604:
7600:
7553:
7546:
7493:
7489:
7482:
7460:
7456:
7425:
7421:
7374:
7370:
7361:
7359:
7347:
7343:
7306:
7302:
7257:
7253:
7222:
7218:
7207:
7203:
7158:
7154:
7146:
7138:
7134:
7095:
7091:
7076:
7072:
7053:
7049:
7012:
7008:
6959:
6955:
6946:
6944:
6932:
6928:
6883:
6879:
6848:
6844:
6835:
6833:
6823:
6816:
6807:
6805:
6795:
6788:
6761:Physical Review
6757:
6753:
6708:
6704:
6685:
6681:
6668:(1â2): 145â54.
6658:
6654:
6647:
6625:
6621:
6581:
6577:
6546:
6542:
6511:
6507:
6468:
6464:
6441:
6434:
6389:
6385:
6332:
6328:
6282:
6278:
6233:
6229:
6183:
6179:
6134:
6130:
6099:
6095:
6056:
6052:
6007:
6003:
5956:
5952:
5921:
5917:
5906:
5892:
5885:
5881:
5876:
5875:
5862:
5858:
5853:
5831:
5811:
5769:
5737:Brachial plexus
5719:neural networks
5575:
5569:
5560:
5554:
5545:
5539:
5511:
5500:
5494:
5491:
5480:
5468:
5457:
5432:
5428:
5415:
5408:
5404:
5398:
5394:
5388:
5384:
5383:
5381:
5370:
5368:
5365:
5364:
5318:
5317:
5310:
5306:
5282:
5278:
5266:
5262:
5238:
5234:
5222:
5218:
5194:
5190:
5184:
5173:
5171:
5168:
5167:
5137:
5133:
5124:
5120:
5111:
5107:
5106:
5099:
5095:
5083:
5079:
5070:
5066:
5065:
5063:
5054:
5050:
5035:
5031:
5022:
5018:
5009:
5005:
5003:
5000:
4999:
4969:
4965:
4956:
4952:
4943:
4939:
4938:
4931:
4927:
4923:
4921:
4912:
4908:
4906:
4903:
4902:
4882:
4878:
4869:
4865:
4856:
4852:
4850:
4847:
4846:
4819:
4815:
4806:
4802:
4793:
4789:
4788:
4778:
4774:
4765:
4761:
4754:
4752:
4743:
4739:
4737:
4734:
4733:
4713:
4709:
4700:
4696:
4687:
4683:
4681:
4678:
4677:
4650:
4646:
4637:
4633:
4624:
4620:
4619:
4612:
4608:
4599:
4595:
4594:
4592:
4583:
4579:
4577:
4574:
4573:
4553:
4549:
4540:
4536:
4527:
4523:
4521:
4518:
4517:
4487:
4483:
4474:
4470:
4461:
4457:
4455:
4452:
4451:
4448:oblate spheroid
4421:
4412:
4408:
4399:
4395:
4386:
4382:
4359:
4356:
4355:
4329:
4324:
4311:
4306:
4293:
4288:
4267:
4263:
4239:
4235:
4223:
4219:
4195:
4191:
4179:
4175:
4151:
4147:
4135:
4124:
4122:
4119:
4118:
4093:
4088:
4083:
4080:
4079:
4066:
4059:
4053:) + (
4052:
4045:
4039:) + (
4038:
4031:
4024:
4017:
3991:
3988:
3987:
3971:
3968:
3967:
3950:
3949:
3943:
3939:
3924:
3923:
3915:
3898:
3889:
3888:
3880:
3868:
3864:
3847:
3838:
3837:
3829:
3811:
3807:
3796:
3789:
3781:
3764:
3760:
3758:
3755:
3754:
3748:
3722:
3713:
3709:
3700:
3696:
3687:
3683:
3669:
3667:
3664:
3663:
3636:
3632:
3630:
3627:
3626:
3609:
3605:
3603:
3600:
3599:
3582:
3578:
3576:
3573:
3572:
3556:
3553:
3552:
3532:
3528:
3519:
3515:
3506:
3502:
3482:
3480:
3477:
3476:
3449:
3445:
3443:
3440:
3439:
3416:
3407:
3403:
3394:
3390:
3378:
3374:
3372:
3369:
3368:
3360:
3349:
3335:
3328:
3321:
3270:
3266:
3249:
3240:
3236:
3221:
3217:
3202:
3198:
3187:
3184:
3183:
3178:
3172:
3166:
3147:Brownian motion
3096:
3060:Diagonalization
3011:
3007:
3002:
2999:
2998:
2981:
2977:
2972:
2969:
2968:
2951:
2947:
2942:
2939:
2938:
2827:
2823:
2811:
2807:
2795:
2791:
2786:
2783:
2782:
2732:
2728:
2716:
2712:
2700:
2696:
2691:
2688:
2687:
2684:
2600:
2573:
2572:
2561:
2557:
2555:
2546:
2542:
2540:
2531:
2527:
2524:
2523:
2514:
2510:
2508:
2497:
2493:
2491:
2482:
2478:
2475:
2474:
2465:
2461:
2459:
2450:
2446:
2444:
2433:
2429:
2422:
2421:
2407:
2406:
2404:
2401:
2400:
2383:
2377:
2353:
2317:
2307:
2285:
2281:
2279:
2276:
2275:
2258:
2254:
2252:
2249:
2248:
2231:
2227:
2225:
2222:
2221:
2201:
2197:
2195:
2192:
2191:
2166:
2162:
2153:
2149:
2137:
2129:
2125:
2117:
2114:
2113:
2090:
2087:
2086:
2061:
2057:
2050:
2046:
2032:
2028:
2022:
2018:
2012:
2008:
1998:
1994:
1990:
1984:
1979:
1978:
1970:
1968:
1965:
