5540:
726:, 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,
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2615:
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
2910:) 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.
2382:. 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.
952:
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.
593:
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:
5525:
2349:
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
5555:
5644:. 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.
303:
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:
189:(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
2360:-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.
9064:
3088:
5608:
5454:
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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.
5761:
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.
124:) 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
2309:
4338:
1781:
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299:-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.
2577:
2921:" 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
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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
2344:
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
2335:
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
244:
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
5813:
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
3028:
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
2614:
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,
2328:
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
302:
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
5576:
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
3103:
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
3045:" 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.
2654:
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
2348:
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
592:
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"
5760:
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.
951:
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
4109:
182:
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
4066:
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
2662:
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
5793:
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
5781:
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
163:
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
2591:
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
5768:
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
3995:
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
1545:
3145:
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
5143:
4014:). 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 = (
2596:; 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
2582:
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.
5694:). 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.
1944:
6174:
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).
5158:
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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
4434:
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
2391:
3325:. 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
136:(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
5438:
4825:
3426:
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
3285:
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
1512:
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
6273:
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).
4656:
4056:). 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
2066:
1296:
5539:
164:
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.
2655:
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).
227:
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.
4975:
4333:{\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})}}}}
216:. 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
1273:
3535:
1174:
4430:
2925:
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
3727:
7595:
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".
3111:, 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
272:-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
4496:
5697:
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
4885:
6572:
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".
3421:
1776:{\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}}}
211:
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
4716:
4556:
6090:
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".
4990:
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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 (
675:
946:
8073:
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).
1824:
200:
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
3750:
3273:
5554:
2872:
5335:{\displaystyle \mathrm {RA} ={\frac {\sqrt {(\lambda _{1}-\operatorname {E} )^{2}+(\lambda _{2}-\operatorname {E} )^{2}+(\lambda _{3}-\operatorname {E} )^{2}}}{{\sqrt {3}}\operatorname {E} }}}
433:
5572:. 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
3945:{\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}}}
2174:
1074:
3350:. 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
6323:
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).
3095:
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
2741:
2572:{\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}}}
6649:
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".
5764:
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
3452:
5733:
29:
7067:
de Sénarmont HH (1848). "Mémoire sur la conductibilité des substances cristalisées pour la chaleur" [Memoir on the conductivity of crystallized substances for heat].
6047:
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".
4439:
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
4097:
3639:
3612:
3585:
5883:
Le Bihan, Denis; Breton, E. (1985). "Imagerie de diffusion in-vivo par résonance magnétique nucléaire" [In-vivo diffusion imaging by nuclear magnetic resonance].
5355:
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method that produces in vivo magnetic resonance images of biological tissues sensitized with the local characteristics of molecular diffusion, generally water (but other
5524:
1813:
155:
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.
4564:
3558:
1491:{\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}}}}
567:
547:
503:
3014:
2984:
2954:
2207:
1955:
93:. Water molecule diffusion patterns can therefore reveal microscopic details about tissue architecture, either normal or in a diseased state. A special kind of DWI,
6276:"Diffusion Histology Imaging Combining Diffusion Basis Spectrum Imaging (DBSI) and Machine Learning Improves Detection and Classification of Glioblastoma Pathology"
6224:
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).
2288:
2261:
2234:
978:
778:
751:
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297:
270:
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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
7640:
724:
6950:
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".
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998:
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587:
527:
483:
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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
1189:
220:(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.
3467:
2356:
A more precise statement of the image acquisition process is that the image-intensities at each position are attenuated, depending on the strength (
1097:
7703:
Griffiths, Timothy T.; Flather, Robert; Teh, Irvin; Haroon, Hamied A.; Shelley, David; Plein, Sven; Bourke, Grainne; Wade, Ryckie G. (2021-07-22).
4346:
3138:. 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
6876:"Signal evolution and infarction risk for apparent diffusion coefficient lesions in acute ischemic stroke are both time- and perfusion-dependent"
197:
with higher specificity and sensitivity than conventional DTI. DBSI has also been useful in determining microstructure properties of the brain.
7086:
Le Bihan D, Mangin JF, Poupon C, Clark CA, Pappata S, Molko N, Chabriat H (April 2001). "Diffusion tensor imaging: concepts and applications".
6380:
Samara, Amjad; Li, Zhaolong; Rutlin, Jerrel; Raji, Cyrus A.; Sun, Peng; Song, Sheng-Kwei; Hershey, Tamara; Eisenstein, Sarah A. (August 2021).
3654:
8738:
8422:
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5506:
The most common application of conventional DWI (without DTI) is in acute brain ischemia. DWI directly visualizes the ischemic necrosis in
4442:
8564:
5615:
2345:
image. It also provides useful structural information about muscleâincluding heart muscleâas well as other tissues such as the prostate.
6537:
Stejskal EO, Tanner JE (1 January 1965). "Spin
Diffusion Measurements: Spin Echoes in the Presence of a Time-Dependent Field Gradient".
5742:: metrics derived from DTI (FA and RD) can differentiate asymptomatic adults from those with compression of the ulnar nerve at the elbow
4837:
8892:
3359:
8164:
6839:
Le Bihan D (August 2013). "Apparent diffusion coefficient and beyond: what diffusion MR imaging can tell us about tissue structure".
6177:"Characterization of multiple sclerosis neuroinflammation and neurodegeneration with relaxation and diffusion basis spectrum imaging"
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143:
Diffusion-weighted images are very useful to diagnose vascular strokes in the brain. It is also used more and more in the staging of
7213:
Westin CF, Maier SE, Mamata H, Nabavi A, Jolesz FA, Kikinis R (June 2002). "Processing and visualization for diffusion tensor MRI".
9000:
5138:{\displaystyle C_{a}=C_{l}+C_{p}=1-C_{s}={\frac {\lambda _{1}+\lambda _{2}-2\lambda _{3}}{\lambda _{1}+\lambda _{2}+\lambda _{3}}}}
3134:, because the measurement misses all the local effects and treats the attenuation as if all the movement rates were solely due to
8530:
7451:
Takahara, Taro; Kwee, Thomas C. (2010). "Diffusion-Weighted Whole-Body
Imaging with Background Body Signal Suppression (DWIBS)".
3282:
subscript to signify that this is what the isotropic diffusion coefficient would be with the effects of anisotropy averaged out.
5947:
Taylor DG, Bushell MC (April 1985). "The spatial mapping of translational diffusion coefficients by the NMR imaging technique".
7762:"Meta-analysis of the normal diffusion tensor imaging values of the median nerve and how they change in carpal tunnel syndrome"
7981:
Basser PJ, Pajevic S (2007). "Spectral decomposition of a 4th-order covariance tensor: applications to diffusion tensor MRI".
7546:"Diffusion Tensor Imaging for Diagnosing Root Avulsions in Traumatic Adult Brachial Plexus Injuries: A Proof-of-Concept Study"
7486:"Diffusion kurtosis MRI as a predictive biomarker of response to neoadjuvant chemotherapy in high grade serous ovarian cancer"
1939:{\displaystyle {\vec {D}}=D\cdot {\vec {I}}=D\cdot {\begin{bmatrix}1&0&0\\0&1&0\\0&0&1\end{bmatrix}},}
599:
151:
as the 'gold standard' for this type of disease. Diffusion tensor imaging is being developed for studying the diseases of the
8535:
7468:
6633:
786:
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7004:
5732:
can differentiate normal nerves (as shown in the tractogram of the spinal cord and brachial plexus and 3D 4k reconstruction
9103:
8376:
7656:"Diffusion tensor imaging of the roots of the brachial plexus: a systematic review and meta-analysis of normative values"
6701:"Diffusional kurtosis imaging: the quantification of non-gaussian water diffusion by means of magnetic resonance imaging"
5898:
5857:
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85:
and non-invasively. Molecular diffusion in tissues is not random, but reflects interactions with many obstacles, such as
69:
as well as software that generates images from the resulting data that uses the diffusion of water molecules to generate
6459:
King MD, Houseman J, Roussel SA, van
Bruggen N, Williams SR, Gadian DG (December 1994). "q-Space imaging of the brain".