1964:
1937:
1936:
1931:
1926:
1920:
1919:
1914:
1909:
1903:
1902:
1897:
1892:
1882:
1881:
1861:
1860:
1840:
1839:
1837:
1834:
1833:
1805:
1804:
1802:
1799:
1798:
1773:
1772:
1758:
1757:
1740:
1736:
1724:
1723:
1714:
1710:
1701:
1697:
1693:
1691:
1680:
1676:
1664:
1663:
1657:
1653:
1639:
1638:
1632:
1628:
1627:
1625:
1611:
1610:
1596:
1595:
1579:
1567:
1566:
1562:
1560:
1558:
1555:
1554:
1529:
1526:
1525:
1521:Bloch equations
1491:
1487:
1475:
1474:
1465:
1461:
1452:
1448:
1444:
1442:
1431:
1427:
1415:
1414:
1408:
1404:
1390:
1389:
1383:
1379:
1378:
1376:
1362:
1361:
1347:
1346:
1330:
1318:
1317:
1313:
1311:
1309:
1306:
1305:
1292:
1249:
1245:
1229:
1206:
1204:
1202:
1199:
1198:
1137:
1114:
1112:
1110:
1107:
1106:
1090:where D is the
1027:
1024:
1023:
996:
993:
992:
976:
973:
972:
969:
967:Diffusion model
940:
936:
927:
923:
915:
885:
881:
876:
849:
845:
801:
799:
796:
795:
775:
771:
769:
766:
765:
748:
744:
742:
739:
738:
716:
713:
712:
696:
693:
692:
637:
633:
616:
614:
612:
609:
608:
585:
582:
581:
565:
562:
561:
545:
542:
541:
525:
522:
521:
501:
498:
497:
481:
478:
477:
460:
456:
454:
451:
450:
413:
406:
402:
396:
392:
386:
382:
376:
372:
368:
364:
347:
343:
326:
324:
322:
319:
318:
294:
290:
288:
285:
284:
267:
263:
261:
258:
257:
245:
172:cerebral cortex
125:
55:
45:
28:
23:
22:
15:
12:
11:
5:
9128:
9118:
9117:
9112:
9107:
9102:
9085:
9084:
9082:
9081:
9070:
9067:
9066:
9064:
9063:
9058:
9052:
9050:
9046:
9045:
9043:
9042:
9037:
9032:
9026:
9024:
9018:
9017:
9015:
9014:
9009:
9007:Endomicroscopy
9004:
8999:
8998:
8997:
8986:
8984:
8974:
8973:
8970:
8969:
8966:
8965:
8963:
8962:
8957:
8952:
8947:
8942:
8936:
8934:
8923:
8922:
8920:
8919:
8913:
8911:
8897:
8890:
8889:
8886:
8885:
8883:
8882:
8877:
8872:
8867:
8861:
8859:
8855:
8854:
8852:
8851:
8846:
8841:
8836:
8831:
8826:
8821:
8816:
8811:
8806:
8801:
8796:
8791:
8785:
8779:
8769:
8763:
8762:
8760:
8759:
8758:
8757:
8752:
8742:
8737:
8732:
8727:
8722:
8721:
8720:
8715:
8705:
8700:
8695:
8690:
8685:
8680:
8679:
8678:
8673:
8668:
8663:
8653:
8652:
8651:
8646:
8641:
8636:
8631:
8622:
8620:
8614:
8613:
8611:
8610:
8605:
8604:
8603:
8598:
8593:
8583:
8578:
8573:
8568:
8567:
8566:
8556:
8551:
8550:
8549:
8538:
8536:
8530:
8529:
8526:
8525:
8523:
8522:
8517:
8512:
8507:
8502:
8496:
8494:
8490:
8489:
8486:
8485:
8483:
8482:
8481:
8480:
8470:
8465:
8460:
8459:
8458:
8453:
8443:
8442:
8441:
8431:
8430:
8429:
8424:
8415:
8413:
8409:
8408:
8406:
8405:
8400:
8395:
8390:
8385:
8380:
8374:
8372:
8365:
8359:
8358:
8355:
8354:
8352:
8351:
8346:
8341:
8336:
8335:
8334:
8329:
8319:
8314:
8309:
8304:
8299:
8294:
8289:
8280:
8267:
8258:
8253:
8248:
8247:
8246:
8236:
8231:
8226:
8221:
8215:
8209:
8200:
8189:
8188:
8181:
8180:
8173:
8166:
8158:
8152:
8151:
8146:
8139:
8138:External links
8136:
8134:
8133:
8096:(2): 773â783.
8076:
8037:
8000:(2): 220â236.
7984:
7957:
7928:(4): 577â582.
7908:
7880:
7828:
7763:
7706:
7677:(6): 419â431.
7657:
7642:
7615:(6): 951â962.
7598:
7544:
7487:
7480:
7454:
7419:
7368:
7341:
7300:
7271:(3): 316â329.
7251:
7216:
7201:
7152:
7132:
7105:(4): 534â546.
7089:
7070:
7047:
7026:(4): 625â632.
7006:
6969:(6): 412â419.
6953:
6926:
6877:
6842:
6814:
6786:
6767:(3): 563â565.
6751:
6702:
6679:
6652:
6645:
6619:
6592:(6): 847â857.
6575:
6556:(1): 288â292.
6540:
6521:(3): 719â725.
6505:
6478:(6): 707â713.
6462:
6451:(4): 825â829.
6432:
6383:
6326:
6276:
6247:(5): 695â706.
6227:
6198:(3): 418â428.
6177:
6128:
6109:(2): 401â407.
6093:
6050:
6001:
5966:(4): 345â349.
5950:
5931:(3): 479â486.