2773:
2094:. Anisotropic diffusion will have a similar solution for the diffusion tensor, except that what will be measured is the
8928:
8807:
8547:
8456:
6432:
Zhang Y, Wang S, Wu L, Huo Y (Jan 2011). "Multi-channel diffusion tensor image registration via adaptive chaotic PSO".
309:
2675:
At this point, it is helpful to consider the mathematics of ellipsoids. An ellipsoid can be described by the formula:
8386:
5794:
tissue. Consequently, this is the topic of a multinational study directed by the diffusion-study group of the ISMRM.
5656:
5493:
8132:
2378:. Only a special case of the general mathematical notion is relevant to imaging, which is based on the concept of a
2104:
8371:
8366:
8249:
5600:
of the water molecules preferentially along their main direction. Such preferentially oriented diffusion is called
5577:
kurtosis imaging (DKI) have also been shown to predict the response of cancer patients to chemotherapy treatment.
8451:
8434:
245:
this capability of producing contrast related to the structure of tissues at the microscopic level. In a typical
7130:
1014:
8812:
8659:
8559:
8542:
7416:
Koh DM, Collins DJ (June 2007). "Diffusion-weighted MRI in the body: applications and challenges in oncology".
5853:
5475:
5471:
8755:
5640:) that describes the 3-dimensional shape of diffusion. The fiber direction is indicated by the tensor's main
3029:
angles required to get to this equivalent position now appear in the three vectors and can be read out as the
8938:
8792:
8391:
8266:
8258:
8157:
5678:
within white matter. Fiber tracking algorithms can be used to track a fiber along its whole length (e.g. the
6924:
5510:
in the form of a cytotoxic edema, appearing as a high DWI signal within minutes of arterial occlusion. With
2678:
9049:
8983:
8915:
8905:
8883:
8654:
8622:
8253:
5560:
DWI showing cortical ribbon-like high signal consistent with diffusion restriction in a patient with known
1818:
For the simplest case where the diffusion is isotropic the diffusion tensor is a multiple of the identity:
148:
6678:
Relevance of the information about the diffusion distribution in invo given by kurtosis in q-space imaging
6382:"Nucleus accumbens microstructure mediates the relationship between obesity and eating behavior in adults"
3099:
causes a group of water molecules to move out from a central point, and gradually reach the surface of an
2659:
2374:
Diffusion MRI relies on the mathematics and physical interpretations of the geometric quantities known as
8713:
8681:
8664:
4000:. The simplest measure is obtained by dividing the longest axis of the ellipsoid by the shortest = (
3147:
2922:
2667:
showed in the 1850s, diffusion exhibits many of the same patterns as those seen in the transfer of heat.
1001:
5623:
The imaging of this property is an extension of diffusion MRI. If a series of diffusion gradients (i.e.
8787:
8522:
8262:
3430:
144:
6226:"Deep learning with diffusion basis spectrum imaging for classification of multiple sclerosis lesions"
9028:
8696:
8686:
8003:
5433:{\displaystyle \mathrm {VR} ={\frac {\lambda _{1}\lambda _{2}\lambda _{3}}{\operatorname {E} ^{3}}}}
4820:{\displaystyle C_{p}={\frac {2(\lambda _{2}-\lambda _{3})}{\lambda _{1}+\lambda _{2}+\lambda _{3}}}}
9067:
9018:
8797:
8676:
8649:
8410:
8195:
8150:
7138:. Proceedings of the Society for Magnetic Resonance in Medicine. Vol. 11. Berlin. p. 412.
5729:
4070:
2324:, is an MRI image that more specifically shows diffusion than conventional DWI, by eliminating the
39:
7149:
Vaillancourt DE, Spraker MB, Prodoehl J, Abraham I, Corcos DM, Zhou XJ, et al. (April 2009).
6502:
Posse S, Cuenod CA, Le Bihan D (September 1993). "Human brain: proton diffusion MR spectroscopy".
3617:
3590:
3563:
2917:, we can apply a useful technique from standard matrix mathematics and linear algebraâthat is to "
2298:
The standard grayscale of DWI images is to represent increased diffusion restriction as brighter.
8966:
8802:
8743:
8728:
8617:
8439:
8415:
5739:
5546:
5464:
4651:{\displaystyle C_{l}={\frac {\lambda _{1}-\lambda _{2}}{\lambda _{1}+\lambda _{2}+\lambda _{3}}}}
3168:. This leads to the idea of describing the average diffusivity in the voxel which will simply be
5860:, and so on, but the generality, which covers arrays of all sizes, may obscure rather than help.
1789:
8837:
8706:
8637:
8444:
7998:
7907:"High angular resolution diffusion imaging reveals intravoxel white matter fiber heterogeneity"
7151:"High-resolution diffusion tensor imaging in the substantia nigra of de novo Parkinson disease"
5749:
5745:
5699:
2369:
2308:
2061:{\displaystyle {M}={M}_{\text{bloch}}e^{-{\frac {1}{3}}\gamma ^{2}G^{2}t^{3}D}\sim e^{-bD_{0}}}
5912:
Merboldt KD, Hanicke W, Frahm J (1985). "Self-diffusion NMR imaging using stimulated echoes".
3543:
178:
in the brain or muscle fibers in the heartâhas an internal fibrous structure analogous to the
9098:
8701:
8632:
8498:
8381:
8300:
8207:
4058:
3116:
3048:
1535:
1080:
552:
532:
488:
217:
3130:
interfere with the free diffusion, we are measuring an "apparent diffusion coefficient", or
2989:
2959:
2929:
2182:
9044:
8733:
8671:
8493:
7990:
7773:
7497:
6757:
6546:
6336:
5956:
5921:
5787:
5637:
2914:
2600:. A three-dimensional vector can be described with three components: its projection on the
2593:
2266:
2239:
2212:
963:
756:
729:
441:
275:
248:
5748:: Metrics derived from DTI (lower FA and MD) differentiate healthy adults from those with
8:
8990:
8868:
8691:
8427:
7954:
Hext GR (1963). "The estimation of second-order tensors with related tests and designs".
5679:
5668:
5531:
5515:
5507:
3108:
2332:
2325:
1084:
703:
90:
7994:
7777:
7544:
Wade RG, Tanner SF, Teh I, Ridgway JP, Shelley D, Chaka B, et al. (16 April 2020).
7501:
6761:
6550:
6340:
6000:"Mapping complex tissue architecture with diffusion spectrum magnetic resonance imaging"
5960:
5925:
5806:, matrix mathematics and vector mathematics to process the analysis of the tensor data.
8978:
8827:
8822:
8723:
8606:
8461:
8212:
8112:
8075:"Comparison of distortion correction preprocessing pipelines for DTI in the upper limb"
8055:
8016:
7936:
7859:
7802:
7761:
7737:
7704:
7680:
7655:
7621:
7572:
7545:
7518:
7485:
7484:
Deen SS, Priest AN, McLean MA, Gill AB, Brodie C, Crawford R, et al. (July 2019).
7393:
7366:
7274:
7249:
7175:
7166:
7150:
7111:
6985:
6963:
6900:
6875:
6730:
6625:
6618:
Principles and Application of Self-Diffusion Measurements by Nuclear Magnetic Resonance
6598:
6484:
6406:
6381:
6357:
6324:
6300:
6275:
6250:
6225:
6201:
6176:
6151:
6126:
6072:
6029:
5980:
3978:
3958:
2077:
1516:
1180:
983:
683:
572:
512:
506:
468:
241:
204:. Clinically, trace-weighted images have proven to be very useful to diagnose vascular
194:
78:
7226:
2611:
axes. Vectors of this sort can be considered tensors of rank 1, or 1st-order tensors.
8817:
8782:
8777:
8718:
8466:
8315:
8116:
8104:
8096:
8059:
7928:
7887:
7851:
7807:
7789:
7742:
7724:
7685:
7654:
Wade RG, Whittam A, Teh I, Andersson G, Yeh FC, Wiberg M, Bourke G (9 October 2020).