5915:
5882:
5880:
5877:
5874:
5873:
5869:component free
5855:
5854:
5852:
5849:
5848:
5847:
5842:
5837:
5830:
5827:
5815:linear algebra
5810:
5807:
5768:
5765:
5764:
5763:
5754:
5748:
5730:
5729:
5706:
5683:
5636:magnetic field
5577:
5576:
5573:MELAS syndrome
5570:
5563:
5561:
5555:
5548:
5546:
5540:
5533:
5513:
5512:
5471:
5469:
5462:
5456:
5453:
5452:
5451:
5435:
5431:
5427:
5424:
5421:
5418:
5411:
5407:
5401:
5397:
5391:
5387:
5380:
5376:
5373:
5354:
5353:
5339:
5336:
5333:
5330:
5327:
5322:
5313:
5309:
5305:
5302:
5299:
5296:
5293:
5290:
5285:
5281:
5277:
5274:
5269:
5265:
5261:
5258:
5255:
5252:
5249:
5246:
5241:
5237:
5233:
5230:
5225:
5221:
5217:
5214:
5211:
5208:
5205:
5202:
5197:
5193:
5189:
5183:
5179:
5176:
5157:
5156:
5140:
5136:
5132:
5127:
5123:
5119:
5114:
5110:
5102:
5098:
5094:
5091:
5086:
5082:
5078:
5073:
5069:
5062:
5057:
5053:
5049:
5046:
5043:
5038:
5034:
5030:
5025:
5021:
5017:
5012:
5008:
4989:
4988:
4972:
4968:
4964:
4959:
4955:
4951:
4946:
4942:
4934:
4930:
4926:
4920:
4915:
4911:
4885:
4881:
4877:
4872:
4868:
4864:
4859:
4855:
4843:spherical case
4839:
4838:
4822:
4818:
4814:
4809:
4805:
4801:
4796:
4792:
4786:
4781:
4777:
4773:
4768:
4764:
4760:
4757:
4751:
4746:
4742:
4716:
4712:
4708:
4703:
4699:
4695:
4690:
4686:
4670:
4669:
4653:
4649:
4645:
4640:
4636:
4632:
4627:
4623:
4615:
4611:
4607:
4602:
4598:
4591:
4586:
4582:
4556:
4552:
4548:
4543:
4539:
4535:
4530:
4526:
4498:
4495:
4490:
4486:
4482:
4477:
4473:
4469:
4464:
4460:
4432:
4428:
4424:
4420:
4415:
4411:
4407:
4402:
4398:
4394:
4389:
4385:
4381:
4378:
4375:
4372:
4369:
4366:
4363:
4352:
4351:
4337:
4332:
4327:
4323:
4319:
4314:
4309:
4305:
4301:
4296:
4291:
4287:
4283:
4280:
4275:
4270:
4266:
4262:
4259:
4256:
4253:
4250:
4247:
4242:
4238:
4234:
4231:
4226:
4222:
4218:
4215:
4212:
4209:
4206:
4203:
4198:
4194:
4190:
4187:
4182:
4178:
4174:
4171:
4168:
4165:
4162:
4159:
4154:
4150:
4146:
4143:
4140:
4134:
4130:
4127:
4097:
4091:
4087:
4064:
4057:
4050:
4043:
4036:
4029:
4022:
4015:
3995:
3975:
3964:
3963:
3946:
3942:
3938:
3935:
3932:
3929:
3927:
3925:
3922:
3918:
3914:
3911:
3908:
3904:
3901:
3897:
3894:
3892:
3890:
3887:
3883:
3879:
3874:
3871:
3867:
3863:
3860:
3857:
3853:
3850:
3846:
3843:
3841:
3839:
3836:
3832:
3828:
3825:
3822:
3817:
3814:
3810:
3806:
3802:
3799:
3795:
3792:
3790:
3788:
3784:
3780:
3777:
3774:
3770:
3767:
3763:
3762:
3746:
3741:
3740:
3729:
3725:
3721:
3716:
3712:
3708:
3703:
3699:
3695:
3690:
3686:
3682:
3679:
3675:
3672:
3639:
3635:
3612:
3608:
3585:
3581:
3560:
3549:
3548:
3535:
3531:
3527:
3522:
3518:
3514:
3509:
3505:
3501:
3498:
3495:
3492:
3488:
3485:
3452:
3448:
3435:
3434:
3423:
3419:
3415:
3410:
3406:
3402:
3397:
3393:
3389:
3386:
3381:
3377:
3358:
3347:
3333:
3326:
3319:
3287:
3286:
3273:
3269:
3265:
3262:
3259:
3256:
3252:
3248:
3243:
3239:
3235:
3232:
3229:
3224:
3220:
3216:
3213:
3210:
3205:
3201:
3197:
3194:
3191:
3174:
3168:
3162:
3135:cell membranes
3095:
3092:
3014:
3010:
3006:
2984:
2980:
2976:
2954:
2950:
2946:
2874:
2871:
2868:
2865:
2862:
2859:
2856:
2853:
2850:
2847:
2844:
2841:
2838:
2835:
2830:
2826:
2822:
2819:
2814:
2810:
2806:
2803:
2798:
2794:
2790:
2743:
2740:
2735:
2731:
2727:
2724:
2719:
2715:
2711:
2708:
2703:
2699:
2695:
2683:
2680:
2674:ellipsoid. As
2599:
2596:
2591:
2590:
2577:
2568:
2565:
2560:
2556:
2552:
2549:
2545:
2541:
2537:
2534:
2530:
2526:
2525:
2520:
2517:
2513:
2509:
2504:
2501:
2496:
2492:
2488:
2485:
2481:
2477:
2476:
2471:
2468:
2464:
2460:
2456:
2453:
2449:
2445:
2440:
2437:
2432:
2428:
2427:
2425:
2420:
2414:
2411:
2376:
2373:
2352:
2349:
2316:
2313:
2306:
2303:
2288:
2284:
2261:
2257:
2234:
2230:
2207:
2204:
2200:
2188:
2187:
2172:
2169:
2165:
2159:
2156:
2152:
2146:
2143:
2140:
2136:
2132:
2128:
2124:
2121:
2094:
2080:
2079:
2064:
2060:
2056:
2053:
2049:
2045:
2040:
2035:
2031:
2025:
2021:
2015:
2011:
2005:
2002:
1997:
1993:
1982:
1977:
1973:
1958:
1957:
1946:
1941:
1935:
1932:
1930:
1927:
1925:
1922:
1921:
1918:
1915:
1913:
1910:
1908:
1905:
1904:
1901:
1898:
1896:
1893:
1891:
1888:
1887:
1885:
1880:
1877:
1874:
1868:
1865:
1859:
1856:
1853:
1847:
1844:
1812:
1809:
1795:
1794:
1780:
1777:
1771:
1765:
1762:
1756:
1753:
1750:
1743:
1739:
1731:
1728:
1722:
1717:
1713:
1709:
1704:
1700:
1696:
1690:
1683:
1679:
1671:
1668:
1660:
1656:
1652:
1646:
1643:
1635:
1631:
1624:
1618:
1615:
1609:
1603:
1600:
1594:
1591:
1585:
1582:
1574:
1571:
1565:
1533:
1510:
1509:
1494:
1490:
1482:
1479:
1473:
1468:
1464:
1460:
1455:
1451:
1447:
1441:
1434:
1430:
1422:
1419:
1411:
1407:
1403:
1397:
1394:
1386:
1382:
1375:
1369:
1366:
1360:
1354:
1351:
1345:
1342:
1336:
1333:
1325:
1322:
1316:
1300:Bloch equation
1291:
1288:
1287:
1286:
1275:
1272:
1269:
1266:
1263:
1260:
1257:
1252:
1248:
1244:
1241:
1235:
1232:
1227:
1224:
1221:
1218:
1215:
1212:
1209:
1188:
1187:
1176:
1173:
1170:
1167:
1164:
1161:
1158:
1155:
1152:
1149:
1143:
1140:
1135:
1132:
1129:
1126:
1123:
1120:
1117:
1088:
1087:
1076:
1073:
1070:
1067:
1064:
1061:
1058:
1055:
1052:
1049:
1046:
1043:
1040:
1037:
1034:
1031:
1000:
980:
968:
965:
960:
959:
948:
943:
939:
935:
930:
926:
922:
918:
914:
911:
908:
905:
902:
899:
896:
893:
888:
884:
879:
875:
872:
869:
866:
863:
860:
857:
852:
848:
844:
841:
838:
835:
832:
829:
826:
823:
820:
817:
814:
810:
807:
804:
778:
774:
751:
747:
726:
723:
720:
700:
689:
688:
677:
674:
671:
668:
665:
662:
659:
656:
653:
650:
647:
640:
636:
631:
628:
625:
622:
619:
589:
569:
549:
529:
505:
485:
463:
459:
447:
446:
434:
430:
426:
420:
417:
412:
409:
405:
399:
395:
389:
385:
379:
375:
371:
367:
363:
360:
357:
350:
346:
341:
338:
335:
332:
329:
297:
293:
270:
266:
244:
241:
124:
121:
119:in the brain.
100:, fibers, and
98:macromolecules
61:
60:
53:
47:
46:
43:
35:
34:
26:
9:
6:
4:
3:
2:
9127:
9116:
9113:
9111:
9108:
9106:
9103:
9101:
9098:
9097:
9095:
9080:
9072:
9071:
9068:
9062:
9059:
9057:
9054:
9053:
9051:
9047:
9041:
9038:
9036:
9033:
9031:
9028:
9027:
9025:
9023:
9019:
9013:
9010:
9008:
9005:
9003:
9000:
8996:
8993:
8992:
8991:
8988:
8987:
8985:
8983:
8979:
8975:
8961:
8958:
8956:
8953:
8951:
8948:
8946:
8943:
8941:
8938:
8937:
8935:
8932:
8928:
8924:
8918:
8915:
8914:
8912:
8909:
8905:
8901:
8898:
8896:
8891:
8881:
8878:
8876:
8875:Ga-68-DOTATOC
8873:
8871:
8868:
8866:
8863:
8862:
8860:
8856:
8850:
8847:
8845:
8842:
8840:
8837:
8835:
8832:
8830:
8827:
8825:
8822:
8820:
8817:
8815:
8812:
8810:
8807:
8805:
8802:
8800:
8797:
8795:
8792:
8790:
8787:
8786:
8783:
8780:
8778:
8773:
8770:
8768:
8764:
8756:
8753:
8751:
8748:
8747:
8746:
8743:
8741:
8738:
8736:
8733:
8731:
8728:
8726:
8723:
8719:
8716:
8714:
8711:
8710:
8709:
8706:
8704:
8701:
8699:
8696:
8694:
8691:
8689:
8688:Intravascular
8686:
8684:
8681:
8677:
8674:
8672:
8669:
8667:
8664:
8662:
8659:
8658:
8657:
8654:
8650:
8647:
8645:
8642:
8640:
8637:
8635:
8632:
8630:
8627:
8626:
8624:
8623:
8621:
8619:
8615:
8609:
8608:Synthetic MRI
8606:
8602:
8599:
8597:
8594:
8592:
8589:
8588:
8587:
8584:
8582:
8579:
8577:
8574:
8572:
8569:
8565:
8562:
8561:
8560:
8557:
8555:
8552:
8548:
8545:
8544:
8543:
8540:
8539:
8537:
8535:
8531:
8521:
8518:
8516:
8513:
8511:
8508:
8506:
8503:
8501:
8498:
8497:
8495:
8491:
8479:
8476:
8475:
8474:
8471:
8469:
8466:
8464:
8461:
8457:
8454:
8452:
8449:
8448:
8447:
8444:
8440:
8437:
8436:
8435:
8432:
8428:
8425:
8423:
8420:
8419:
8417:
8416:
8414:
8410:
8404:
8401:
8399:
8398:Electron beam
8396:
8394:
8391:
8389:
8386:
8384:
8381:
8379:
8376:
8375:
8373:
8369:
8366:
8364:
8360:
8350:
8349:Orbital x-ray
8347:
8345:
8342:
8340:
8337:
8333:
8330:
8328:
8325:
8324:
8323:
8320:
8318:
8315:
8313:
8310:
8308:
8305:
8303:
8300:
8298:
8295:
8293:
8290:
8288:
8284:
8281:
8279:
8275:
8271:
8268:
8266:
8262:
8259:
8257:
8254:
8252:
8249:
8245:
8244:Bronchography
8242:
8241:
8240:
8237:
8235:
8232:
8230:
8227:
8225:
8222:
8220:
8217:
8216:
8213:
8210:
8208:
8204:
8201:
8199:
8194:
8190:
8186:
8179:
8174:
8172:
8167:
8165:
8160:
8159:
8156:
8150:
8147:
8145:
8142:
8141:
8129:
8125:
8121:
8117:
8113:
8109:
8104:
8099:
8095:
8091:
8087:
8080:
8072:
8068:
8064:
8060:
8056:
8052:
8048:
8041:
8033:
8029:
8025:
8021:
8016:
8011:
8007:
8003:
7999:
7995:
7988:
7980:
7976:
7972:
7968:
7961:
7953:
7949:
7945:
7941:
7936:
7931:
7927:
7923:
7919:
7912:
7904:
7900:
7893:
7892:
7884:
7876:
7872:
7868:
7864:
7859:
7854:
7850:
7846:
7842:
7835:
7833:
7824:
7820:
7815:
7810:
7806:
7802:
7798:
7794:
7790:
7786:
7782:
7778:
7774:
7767:
7759:
7755:
7750:
7745:
7741:
7737:
7733:
7729:
7725:
7721:
7717:
7710:
7702:
7698:
7693:
7688:
7684:
7680:
7676:
7672:
7668:
7661:
7653:
7646:
7638:
7634:
7630:
7626:
7622:
7618:
7614:
7610:
7602:
7594:
7590:
7585:
7580:
7575:
7570:
7566:
7562:
7558:
7551:
7549:
7540:
7536:
7531:
7526:
7522:
7518:
7514:
7510:
7506:
7502:
7498:
7491:
7483:
7477:
7473:
7469:
7465:
7458:
7450:
7446:
7442:
7438:
7434:
7430:
7423:
7415:
7411:
7406:
7401:
7396:
7391:
7387:
7383:
7379:
7372:
7358:
7357:
7352:
7345:
7337:
7333:
7328:
7323:
7319:
7315:
7311:
7304:
7296:
7292:
7287:
7282:
7278:
7274:
7270:
7266:
7262:
7255:
7247:
7243:
7239:
7235:
7232:(2): 93â108.
7231:
7227:
7220:
7212:
7205:
7197:
7193:
7188:
7183:
7179:
7175:
7171:
7167:
7163:
7156:
7145:
7144:
7136:
7128:
7124:
7120:
7116:
7112:
7108:
7104:
7100:
7093:
7085:
7082:(in French).
7081:
7074:
7066:
7062:
7058:
7051:
7043:
7039:
7034:
7029:
7025:
7021:
7017:
7010:
7002:
6998:
6994:
6990:
6985:
6980:
6976:
6972:
6968:
6964:
6957:
6943:
6942:
6937:
6930:
6922:
6918:
6913:
6908:
6904:
6900:
6896:
6892:
6888:
6881:
6873:
6869:
6865:
6861:
6858:(2): 318â22.
6857:
6853:
6846:
6832:
6828:
6821:
6819:
6804:
6800:
6793:
6791:
6782:
6778:
6774:
6770:
6766:
6762:
6755:
6747:
6743:
6739:
6735:
6730:
6725:
6721:
6717:
6713:
6706:
6698:
6694:
6690:
6683:
6675:
6671:
6667:
6663:
6656:
6648:
6642:
6638:
6634:
6630:
6623:
6615:
6611:
6607:
6603:
6599:
6595:
6591:
6587:
6579:
6571:
6567:
6563:
6559:
6555:
6551:
6544:
6536:
6532:
6528:
6524:
6520:
6516:
6509:
6501:
6497:
6493:
6489:
6485:
6481:
6477:
6473:
6466:
6458:
6454:
6450:
6446:
6439:
6437:
6428:
6424:
6419:
6414:
6410:
6406:
6402:
6398:
6394:
6387:
6379:
6375:
6370:
6365:
6361:
6357:
6353:
6349:
6345:
6341:
6337:
6330:
6322:
6318:
6313:
6308:
6304:
6300:
6296:
6292:
6288:
6280:
6272:
6268:
6263:
6258:
6254:
6250:
6246:
6242:
6238:
6231:
6223:
6219:
6214:
6209:
6205:
6201:
6197:
6193:
6189:
6181:
6173:
6169:
6164:
6159:
6155:
6151:
6147:
6143:
6139:
6132:
6124:
6120:
6116:
6112:
6108:
6104:
6097:
6089:
6085:
6081:
6077:
6073:
6069:
6065:
6061:
6054:
6046:
6042:
6038:
6034:
6029:
6024:
6020:
6016:
6012:
6005:
5997:
5993:
5989:
5985:
5981:
5977:
5973:
5969:
5965:
5961:
5954:
5946:
5942:
5938:
5934:
5930:
5926:
5919:
5911:
5905:
5901:
5898:(in French).