7613:
7577:
7523:
7464:
7433:
7398:
7367:"Magnetic resonance diffusion-perfusion mismatch in acute ischemic stroke: An update"
7320:
7279:
7230:
7180:
7103:
7049:
7026:
6977:
6905:
6856:
6722:
6681:
6629:
6590:
6519:
6476:
6411:
6362:
6325:"Diffusion histology imaging differentiates distinct pediatric brain tumor histology"
6305:
6255:
6206:
6156:
6107:
6064:
6060:
6021:
5984:
5972:
5968:
5933:
4970:{\displaystyle C_{s}={\frac {3\lambda _{3}}{\lambda _{1}+\lambda _{2}+\lambda _{3}}}}
70:
7641:"DTI (Quantitative), a new and advanced MRI procedure for evaluation of Concussions"
7625:
7339:
6989:
6734:
6602:
6515:
6488:
6103:
2651:. These can be written as a matrix similar to the one at the start of this section.
208:
in the brain, by early detection (within a couple of minutes) of the hypoxic edema.
9093:
9088:
8644:
8503:
8275:
8227:
8086:
8047:
8020:
8008:
7963:
7940:
7918:
7863:
7841:
7797:
7781:
7732:
7716:
7675:
7667:
7605:
7567:
7557:
7513:
7505:
7456:
7425:
7388:
7378:
7310:
7269:
7261:
7222:
7170:
7162:
7115:
7095:
7016:
6967:
6959:
6895:
6887:
6848:
6765:
6712:
6658:
6621:
6582:
6554:
6511:
6468:
6441:
6401:
6393:
6352:
6344:
6295:
6287:
6245:
6237:
6196:
6188:
6146:
6138:
6099:
6076:
6056:
6033:
6011:
5964:
5929:
3997:
2664:
2597:
2379:
6291:
8853:
8305:
8271:
8244:
8239:
8173:
8012:
7905:
Tuch DS, Reese TG, Wiegell MR, Makris N, Belliveau JW, Wedeen VJ (October 2002).
7460:
6891:
6787:
5725:
4436:
3135:
2312:
ADC image of the same case of cerebral infarction as seen on DWI in section above
1509:
1268:{\displaystyle {\frac {\partial \rho (x,t)}{\partial t}}=D\nabla ^{2}\rho (x,t).}
160:
45:
7967:
7878:
6662:
8995:
7785:
7720:
7671:
7509:
7315:
7298:
7297:
Grand S, Tahon F, Attye A, Lefournier V, Le Bas JF, Krainik A (December 2013).
7265:
6348:
5803:
5707:
5624:
5573:
5561:
1288:
7760:
Rojoa, Djamila; Raheman, Firas; Rassam, Joseph; Wade, Ryckie G. (2021-10-22).
7132:
Vector analysis of diffusion images in experimental allergic encephalomyelitis
6192:
6125:
Wang Y, Wang Q, Haldar JP, Yeh FC, Xie M, Sun P, et al. (December 2011).
5545:
DWI showing restricted diffusion in the medial dorsal thalami consistent with
3530:{\displaystyle \mathrm {tr} (\Lambda )=\lambda _{1}+\lambda _{2}+\lambda _{3}}
1169:{\displaystyle {\frac {\partial \rho (x,t)}{\partial t}}=-\nabla \cdot J(x,t)}
9082:
8627:
8596:
8584:
8337:
8232:
8100:
7793:
7728:
7562:
7053:
7046:
Physical Properties of Crystals: Their Representation by Tensors and Matrices
6571:
5774:
5511:
4425:{\textstyle \operatorname {E} =(\lambda _{1}+\lambda _{2}+\lambda _{3})/3\,.}
3123:
1284:
129:
125:
86:
7891:
6852:
6769:
6685:
6445:
9010:
8863:
8858:
8765:
8589:
8574:
8508:
8108:
7932:
7855:
7811:
7746:
7689:
7581:
7527:
7437:
7402:
7324:
7283:
7234:
7184:
7107:
7030:
6981:
6909:
6860:
6726:
6620:. Advances in Magnetic and Optical Resonance. Vol. 12. pp. 1â89.
6586:
6472:
6415:
6366:
6309:
6259:
6210:
6160:
6142:
6127:"Quantification of increased cellularity during inflammatory demyelination"
6068:
6025:
5833:
5823:
5810:
5714:
5683:
5675:
5664:
5581:
3457:
Another commonly used measure that summarizes the total diffusivity is the
3288:
3042:
2350:
1533:
is a vector, there are 3 diffusion equations, one for each dimension. The
224:; trying to see which part of the brain is connected to which other part).
221:
175:
152:
105:
102:
66:
7617:
7609:
6594:
6523:
6480:
6111:
5976:
5809:
In some cases, the full set of tensor properties is of interest, but for
8933:
8488:
8320:
8310:
8290:
8280:
8222:
8217:
8186:
7383:
7344:
6929:
5765:
5687:
5667:, surgery is aided by knowing the proximity and relative position of the
5641:
3294:
3127:
3112:
3076:
2764:
190:
156:
7429:
5998:
Wedeen VJ, Hagmann P, Tseng WY, Reese TG, Weisskoff RM (December 2005).
3722:{\displaystyle \mathrm {MD} =(\lambda _{1}+\lambda _{2}+\lambda _{3})/3}
8569:
8332:
8327:
8295:
8051:
6241:
5828:
5783:
5585:
5478: in this section. Unsourced material may be challenged and removed.
4980:
Each measure lies between 0 and 1 and they sum to unity. An additional
3087:
1088:
179:
8091:
8074:
7923:
7906:
7846:
7829:
7099:
7003:
Basser PJ, Pajevic S, Pierpaoli C, Duda J, Aldroubi A (October 2000).
6972:
6815:
6717:
6700:
6558:
6397:
6016:
5999:
5607:
2385:
The following matrix displays the components of the diffusion tensor:
8896:
8832:
8285:
5619:
DTI of a healthy human brachial plexus. Taken from Wade et al., 2020.
5597:
3100:
3096:
2760:
74:
8035:
7886:. Proceedings of the 7th Annual Meeting of the ISMRM. Philadelphia.
5852:
Several full mathematical treatments of general tensors exist, e.g.
5453:
4491:{\displaystyle \lambda _{1}\geq \lambda _{2}\geq \lambda _{3}\geq 0}
8919:
8036:"Uber eine geometrische Anwendung der Abelschen Integralgleichnung"
7069:
Comptes Rendus Hebdomadaires des Séances de l'Académie des Sciences
7022:
10.1002/1522-2594(200010)44:4<625::AID-MRM17>3.0.CO;2-O
5703:
5569:
3063:) components have numbers in them but the off-diagonal components (
1005:
8142:
7148:
4880:{\displaystyle \lambda _{1}\simeq \lambda _{2}\simeq \lambda _{3}}
1501:
which has terms for precession, T2 relaxation, and T1 relaxation.
8948:
8351:
7200:
Geometrical diffusion measures for MRI from tensor basis analysis
7198:
Westin CF, Peled S, Gudbjartsson H, Kikinis R, Jolesz FA (1997).
5814:
aligned with the main axon direction to accomplish tractography.
5710:
2744:
82:
7594:
6699:
Jensen JH, Helpern JA, Ramani A, Lu H, Kaczynski K (June 2005).
3416:{\displaystyle \lambda _{\perp }=(\lambda _{2}+\lambda _{3})/2.}
2770:
As it turns out, three more components can be added as follows:
8943:
6680:. Proceedings, 12th ISMRM Annual Meeting. Kyoto. p. 1238.
5652:
5633:
5593:
4711:{\displaystyle \lambda _{1}\simeq \lambda _{2}\gg \lambda _{3}}
4551:{\displaystyle \lambda _{1}\gg \lambda _{2}\simeq \lambda _{3}}
3300:
2375:
213:
205:
7197:
6949:
6458:
6089:
5706:. Applications in brain research include the investigation of
3329:
is an actual measurement of the ellipsoid we have calculated.
8970:
8181:
5691:
5660:
5629:
201:
171:
121:
7880:
High angular resolution diffusion imaging of the human brain
7876:
4984:
can used to describe the deviation from the spherical case:
3041:
components of each of them. Those three vectors are called "
1004:
gives a relationship between the flux and the concentration
28:
5589:
4099:
so that FA has a maximum value of 1. The whole formula for
3139:
3016:
are along the diagonal (red subscripts), but the variables
137:
8137:
7128:
7002:
6874:
An H, Ford AL, Vo K, Powers WJ, Lee JM, Lin W (May 2011).