5897:
5890:
5888:
5883:
5870:
5866:
5860:
5856:
5846:
5843:
5841:
5838:
5836:
5833:
5832:
5826:
5823:
5818:
5816:
5806:
5802:
5800:
5796:
5790:
5787:
5786:tessellations
5783:
5778:
5773:
5762:
5758:
5755:
5752:
5749:
5746:
5742:
5738:
5735:
5734:
5733:
5727:
5723:
5720:
5716:
5712:
5707:
5704:
5700:
5696:
5692:
5688:
5684:
5681:
5677:
5673:
5669:
5665:
5661:
5660:
5659:
5656:
5654:
5650:
5646:
5642:
5637:
5628:
5620:
5616:
5614:
5610:
5606:
5602:
5598:
5594:
5589:
5586:
5582:
5574:
5567:
5562:
5559:
5552:
5547:
5544:
5537:
5532:
5531:
5530:
5528:
5524:
5523:perfusion MRI
5520:
5509:
5506:
5498:
5495:December 2013
5488:
5484:
5478:
5477:
5472:This section
5470:
5466:
5461:
5460:
5433:
5425:
5419:
5409:
5405:
5399:
5395:
5389:
5385:
5378:
5363:
5362:
5361:
5359:
5334:
5328:
5320:
5311:
5300:
5294:
5288:
5283:
5279:
5272:
5267:
5256:
5250:
5244:
5239:
5235:
5228:
5223:
5212:
5206:
5200:
5195:
5191:
5181:
5166:
5165:
5164:
5162:
5138:
5134:
5130:
5125:
5121:
5117:
5112:
5108:
5100:
5096:
5092:
5089:
5084:
5080:
5076:
5071:
5067:
5060:
5055:
5051:
5047:
5044:
5041:
5036:
5032:
5028:
5023:
5019:
5015:
5010:
5006:
4998:
4997:
4996:
4994:
4970:
4966:
4962:
4957:
4953:
4949:
4944:
4940:
4932:
4928:
4924:
4918:
4913:
4909:
4901:
4900:
4899:
4883:
4879:
4875:
4870:
4866:
4862:
4857:
4853:
4844:
4820:
4816:
4812:
4807:
4803:
4799:
4794:
4790:
4779:
4775:
4771:
4766:
4762:
4755:
4749:
4744:
4740:
4732:
4731:
4730:
4714:
4710:
4706:
4701:
4697:
4693:
4688:
4684:
4675:
4651:
4647:
4643:
4638:
4634:
4630:
4625:
4621:
4613:
4609:
4605:
4600:
4596:
4589:
4584:
4580:
4572:
4571:
4570:
4554:
4550:
4546:
4541:
4537:
4533:
4528:
4524:
4515:
4510:
4496:
4493:
4488:
4484:
4480:
4475:
4471:
4467:
4462:
4458:
4449:
4443:
4430:
4426:
4422:
4413:
4409:
4405:
4400:
4396:
4392:
4387:
4383:
4376:
4370:
4364:
4330:
4325:
4321:
4317:
4312:
4307:
4303:
4299:
4294:
4289:
4285:
4278:
4268:
4257:
4251:
4245:
4240:
4236:
4229:
4224:
4213:
4207:
4201:
4196:
4192:
4185:
4180:
4169:
4163:
4157:
4152:
4148:
4138:
4132:
4117:
4116:
4115:
4113:
4095:
4089:
4085:
4076:
4072:
4071:
4063:
4060: â
4056:
4049:
4046: â
4042:
4035:
4032: â
4028:
4021:
4014:
4010:
3993:
3973:
3944:
3940:
3936:
3933:
3930:
3928:
3920:
3916:
3909:
3895:
3893:
3885:
3881:
3872:
3869:
3865:
3858:
3844:
3842:
3834:
3830:
3820:
3815:
3812:
3808:
3793:
3791:
3786:
3782:
3753:
3752:
3751:
3749:
3743:which equals
3727:
3723:
3714:
3710:
3706:
3701:
3697:
3693:
3688:
3684:
3677:
3662:
3661:
3660:
3658:
3653:
3637:
3633:
3610:
3606:
3583:
3579:
3533:
3529:
3525:
3520:
3516:
3512:
3507:
3503:
3499:
3475:
3474:
3473:
3471:
3466:
3450:
3446:
3421:
3417:
3408:
3404:
3400:
3395:
3391:
3384:
3379:
3375:
3367:
3366:
3365:
3364:
3357:
3353:
3346:
3341:
3339:
3332:
3325:
3318:
3314:
3313:
3308:
3307:
3302:
3301:
3294:
3292:
3271:
3267:
3263:
3260:
3257:
3254:
3250:
3241:
3237:
3233:
3230:
3227:
3222:
3218:
3214:
3211:
3208:
3203:
3199:
3195:
3192:
3182:
3181:
3180:
3177:
3171:
3165:
3160:
3154:
3152:
3148:
3144:
3140:
3136:
3132:
3129:
3125:
3121:
3116:
3113:
3109:
3100:
3091:
3090:information.
3089:
3085:
3081:
3077:
3073:
3069:
3065:
3061:
3057:
3055:
3051:
3047:
3043:
3038:
3034:
3030:
3012:
3008:
3004:
2982:
2978:
2974:
2952:
2948:
2944:
2935:
2931:
2927:
2922:
2920:
2916:
2912:
2908:
2904:
2900:
2896:
2892:
2888:
2872:
2869:
2866:
2863:
2860:
2857:
2854:
2851:
2848:
2845:
2842:
2839:
2836:
2833:
2828:
2824:
2820:
2817:
2812:
2808:
2804:
2801:
2796:
2792:
2788:
2779:
2777:
2773:
2769:
2765:
2761:
2757:
2741:
2738:
2733:
2729:
2725:
2722:
2717:
2713:
2709:
2706:
2701:
2697:
2693:
2679:
2677:
2672:
2667:
2663:
2661:
2657:
2653:
2649:
2645:
2641:
2637:
2633:
2629:
2623:
2621:
2617:
2614:
2610:
2606:
2595:
2575:
2566:
2563:
2558:
2550:
2547:
2543:
2535:
2532:
2528:
2518:
2515:
2511:
2502:
2499:
2494:
2486:
2483:
2479:
2469:
2466:
2462:
2454:
2451:
2447:
2438:
2435:
2430:
2423:
2418:
2409:
2399:
2398:
2397:
2394:
2392:
2388:
2382:
2372:
2370:
2365:
2363:
2357:
2348:
2345:
2341:
2338:
2334:
2330:
2321:
2312:
2310:
2302:
2286:
2282:
2259:
2255:
2232:
2228:
2205:
2202:
2198:
2170:
2167:
2163:
2157:
2154:
2150:
2144:
2141:
2138:
2134:
2130:
2126:
2122:
2119:
2112:
2111:
2110:
2108:
2092:
2085:
2062:
2058:
2054:
2051:
2047:
2043:
2038:
2033:
2029:
2023:
2019:
2013:
2009:
2003:
2000:
1995:
1991:
1980:
1975:
1971:
1963:
1962:
1961:
1944:
1939:
1933:
1928:
1923:
1916:
1911:
1906:
1899:
1894:
1889:
1883:
1878:
1875:
1872:
1863:
1857:
1854:
1851:
1842:
1832:
1831:
1830:
1827:
1807:
1775:
1760:
1754:
1748:
1741:
1737:
1726:
1715:
1711:
1707:
1702:
1698:
1688:
1681:
1677:
1666:
1658:
1654:
1650:
1641:
1633:
1629:
1622:
1613:
1607:
1598:
1592:
1589:
1583:
1580:
1569:
1563:
1553:
1552:
1551:
1549:
1548:
1531:
1522:
1518:
1513:
1492:
1488:
1477:
1466:
1462:
1458:
1453:
1449:
1439:
1432:
1428:
1417:
1409:
1405:
1401:
1392:
1384:
1380:
1373:
1364:
1358:
1349:
1343:
1340:
1334:
1331:
1320:
1314:
1304:
1303:
1302:
1301:
1297:
1296:magnetization
1273:
1267:
1264:
1261:
1255:
1250:
1242:
1239:
1233:
1222:
1219:
1216:
1210:
1197:
1196:
1195:
1193:
1171:
1168:
1165:
1159:
1156:
1150:
1147:
1141:
1130:
1127:
1124:
1118:
1105:
1104:
1103:
1102:of the flux:
1101:
1097:
1093:
1071:
1068:
1065:
1059:
1053:
1050:
1047:
1041:
1038:
1035:
1029:
1022:
1021:
1020:
1018:
1014:
998:
978:
964:
941:
937:
933:
928:
924:
916:
906:
903:
900:
897:
894:
886:
882:
877:
870:
867:
864:
861:
858:
850:
846:
839:
836:
833:
827:
824:
821:
818:
815:
794:
793:
792:
776:
772:
749:
745:
724:
721:
718:
698:
672:
669:
666:
663:
660:
657:
651:
648:
645:
638:
634:
626:
623:
617:
607:
606:
605:
601:
587:
547:
527:
519:
503:
483:
461:
457:
432:
428:
424:
418:
415:
410:
403:
397:
393:
387:
383:
377:
373:
369:
365:
361:
358:
355:
348:
344:
336:
333:
327:
317:
316:
315:
311:
295:
291:
268:
264:
254:
250:
240:
236:
234:
230:
226:
220:
218:
214:
209:
207:
203:
199:
195:
192:
188:
184:
180:
176:
173:
169:
165:
161:
157:
152:
150:
146:
142:
141:T2 relaxation
138:
134:
130:
120:
118:
115:
111:
107:
103:
99:
95:
91:
87:
83:
79:
78:MRI sequences
75:
71:
67:
58:
54:
52:
48:
44:DTI Color Map
41:
36:
33:Diffusion MRI
31:
19:
9110:Neuroimaging
9056:Acute stroke
9022:Thermography
8777:scintigraphy
8767:Radionuclide
8755:pre-hospital
8601:Tractography
8590:
8520:Radiodensity
8422:calcium scan
8383:Quantitative
8093:
8089:
8079:
8054:
8050:
8040:
7997:
7993:
7987:
7970:
7966:
7960:
7925:
7921:
7911:
7890:
7883:
7848:
7844:
7783:(1): 20935.
7780:
7776:
7766:
7726:(1): 14982.
7723:
7719:
7709:
7674:
7670:
7660:
7645:
7612:
7608:
7601:
7564:
7560:
7507:(1): 10742.
7504:
7500:
7490:
7463:
7457:
7432:
7428:
7422:
7388:(3): 63â74.
7385:
7381:
7371:
7360:. Retrieved
7354:
7344:
7317:
7313:
7303:
7268:
7264:
7254:
7229:
7225:
7219:
7210:
7204:
7169:
7165:
7155:
7142:
7135:
7102:
7098:
7092:
7083:
7079:
7073:
7056:
7050:
7023:
7019:
7009:
6966:
6962:
6956:
6945:. Retrieved
6939:
6929:
6894:
6890:
6880:
6855:
6851:
6845:
6834:. Retrieved
6830:
6806:. Retrieved
6802:
6764:
6760:
6754:
6719:
6715:
6705:
6688:
6682:
6665:
6661:
6655:
6628:
6622:
6589:
6585:
6578:
6553:
6549:
6543:
6518:
6514:
6508:
6475:
6471:
6465:
6448:
6444:
6400:
6396:
6386:
6343:
6339:
6329:
6294:
6290:
6279:
6244:
6240:
6230:
6195:
6191:
6180:
6145:
6141:
6131:
6106:
6102:
6096:
6063:
6059:
6053:
6018:
6014:
6004:
5963:
5959:
5953:
5928:
5924:
5918:
5899:
5895:
5859:
5845:Tractography
5835:Connectogram
5822:tractography
5819:
5812:
5803:
5791:
5781:
5774:
5770:
5731:
5726:connectomics
5695:motor cortex
5687:tractography
5682:and a tumor.
5676:brain tumors
5657:
5633:
5612:
5593:white matter
5590:
5578:
5516:
5501:
5492:
5481:Please help
5476:verification
5473:
5455:Applications
5358:volume ratio
5357:
5355:
5160:
5158:
4992:
4990:
4842:
4840:
4673:
4671:
4513:
4511:
4444:
4353:
4111:
4074:
4068:
4061:
4054:
4047:
4040:
4033:
4026:
4019:
4012:
3965:
3744:
3742:
3656:
3654:
3550:
3469:
3467:
3436:
3362:
3355:
3351:
3344:
3342:
3337:
3330:
3323:
3316:
3310:
3304:
3300:eigenvectors
3298:
3295:
3290:
3288:
3175:
3169:
3163:
3155:
3142:
3139:microtubules
3130:
3117:
3105:
3083:
3079:
3075:
3071:
3067:
3063:
3058:
3054:eigenvectors
3049:
3045:
3041:
3036:
3032:
3028:
2923:
2918:
2914:
2910:
2906:
2902:
2898:
2894:
2890:
2886:
2780:
2767:
2763:
2759:
2685:
2668:
2664:
2659:
2655:
2651:
2647:
2643:
2639:
2635:
2631:
2627:
2624:
2619:
2615:
2612:
2601:
2592:
2395:
2384:
2368:
2366:
2362:tractography
2358:
2354:
2342:
2337:T2 weighting
2332:
2328:
2326:
2311:
2308:
2189:
2106:
2083:
2081:
1959:
1828:
1796:
1545:
1514:
1511:
1293:
1189:
1089:
970:
961:
690:
602:
448:
312:
246:
237:
233:tractography
221:
210:
202:brain tumors
197:
196:
187:white matter
178:
177:
164:white matter
153:
144:
128:
126:
123:Introduction
117:tractography
114:white matter
109:
105:
73:
69:
65:
64:
8945:Cardiac PET
8718:renal tract
8693:Gynecologic
8625:Techniques
8596:restriction
8571:Angiography
8554:Neurography
8500:Fluoroscopy
8446:Angiography
8427:angiography
8371:Techniques:
8332:Aortography
8322:Angiography
8302:Cystography
8292:Mammography
8234:Myelography
8229:Sialography
8198:radiography
8057:: 129â135.