6748:
Torrey HC (1956). "Bloch Equations with Diffusion Terms".
6046:
5786:
and vector mathematics developed nearly 100 years agoâthe
5611:
Tractographic reconstruction of neural connections via DTI
5148:
There are other metrics of anisotropy used, including the
3115:
of an MRI voxel into a numerical measure of diffusionâthe
2898:
still describe ellipsoids, but the additional components (
1087:
relates the time derivative of the concentration with the
7702:
7296:
7085:
5518:, the latter can be quantified by DWI and perfusion MRI.
3304:. The long one pointing along the axon direction will be
3082:
465:
is the signal intensity without the diffusion weighting,
237:
3122:. When various barriers and restricting factors such as
670:{\displaystyle {\frac {S(TE)}{S_{0}}}=\exp(-b\cdot ADC)}
7877:
Tuch DS, Weisskoff RM, Belliveau JW, Wedeen VJ (1999).
7653:
7212:
5997:
5682:, through which the motor information transit from the
5588:
of the tracts can be measured. The architecture of the
941:{\displaystyle \mathrm {ADC} (x,y,z)=\ln/(b_{1}-b_{2})}
16:
Method of utilizing water in magnetic resonance imaging
7904:
7759:
7247:
6616:
KÀrger, Jörg; Pfeifer, Harry; Heink, Wilfried (1988).
5514:
detecting both the infarcted core and the salvageable
4349:
2415:
1949:
then the Bloch-Torrey equation will have the solution
1875:
140:
trajectories than other diffusion imaging approaches.
7705:"Diffusion tensor imaging in cubital tunnel syndrome"
7248:
Alexander AL, Lee JE, Lazar M, Field AS (July 2007).
7129:
Richards TL, Heide AC, Tsuruda JS, Alvord EC (1992).
6698:
5674:
Diffusion tensor imaging data can be used to perform
5358:
5161:
4993:
4896:
4840:
4727:
4671:
4567:
4511:
4445:
4112:
4073:
3981:
3961:
3748:
3657:
3620:
3593:
3566:
3546:
3470:
3433:
3362:
3177:
2992:
2962:
2932:
2776:
2681:
2394:
2269:
2242:
2215:
2185:
2107:
2080:
1958:
1827:
1792:
1548:
1519:
1299:
1192:
1100:
1017:
986:
966:
789:
759:
732:
706:
686:
602:
575:
555:
535:
515:
491:
471:
444:
312:
278:
251:
8072:
7543:
7483:
6675:
5911:
700:, is replaced by an apparent diffusion coefficient,
170:(DTI) is important when a tissueâsuch as the neural
6501:
6322:
3339:is also called the longitudinal diffusivity or the
3268:{\displaystyle (ADC_{x}+ADC_{y}+ADC_{z})/3=ADC_{i}}
6615:
6379:
6173:
5432:
5334:
5137:
4969:
4879:
4819:
4710:
4650:
4550:
4498:, then the measures can be calculated as follows:
4490:
4424:
4332:
4091:
3987:
3967:
3944:
3721:
3633:
3606:
3579:
3552:
3529:
3446:
3415:
3267:
3008:
2978:
2948:
2867:{\displaystyle ax^{2}+by^{2}+cz^{2}+dyz+ezx+fxy=1}
2866:
2735:
2571:
2363:
2282:
2255:
2228:
2201:
2168:
2086:
2060:
1938:
1807:
1775:
1525:
1490:
1267:
1168:
1068:
992:
972:
940:
772:
745:
718:
692:
669:
581:
561:
541:
521:
497:
477:
457:
427:
291:
264:
7823:
7821:
6124:
3298:. The lengths are symbolized by the Greek letter
1283:With no diffusion present, the change in nuclear
569:is the time between the two pulses, and finally,
428:{\displaystyle {\frac {S(TE)}{S_{0}}}=\exp \left}
9080:
6648:
2071:The exponential term will be referred to as the
132:rates. A variant of diffusion weighted imaging,
6781:
6779:
5655:such as trauma and in defining the severity of
5580:The principal application is in the imaging of
7818:
7337:
7290:
7005:"In vivo fiber tractography using DT-MRI data"
6873:
6536:
6431:
6230:Annals of Clinical and Translational Neurology
5882:
5530:DWI showing necrosis (shown as brighter) in a
2913:When the ellipsoid/tensor is represented by a
2169:{\displaystyle A=e^{-\sum _{i,j}b_{ij}D_{ij}}}
8158:
7980:
6427:
6425:
6272:
6223:
5946:
5878:
5876:
2336:DWI is a result of its partial T2 weighting.
183:direction from which an observer is looking.
147:, where it is a serious candidate to replace
55:Diffusion-weighted magnetic resonance imaging
7450:
7206:
7202:. ISMRM '97. Vancouver Canada. p. 1742.
7122:
7066:
6832:
6776:
5651:Tract-specific localization of white matter
3461:âwhich is the sum of the three eigenvalues,
2670:
120:(DWI), the intensity of each image element (
6669:
3644:If we divide this sum by three we have the
2339:
73:in MR images. It allows the mapping of the
8165:
8151:
7415:
6867:
6422:
6083:
5873:
3332:The diffusivity along the principal axis,
3104:of diffusion in each voxel of the tissue.
3091:Visualization of DTI data with ellipsoids.
2329:smaller magnitude of diffusion as darker.
1069:{\displaystyle J(x,t)=-D\nabla \rho (x,t)}
27:
8090:
8002:
7922:
7845:
7801:
7736:
7679:
7571:
7561:
7539:
7537:
7517:
7392:
7382:
7314:
7273:
7174:
7020:
6971:
6899:
6816:"MRI Physics: Diffusion-Weighted Imaging"
6809:
6807:
6716:
6405:
6356:
6299:
6249:
6200:
6150:
6015:
5885:Comptes Rendus de l'Académie des Sciences
5802:For DTI, it is generally possible to use
5736:) from traumatically injured nerve roots.
5494:Learn how and when to remove this message
4418:
3311:and the two small axes will have lengths
1278:
9001:Orthogonal polarization spectral imaging
7358:
6925:"Diffusion weighted MRI in acute stroke"
6838:
6676:Chabert S, Mecca CC, Le Bihan D (2004).
5636:(i.e. a symmetric positive definite 3Ă3
5614:
5606:
3142:, and slower perpendicularly across it.
3086:
2307:
1083:. Then, given conservation of mass, the
7455:. Medical Radiology. pp. 227â252.
7250:"Diffusion tensor imaging of the brain"
2098:(ADC). In general, the attenuation is:
529:is the strength of the gradient pulse,
77:process of molecules, mainly water, in
9081:
7534:
7418:AJR. American Journal of Roentgenology
7364:
7338:Weerakkody Y, Gaillard F, et al.
6804:
6785:
6747:
5568:Another application area of DWI is in
3560:is a diagonal matrix with eigenvalues
3083:Measures of anisotropy and diffusivity
2759:determine if the quadric describes an
2736:{\displaystyle ax^{2}+by^{2}+cz^{2}=1}
2209:terms incorporate the gradient fields
8146:
7303:Diagnostic and Interventional Imaging
7088:Journal of Magnetic Resonance Imaging
5584:where the location, orientation, and
1179:Putting the two together, we get the
8033:
7953:
7827:
7638:
7299:"Perfusion imaging in brain disease"
5839:
5721:Applications for peripheral nerves:
5476:adding citations to reliable sources
5447:
3075:) are 0. The second matrix provides
1287:over time is given by the classical
101:), has been used extensively to map
8172:
7043:
2586:
13:
8808:Sestamibi parathyroid scintigraphy
7660:Clinical and Translational Imaging
7167:10.1212/01.wnl.0000340982.01727.6e
6964:10.1097/01.rli.0000264059.46444.bf
6813:
6626:10.1016/b978-0-12-025512-2.50004-x
5405:
5363:
5360:
5314:
5280:
5236:
5192:
5166:
5163:
4350:
4237:
4193:
4149:
4117:
4114:
3975:is the matrix of eigenvectors and
3891:
3888:
3850:
3840:
3837:
3812:
3789:
3786:
3764:
3757:
3754:
3662:
3659:
3547:
3482:
3475:
3472:
3292:and the measures of their lengths
1758:
1740:
1235:
1219:
1196:
1142:
1127:
1104:
1045:
955:
797:
794:
791:
556:
396:
14:
9115:
8126:
6922:
3447:{\displaystyle \lambda _{\perp }}
3343:or even the parallel diffusivity
2551:
2487:
2423:
680:Also, the diffusion coefficient,
485:is the signal with the gradient,
9063:
9062:
6061:10.1016/j.neuroimage.2008.03.036
5553:
5538:
5523:
5452:
2747:surface. The relative values of
780:, of the b factor according to:
187:Diffusion Basis Spectrum Imaging
8565:Cholangiopancreatography (MRCP)
8066:
8027:
7974:
7947:
7898:
7870:
7753:
7696:
7647:
7632:
7588:
7477:
7444:
7409:
7331:
7241:
7191:
7142:
7079:
7060:
7037:
6996:
6943:
6916:
6741:
6692:
6642:
6609:
6565:
6539:The Journal of Chemical Physics
6530:
6516:10.1148/radiology.188.3.8351339
6495:
6452:
6373:
6316:
6266:
6104:10.1148/radiology.161.2.3763909
5949:Physics in Medicine and Biology
5846:
5592:in parallel bundles, and their
5463:needs additional citations for
5443:
3107:In an isotropic medium such as
2364:Mathematical foundationâtensors
111:
8813:Radioactive iodine uptake test
8079:Magnetic Resonance in Medicine
7911:Magnetic Resonance in Medicine
7834:Magnetic Resonance in Medicine
7009:Magnetic Resonance in Medicine
6705:Magnetic Resonance in Medicine
6575:Magnetic Resonance in Medicine
6461:Magnetic Resonance in Medicine
6217:
6167:
6118:
6040:
6004:Magnetic Resonance in Medicine
5991:
5940:
5905:
5657:diffuse traumatic brain injury
5418:
5411:
5326:
5320:
5296:
5292:
5286:
5264:
5252:
5248:
5242:
5220:
5208:
5204:
5198:
5176:
4773:
4747:
4407:
4368:
4362:
4356:
4324:
4270:
4262:
4253:
4249:
4243:
4221:
4209:
4205:
4199:
4177:
4165:
4161:
4155:
4133:
4130:
3901:
3895:
3866:
3844:
3815:
3793:
3767:
3761:
3708:
3669:
3485:
3479:
3402:
3376:
3235:
3178:
2401:
2318:apparent diffusion coefficient
2096:apparent diffusion coefficient
1855:
1834:
1799:
1767:
1752:
1718:
1709:
1683:
1658:
1633:
1605:
1590:
1561:
1508:mathematically showed how the
1469:
1460:
1434:
1409:
1384:
1356:
1341:
1312:
1259:
1247:
1214:
1202:
1163:
1151:
1122:
1110:
1063:
1051:
1033:
1021:
935:
909:
901:
898:
880:
862:
844:
831:
819:
801:
664:
643:
618:
609:
589:is the diffusion-coefficient.
549:is the duration of the pulse,
328:
319:
1:
8793:Radionuclide ventriculography
8267:Lower gastrointestinal series
8259:Upper gastrointestinal series
7453:Diffusion-Weighted MR Imaging
7227:10.1016/s1361-8415(02)00053-1
6292:10.1158/1078-0432.ccr-20-0736
5914:Journal of Magnetic Resonance
5867:
4092:{\displaystyle 1/{\sqrt {2}}}
1815:is now the diffusion tensor.
8984:Optical coherence tomography
8906:Myocardial perfusion imaging
8494:Dental panoramic radiography
8013:10.1016/j.sigpro.2006.02.050
7461:10.1007/978-3-540-78576-7_14
7365:Chen F, Ni YC (March 2012).
7044:Nye, John Frederick (1957).
6892:10.1161/STROKEAHA.110.610501
5934:10.1016/0022-2364(85)90111-8
3634:{\displaystyle \lambda _{3}}
3607:{\displaystyle \lambda _{2}}
3580:{\displaystyle \lambda _{1}}
2743:. This equation describes a
2303:
2293:
231:
149:positron emission tomography
7:
6663:10.1560/HB5H-6XBR-1AW1-LNX9
6651:Israel Journal of Chemistry
5817:
5755:
5647:Applications in the brain:
10:
9120:
9104:Magnetic resonance imaging
8788:Ventilation/perfusion scan
8263:Small-bowel follow-through
7786:10.1038/s41598-021-00353-z
7721:10.1038/s41598-021-94211-7
7672:10.1007/s40336-020-00393-x
7510:10.1038/s41598-019-47195-4
7371:World Journal of Radiology
7316:10.1016/j.diii.2013.06.009
7266:10.1016/j.nurt.2007.05.011
6349:10.1038/s41598-021-84252-3
6181:Multiple Sclerosis Journal
5969:10.1088/0031-9155/30/4/009
5797:
2367:
1808:{\displaystyle {\vec {D}}}
145:non-small-cell lung cancer
134:diffusion spectrum imaging
118:diffusion weighted imaging
9058:
9037:
9029:Dynamic angiothermography
9009:
8965:
8914:
8891:
8881:
8846:
8773:
8763:
8754:
8697:Abdominal ultrasonography
8605:
8521:
8481:
8400:
8359:
8350:
8203:
8194:
8180:
8133:PNRC: About Diffusion MRI
7968:10.1093/biomet/50.3-4.353
7828:Tuch DS (December 2004).
6193:10.1177/13524585211023345
2671:Mathematics of ellipsoids
2660:Henri Hureau de SĂ©narmont
65:) is the use of specific
38:
26:
21:
9019:Non-contact thermography
8798:Radionuclide angiography
8650:Doppler echocardiography
7563:10.3389/fsurg.2020.00019
6280:Clinical Cancer Research
5596:sheaths, facilitate the
3553:{\displaystyle \Lambda }
2340:Diffusion tensor imaging
960:Given the concentration
236:Diffusion imaging is an
168:Diffusion tensor imaging
95:diffusion tensor imaging
8803:Radioisotope renography
6952:Investigative Radiology
6853:10.1148/radiol.13130420
6770:10.1103/PhysRev.104.563
6446:10.4304/jcp.6.4.825-829
5740:Cubital Tunnel Syndrome
5547:Wernicke encephalopathy
2874:. Many combinations of
562:{\displaystyle \Delta }
542:{\displaystyle \delta }
498:{\displaystyle \gamma }
8838:Gastric emptying study
7215:Medical Image Analysis
6788:"Restricted Diffusion"
6587:10.1002/mrm.1910360607
6473:10.1002/mrm.1910320605
5750:carpal tunnel syndrome
5746:Carpal Tunnel Syndrome
5659:. The localization of
5620:
5612:
5434:
5336:
5139:
4971:
4881:
4821:
4712:
4652:
4552:
4492:
4426:
4334:
4093:
3989:
3969:
3946:
3723:
3635:
3608:
3581:
3554:
3531:
3448:
3417:
3269:
3092:
3010:
3009:{\displaystyle cz^{2}}
2980:
2979:{\displaystyle by^{2}}
2950:
2949:{\displaystyle ax^{2}}
2868:
2737:
2573:
2370:Principal axis theorem
2313:
2284:
2257:
2230:
2203:
2202:{\displaystyle b_{ij}}
2170:
2088:
2062:
1940:
1809:
1777:
1527:
1492:
1279:Magnetization dynamics
1269:
1170:
1070:
994:
974:
942:
774:
747:
720:
694:
671:
583:
563:
543:
523:
499:
479:
459:
429:
293:
266:
8499:X-ray motion analysis
8382:X-ray microtomography
8301:Hysterosalpingography
8208:Pneumoencephalography
7610:10.1002/ana.410420617
5618:
5610:
5602:anisotropic diffusion
5435:
5337:
5140:
4972:
4882:
4822:
4713:
4653:
4553:
4493:
4427:
4335:
4094:
4059:fractional anisotropy
3990:
3970:
3947:
3724:
3636:
3609:
3582:
3555:
3532:
3449:
3418:
3270:
3117:diffusion coefficient
3090:
3011:
2981:
2951:
2869:
2738:
2574:
2311:
2285:
2283:{\displaystyle G_{z}}
2258:
2256:{\displaystyle G_{y}}
2231:
2229:{\displaystyle G_{x}}
2204:
2171:
2089:
2063:
1941:
1810:
1778:
1536:Bloch-Torrey equation
1528:
1493:
1270:
1171:
1081:diffusion coefficient
1071:
995:
975:
973:{\displaystyle \rho }
943:
775:
773:{\displaystyle b_{2}}
748:
746:{\displaystyle b_{1}}
721:
695:
672:
584:
564:
544:
524:
500:
480:
460:
458:{\displaystyle S_{0}}
430:
294:
292:{\displaystyle T_{2}}
267:
265:{\displaystyle T_{1}}
218:fractional anisotropy
9024:Contact thermography
8734:Emergency ultrasound
8672:Transcranial Doppler
8423:Abdominal and pelvis
7550:Frontiers in Surgery
7384:10.4329/wjr.v4.i3.63
6434:Journal of Computers
6143:10.1093/brain/awr307
6137:(Pt 12): 3590â3601.