7903:10027851300
7356:Radiopaedia
6941:Radiopaedia
6831:XRayPhysics
6697:10018514722
6346:(1): 4749.
5777:icosahedron
5699:spinal cord
5653:eigenvector
4674:planar case
4514:linear case
3306:eigenvalues
3289:We use the
3124:attenuation
3088:eigenvector
2930:diagonalize
2776:hyperboloid
2084:attenuation
1517:H.C. Torrey
168:grey matter
9094:Categories
8858:Full body:
8644:endoscopic
8618:Ultrasound
8547:functional
8344:Lymphogram
8339:Venography
8307:Arthrogram
7967:Biometrika
7362:2017-10-15
7086:: 459â461.
6984:2108/34611
6947:2017-10-15
6836:2017-10-15
6825:Hammer M.
6808:2018-03-15
6060:NeuroImage
5879:References
5840:Connectome
5795:tomography
5597:anisotropy
2676:Adolf Fick
2379:See also:
2190:where the
1100:divergence
191:anisotropy
9061:Pregnancy
8940:Brain PET
8908:gamma ray
8844:DMSA scan
8698:Obstetric
8591:diffusion
8586:Sequences
8564:perfusion
8456:Pulmonary
8403:Cone beam
8297:Pyelogram
8128:264099038
8112:0740-3194
8071:121103537
8051:Math. Ann
8010:CiteSeerX
7805:2045-2322
7740:2045-2322
7166:Neurology
7065:576214706
6934:Bhuta S.
6852:Radiology
6515:Radiology
6103:Radiology
5996:250787827
5865:classical
5609:diffusion
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2772:ellipsoid
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8363:CT scan
8032:6080176
8002:Bibcode
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2858:+
2855:x
2852:z
2849:e
2846:+
2843:z
2840:y
2837:d
2834:+
2829:2
2825:z
2821:c
2818:+
2813:2
2809:y
2805:b
2802:+
2797:2
2793:x
2789:a
2768:c
2764:b
2760:a
2742:1
2739:=
2734:2
2730:z
2726:c
2723:+
2718:2
2714:y
2710:b
2707:+
2702:2
2698:x
2694:a
2620:z
2616:y
2576:|
2567:z
2564:z
2559:D
2551:z
2548:y
2544:D
2536:z
2533:x
2529:D
2519:z
2516:y
2512:D
2503:y
2500:y
2495:D
2487:y
2484:x
2480:D
2470:z
2467:x
2463:D
2455:y
2452:x
2448:D
2439:x
2436:x
2431:D
2424:|
2419:=
2410:D
2369:b
2287:z
2283:G
2260:y
2256:G
2233:x
2229:G
2206:j
2203:i
2199:b
2171:j
2168:i
2164:D
2158:j
2155:i
2151:b
2145:j
2142:,
2139:i
2127:e
2123:=
2120:A
2093:A
2063:0
2059:D
2055:b
2048:e
2039:D
2034:3
2030:t
2024:2
2020:G
2014:2
2004:3
2001:1
1992:e
1981:M
1976:=
1972:M
1945:,
1940:]
1934:1
1929:0
1924:0
1917:0
1912:1
1907:0
1900:0
1895:0
1890:1
1884:[
1876:D
1873:=
1864:I
1855:D
1852:=
1843:D
1808:D
1776:M
1761:D
1749:+
1742:1
1738:T
1727:k
1721:)
1716:0
1712:M
1703:z
1699:M
1695:(
1682:2
1678:T
1667:j
1659:y
1655:M
1651:+
1642:i
1634:x
1630:M
1614:B
1599:M
1590:=
1584:t
1581:d
1570:M
1564:d
1532:M
1493:1
1489:T
1478:k
1472:)
1467:0
1463:M
1454:z
1450:M
1446:(
1433:2
1429:T
1418:j
1410:y
1406:M
1402:+
1393:i
1385:x
1381:M
1365:B
1350:M
1341:=
1335:t
1332:d
1321:M
1315:d
1274:.
1271:)
1268:t
1265:,
1262:x
1259:(
1251:2
1243:D
1240:=
1234:t
1226:)
1223:t
1220:,
1217:x
1214:(
1175:)
1172:t
1169:,
1166:x
1163:(
1160:J
1148:=
1142:t
1134:)
1131:t
1128:,
1125:x
1122:(
1075:)
1072:t
1069:,
1066:x
1063:(
1054:D
1048:=
1045:)
1042:t
1039:,
1036:x
1033:(
1030:J
999:J
947:)
942:2
938:b
929:1
925:b
921:(
917:/
913:]
910:)
907:z
904:,
901:y
898:,
895:x
892:(
887:1
883:S
878:/
874:)
871:z
868:,
865:y
862:,
859:x
856:(
851:2
847:S
843:[
834:=
831:)
828:z
825:,
822:y
819:,
816:x
813:(
809:C
806:D
803:A
777:2
773:b
750:1
746:b
725:C
722:D
719:A
699:D
676:)
673:C
670:D
667:A
661:b
655:(
646:=
639:0
635:S
630:)
627:E
624:T
621:(
618:S
588:D
528:G
484:S
462:0
458:S
433:]
429:D
425:)
419:3
404:(
398:2
388:2
384:G
378:2
366:[
356:=
349:0
345:S
340:)
337:E
334:T
331:(
328:S
296:2
292:T
269:1
265:T
108:(
68:(
20:)
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