5788:Funk Radon Transform
5702:from other types of
5472:improve this article
5356:
5159:
4991:
4894:
4838:
4725:
4669:
4565:
4509:
4443:
4347:
4110:
4071:
3979:
3959:
3746:
3655:
3618:
3591:
3564:
3544:
3468:
3431:
3360:
3175:
2990:
2960:
2930:
2774:
2679:
2392:
2267:
2240:
2213:
2183:
2105:
2078:
1956:
1825:
1790:
1546:
1517:
1297:
1190:
1098:
1015:
984:
964:
787:
757:
730:
704:
684:
600:
573:
553:
533:
513:
489:
469:
442:
310:
276:
249:
8991:Confocal microscopy
8869:Indium-111 WBC scan
8692:Echoencephalography
8428:Virtual colonoscopy
7995:2007SigPr..87..220B
7778:2021NatSR..1120935R
7598:Annals of Neurology
7502:2019NatSR...910742D
7430:10.2214/AJR.06.1403
7048:. Clarendon Press.
6762:1956PhRv..104..563T
6551:1965JChPh..42..288S
6341:2021NatSR..11.4749Y
5961:1985PMB....30..345T
5926:1985JMagR..64..479M
5700:Alzheimer's disease
5690:and the peripheral
5680:corticospinal tract
5669:corticospinal tract
5532:cerebral infarction
5508:cerebral infarction
5150:relative anisotropy
4323:
4305:
4287:
3109:cerebrospinal fluid
2333:Cerebral infarction
2320:(ADC) image, or an
1085:continuity equation
719:{\displaystyle ADC}
8979:Optical tomography
8828:Dacryoscintigraphy
8823:Immunoscintigraphy
8462:Whole body imaging
8213:Dental radiography
8138:White Matter Atlas
8052:10.1007/BF01456824
7766:Scientific Reports
7709:Scientific Reports
7490:Scientific Reports
7340:"Ischaemic stroke"
6786:Elster AD (2021).
6329:Scientific Reports
6242:10.1002/acn3.51037
5621:
5613:
5430:
5332:
5135:
4982:anisotropy measure
4967:
4877:
4817:
4708:
4648:
4548:
4488:
4422:
4330:
4309:
4291:
4273:
4089:
3985:
3965:
3942:
3940:
3719:
3631:
3604:
3577:
3550:
3527:
3444:
3413:
3352:radial diffusivity
3265:
3093:
3006:
2976:
2946:
2864:
2733:
2569:
2563:
2558:
2494:
2430:
2314:
2280:
2253:
2226:
2199:
2166:
2137:
2084:
2058:
1936:
1927:
1805:
1773:
1523:
1488:
1265:
1181:diffusion equation
1166:
1066:
990:
970:
938:
770:
743:
716:
690:
667:
579:
559:
539:
519:
507:gyromagnetic ratio
495:
475:
455:
425:
289:
262:
195:multiple sclerosis
79:biological tissues
9076:
9075:
9038:Target conditions
8961:
8960:
8957:
8956:
8877:
8876:
8818:Bone scintigraphy
8783:Scintimammography
8778:Cholescintigraphy
8623:contrast-enhanced
8517:
8516:
8477:
8476:
8467:Full-body CT scan
8367:General operation
8346:
8345:
8316:Angiocardiography
8092:10.1002/mrm.29881
7983:Signal Processing
7924:10.1002/mrm.10268
7847:10.1002/mrm.20279
7470:978-3-540-78575-0
7309:(12): 1241â1257.
7254:Neurotherapeutics
7161:(16): 1378â1384.
7100:10.1002/jmri.1076
6792:mriquestions.com/
6718:10.1002/mrm.20508
6635:978-0-12-025512-2
6559:10.1063/1.1695690
6398:10.1002/oby.23201
6286:(20): 5388â5399.
6017:10.1002/mrm.20642
5891:(15): 1109â1112.
5840:Explanatory notes
5504:
5503:
5496:
5428:
5330:
5312:
5305:
5133:
4965:
4815:
4646:
4328:
4327:
4265:
4103:looks like this:
4087:
3988:{\displaystyle D}
3968:{\displaystyle V}
3641:on its diagonal.
3341:axial diffusivity
2404:
2122:
2087:{\displaystyle A}
1995:
1976:
1858:
1837:
1802:
1770:
1755:
1735:
1721:
1675:
1661:
1636:
1608:
1593:
1576:
1564:
1526:{\displaystyle M}
1486:
1472:
1426:
1412:
1387:
1359:
1344:
1327:
1315:
1226:
1134:
993:{\displaystyle J}
693:{\displaystyle D}
632:
582:{\displaystyle D}
522:{\displaystyle G}
478:{\displaystyle S}
410:
342:
52:
51:
9111:
9066:
9065:
8889:
8888:
8771:
8770:
8761:
8760:
8645:Echocardiography
8504:Hounsfield scale
8357:
8356:
8276:Cholecystography
8201:
8200:
8192:
8191:
8167:
8160:
8153:
8144:
8143:
8121:
8120:
8094:
8070:
8064:
8063:
8031:
8025:
8024:
8006:
7978:
7972:
7971:
7962:(3â4): 353â373.
7951:
7945:
7944:
7926:
7902:
7896:
7895:
7885:
7874:
7868:
7867:
7849:
7840:(6): 1358â1372.
7830:"Q-ball imaging"
7825:
7816:
7815:
7805:
7757:
7751:
7750:
7740:
7700:
7694:
7693:
7683:
7651:
7645:
7644:
7636:
7630:
7629:
7592:
7586:
7585:
7575:
7565:
7541:
7532:
7531:
7521:
7481:
7475:
7474:
7448:
7442:
7441:
7424:(6): 1622â1635.
7413:
7407:
7406:
7396:
7386:
7362:
7356:
7355:
7353:
7352:
7335:
7329:
7328:
7318:
7294:
7288:
7287:
7277:
7245:
7239:
7238:
7210:
7204:
7203:
7195:
7189:
7188:
7178:
7146:
7140:
7139:
7137:
7126:
7120:
7119:
7083:
7077:
7076:
7064:
7058:
7057:
7041:
7035:
7034:
7024:
7000:
6994:
6993:
6975:
6947:
6941:
6940:
6938:
6937:
6920:
6914:
6913:
6903:
6886:(5): 1276â1281.
6871:
6865:
6864:
6836:
6830:
6829:
6827:
6826:
6811:
6802:
6801:
6799:
6798:
6783:
6774:
6773:
6745:
6739:
6738:
6720:
6711:(6): 1432â1440.
6696:
6690:
6689:
6673:
6667:
6666:
6646:
6640:
6639:
6613:
6607:
6606:
6569:
6563:
6562:
6534:
6528:
6527:
6499:
6493:
6492:
6456:
6450:
6449:
6429:
6420:
6419:
6409:
6392:(8): 1328â1337.
6377:
6371:
6370:
6360:
6320:
6314:
6313:
6303:
6270:
6264:
6263:
6253:
6221:
6215:
6214:
6204:
6171:
6165:
6164:
6154:
6122:
6116:
6115:
6087:
6081:
6080:
6055:(4): 1267â1277.
6044:
6038:
6037:
6019:
6010:(6): 1377â1386.
5995:
5989:
5988:
5944:
5938:
5937:
5909:
5903:
5902:
5897:
5880:
5861:
5850:
5713:, as well as in
5557:
5542:
5527:
5499:
5492:
5488:
5485:
5479:
5456:
5448:
5439:
5437:
5436:
5431:
5429:
5427:
5426:
5425:
5403:
5402:
5401:
5392:
5391:
5382:
5381:
5371:
5366:
5341:
5339:
5338:
5333:
5331:
5329:
5313:
5308:
5304:
5303:
5276:
5275:
5260:
5259:
5232:
5231:
5216:
5215:
5188:
5187:
5175:
5174:
5169:
5144:
5142:
5141:
5136:
5134:
5132:
5131:
5130:
5118:
5117:
5105:
5104:
5094:
5093:
5092:
5077:
5076:
5064:
5063:
5053:
5048:
5047:
5029:
5028:
5016:
5015:
5003:
5002:
4976:
4974:
4973:
4968:
4966:
4964:
4963:
4962:
4950:
4949:
4937:
4936:
4926:
4925:
4924:
4911:
4906:
4905:
4886:
4884:
4883:
4878:
4876:
4875:
4863:
4862:
4850:
4849:
4826:
4824:
4823:
4818:
4816:
4814:
4813:
4812:
4800:
4799:
4787:
4786:
4776:
4772:
4771:
4759:
4758:
4742:
4737:
4736:
4717:
4715:
4714:
4709:
4707:
4706:
4694:
4693:
4681:
4680:
4657:
4655:
4654:
4649:
4647:
4645:
4644:
4643:
4631:
4630:
4618:
4617:
4607:
4606:
4605:
4593:
4592:
4582:
4577:
4576:
4557:
4555:
4554:
4549:
4547:
4546:
4534:
4533:
4521:
4520:
4497:
4495:
4494:
4489:
4481:
4480:
4468:
4467:
4455:
4454:
4431:
4429:
4428:
4423:
4414:
4406:
4405:
4393:
4392:
4380:
4379:
4339:
4337:
4336:
4331:
4329:
4322:
4317:
4304:
4299:
4286:
4281:
4266:
4261:
4260:
4233:
4232:
4217:
4216:
4189:
4188:
4173:
4172:
4145:
4144:
4126:
4125:
4120:
4098:
4096:
4095:
4090:
4088:
4083:
4081:
3998:prolate spheroid
3994:
3992:
3991:
3986:
3974:
3972:
3971:
3966:
3951:
3949:
3948:
3943:
3941:
3937:
3936:
3915:
3908:
3894:
3880:
3873:
3865:
3864:
3843:
3829:
3822:
3808:
3807:
3792:
3774:
3760:
3728:
3726:
3725:
3720:
3715:
3707:
3706:
3694:
3693:
3681:
3680:
3665:
3646:mean diffusivity
3640:
3638:
3637:
3632:
3630:
3629:
3613:
3611:
3610:
3605:
3603:
3602:
3586:
3584:
3583:
3578:
3576:
3575:
3559:
3557:
3556:
3551:
3536:
3534:
3533:
3528:
3526:
3525:
3513:
3512:
3500:
3499:
3478:
3453:
3451:
3450:
3445:
3443:
3442:
3422:
3420:
3419:
3414:
3409:
3401:
3400:
3388:
3387:
3372:
3371:
3274:
3272:
3271:
3266:
3264:
3263:
3242:
3234:
3233:
3215:
3214:
3196:
3195:
3015:
3013:
3012:
3007:
3005:
3004:
2985:
2983:
2982:
2977:
2975:
2974:
2955:
2953:
2952:
2947:
2945:
2944:
2873:
2871:
2870:
2865:
2821:
2820:
2805:
2804:
2789:
2788:
2742:
2740:
2739:
2734:
2726:
2725:
2710:
2709:
2694:
2693:
2587:Physical tensors
2578:
2576:
2575:
2570:
2568:
2567:
2560:
2559:
2543:
2542:
2528:
2527:
2511:
2510:
2496:
2495:
2479:
2478:
2462:
2461:
2447:
2446:
2432:
2431:
2406:
2405:
2397:
2380:symmetric matrix
2289:
2287:
2286:
2281:
2279:
2278:
2262:
2260:
2259:
2254:
2252:
2251:
2235:
2233:
2232:
2227:
2225:
2224:
2208:
2206:
2205:
2200:
2198:
2197:
2175:
2173:
2172:
2167:
2165:
2164:
2163:
2162:
2150:
2149:
2136:
2093:
2091:
2090:
2085:
2067:
2065:
2064:
2059:
2057:
2056:
2055:
2054:
2031:
2030:
2026:
2025:
2016:
2015:
2006:
2005:
1996:
1988:
1978:
1977:
1974:
1972:
1963:
1945:
1943:
1942:
1937:
1932:
1931:
1860:
1859:
1851:
1839:
1838:
1830:
1814:
1812:
1811:
1806:
1804:
1803:
1795:
1782:
1780:
1779:
1774:
1772:
1771:
1763:
1757:
1756:
1748:
1736:
1734:
1733:
1724:
1723:
1722:
1714:
1708:
1707:
1695:
1694:
1681:
1676:
1674:
1673:
1664:
1663:
1662:
1654:
1651:
1650:
1638:
1637:
1629:
1626:
1625:
1615:
1610:
1609:
1601:
1595:
1594:
1586:
1577:
1575:
1567:
1566:
1565:
1557:
1550:
1532:
1530:
1529:
1524:
1497:
1495:
1494:
1489:
1487:
1485:
1484:
1475:
1474:
1473:
1465:
1459:
1458:
1446:
1445:
1432:
1427:
1425:
1424:
1415:
1414:
1413:
1405:
1402:
1401:
1389:
1388:
1380:
1377:
1376:
1366:
1361:
1360:
1352:
1346:
1345:
1337:
1328:
1326:
1318:
1317:
1316:
1308:
1301:
1274:
1272:
1271:
1266:
1243:
1242:
1227:
1225:
1217:
1194:
1175:
1173:
1172:
1167:
1135:
1133:
1125:
1102:
1075:
1073:
1072:
1067:
1002:Fick's first law
999:
997:
996:
991:
979:
977:
976:
971:
947:
945:
944:
939:
934:
933:
921:
920:
908:
879:
878:
869:
843:
842:
800:
779:
777:
776:
771:
769:
768:
752:
750:
749:
744:
742:
741:
725:
723:
722:
717:
699:
697:
696:
691:
676:
674:
673:
668:
633:
631:
630:
621:
604:
588:
586:
585:
580:
568:
566:
565:
560:
548:
546:
545:
540:
528:
526:
525:
520:
504:
502:
501:
496:
484:
482:
481:
476:
464:
462:
461:
456:
454:
453:
434:
432:
431:
426:
424:
420:
416:
412:
411:
403:
390:
389:
380:
379:
370:
369:
343:
341:
340:
331:
314:
298:
296:
295:
290:
288:
287:
271:
269:
268:
263:
261:
260:
48:
31:
19:
18:
9119:
9118:
9114:
9113:
9112:
9110:
9109:
9108:
9079:
9078:
9077:
9072:
9054:
9033:
9005:
8953:
8939:PET mammography
8910:
8873:
8859:Gallium-67 scan
8854:Octreotide scan
8842:
8750:
8601:
8513:
8473:
8396:
8377:High-resolution
8342:
8306:Skeletal survey
8272:Cholangiography
8185:
8176:
8174:Medical imaging
8171:
8129:
8124:
8071:
8067:
8034:Funk P (1919).
8032:
8028:
8004:10.1.1.104.9041
7979:
7975:
7952:
7948:
7903:
7899:
7883:
7875:
7871:
7826:
7819:
7758:
7754:
7701:
7697:
7652:
7648:
7639:Soll E (2015).
7637:
7633:
7593:
7589:
7542:
7535:
7482:
7478:
7471:
7449:
7445:
7414:
7410:
7363:
7359:
7350:
7348:
7336:
7332:
7295:
7291:
7246:
7242:
7211:
7207:
7196:
7192:
7147:
7143:
7135:
7127:
7123:
7084:
7080:
7065:
7061:
7042:
7038:
7001:
6997:
6948:
6944:
6935:
6933:
6921:
6917:
6872:
6868:
6837:
6833:
6824:
6822:
6812:
6805:
6796:
6794:
6784:
6777:
6750:Physical Review
6746:
6742:
6697:
6693:
6674:
6670:
6657:(1â2): 145â54.
6647:
6643:
6636:
6614:
6610:
6570:
6566:
6535:
6531:
6500:
6496:
6457:
6453:
6430:
6423:
6378:
6374:
6321:
6317:
6271:
6267:
6222:
6218:
6172:
6168:
6123:
6119:
6088:
6084:
6045:
6041:
5996:
5992:
5945:
5941:
5910:
5906:
5895:
5881:
5874:
5870:
5865:
5864:
5851:
5847:
5842:
5820:
5800:
5758:
5726:Brachial plexus
5708:neural networks
5564:
5558:
5549:
5543:
5534:
5528:
5500:
5489:
5483:
5480:
5469:
5457:
5446:
5421:
5417:
5404:
5397:
5393:
5387:
5383:
5377:
5373:
5372:
5370:
5359:
5357:
5354:
5353:
5307:
5306:
5299:
5295:
5271:
5267:
5255:
5251:
5227:
5223:
5211:
5207:
5183:
5179:
5173:
5162:
5160:
5157:
5156:
5126:
5122:
5113:
5109:
5100:
5096:
5095:
5088:
5084:
5072:
5068:
5059:
5055:
5054:
5052:
5043:
5039:
5024:
5020:
5011:
5007:
4998:
4994:
4992:
4989:
4988:
4958:
4954:
4945:
4941:
4932:
4928:
4927:
4920:
4916:
4912:
4910:
4901:
4897:
4895:
4892:
4891:
4871:
4867:
4858:
4854:
4845:
4841:
4839:
4836:
4835:
4808:
4804:
4795:
4791:
4782:
4778:
4777:
4767:
4763:
4754:
4750:
4743:
4741:
4732:
4728:
4726:
4723:
4722:
4702:
4698:
4689:
4685:
4676:
4672:
4670:
4667:
4666:
4639:
4635:
4626:
4622:
4613:
4609:
4608:
4601:
4597:
4588:
4584:
4583:
4581:
4572:
4568:
4566:
4563:
4562:
4542:
4538:
4529:
4525:
4516:
4512:
4510:
4507:
4506:
4476:
4472:
4463:
4459:
4450:
4446:
4444:
4441:
4440:
4437:oblate spheroid
4410:
4401:
4397:
4388:
4384:
4375:
4371:
4348:
4345:
4344:
4318:
4313:
4300:
4295:
4282:
4277:
4256:
4252:
4228:
4224:
4212:
4208:
4184:
4180:
4168:
4164:
4140:
4136:
4124:
4113:
4111:
4108:
4107:
4082:
4077:
4072:
4069:
4068:
4055:
4048:
4042:) + (
4041:
4034:
4028:) + (
4027:
4020:
4013:
4006:
3980:
3977:
3976:
3960:
3957:
3956:
3939:
3938:
3932:
3928:
3913:
3912:
3904:
3887:
3878:
3877:
3869:
3857:
3853:
3836:
3827:
3826:
3818:
3800:
3796:
3785:
3778:
3770:
3753:
3749:
3747:
3744:
3743:
3737:
3711:
3702:
3698:
3689:
3685:
3676:
3672:
3658:
3656:
3653:
3652:
3625:
3621:
3619:
3616:
3615:
3598:
3594:
3592:
3589:
3588:
3571:
3567:
3565:
3562:
3561:
3545:
3542:
3541:
3521:
3517:
3508:
3504:
3495:
3491:
3471:
3469:
3466:
3465:
3438:
3434:
3432:
3429:
3428:
3405:
3396:
3392:
3383:
3379:
3367:
3363:
3361:
3358:
3357:
3349:
3338:
3324:
3317:
3310:
3259:
3255:
3238:
3229:
3225:
3210:
3206:
3191:
3187:
3176:
3173:
3172:
3167:
3161:
3155:
3136:Brownian motion
3085:
3049:Diagonalization
3000:
2996:
2991:
2988:
2987:
2970:
2966:
2961:
2958:
2957:
2940:
2936:
2931:
2928:
2927:
2816:
2812:
2800:
2796:
2784:
2780:
2775:
2772:
2771:
2721:
2717:
2705:
2701:
2689:
2685:
2680:
2677:
2676:
2673:
2589:
2562:
2561:
2550:
2546:
2544:
2535:
2531:
2529:
2520:
2516:
2513:
2512:
2503:
2499:
2497:
2486:
2482:
2480:
2471:
2467:
2464:
2463:
2454:
2450:
2448:
2439:
2435:
2433:
2422:
2418:
2411:
2410:
2396:
2395:
2393:
2390:
2389:
2372:
2366:
2342:
2306:
2296:
2274:
2270:
2268:
2265:
2264:
2247:
2243:
2241:
2238:
2237:
2220:
2216:
2214:
2211:
2210:
2190:
2186:
2184:
2181:
2180:
2155:
2151:
2142:
2138:
2126:
2118:
2114:
2106:
2103:
2102:
2079:
2076:
2075:
2050:
2046:
2039:
2035:
2021:
2017:
2011:
2007:
2001:
1997:
1987:
1983:
1979:
1973:
1968:
1967:
1959:
1957:
1954:
1953:
1926:
1925:
1920:
1915:
1909:
1908:
1903:
1898:
1892:
1891:
1886:
1881:
1871:
1870:
1850:
1849:
1829:
1828:
1826:
1823:
1822:
1794:
1793:
1791:
1788:
1787:
1762:
1761:
1747:
1746:
1729:
1725:
1713:
1712:
1703:
1699:
1690:
1686:
1682:
1680:
1669:
1665:
1653:
1652:
1646:
1642:
1628:
1627:
1621:
1617:
1616:
1614:
1600:
1599:
1585:
1584:
1568:
1556:
1555:
1551:
1549:
1547:
1544:
1543:
1518:
1515:
1514:
1510:Bloch equations
1480:
1476:
1464:
1463:
1454:
1450:
1441:
1437:
1433:
1431:
1420:
1416:
1404:
1403:
1397:
1393:
1379:
1378:
1372:
1368:
1367:
1365:
1351:
1350:
1336:
1335:
1319:
1307:
1306:
1302:
1300:
1298:
1295:
1294:
1281:
1238:
1234:
1218:
1195:
1193:
1191:
1188:
1187:
1126:
1103:
1101:
1099:
1096:
1095:
1079:where D is the
1016:
1013:
1012:
985:
982:
981:
965:
962:
961:
958:
956:Diffusion model
929:
925:
916:
912:
904:
874:
870:
865:
838:
834:
790:
788:
785:
784:
764:
760:
758:
755:
754:
737:
733:
731:
728:
727:
705:
702:
701:
685:
682:
681:
626:
622:
605:
603:
601:
598:
597:
574:
571:
570:
554:
551:
550:
534:
531:
530:
514:
511:
510:
490:
487:
486:
470:
467:
466:
449:
445:
443:
440:
439:
402:
395:
391:
385:
381:
375:
371:
365:
361:
357:
353:
336:
332:
315:
313:
311:
308:
307:
283:
279:
277:
274:
273:
256:
252:
250:
247:
246:
234:
161:cerebral cortex
114:
44:
34:
17:
12:
11:
5:
9117:
9107:
9106:
9101:
9096:
9091:
9074:
9073:
9071:
9070:
9059:
9056:
9055:
9053:
9052:
9047:
9041:
9039:
9035:
9034:
9032:
9031:
9026:
9021:
9015:
9013:
9007:
9006:
9004:
9003:
8998:
8996:Endomicroscopy
8993:
8988:
8987:
8986:
8975:
8973:
8963:
8962:
8959:
8958:
8955:
8954:
8952:
8951:
8946:
8941:
8936:
8931:
8925:
8923:
8912:
8911:
8909:
8908:
8902:
8900:
8886:
8879:
8878:
8875:
8874:
8872:
8871:
8866:
8861:
8856:
8850:
8848:
8844:
8843:
8841:
8840:
8835:
8830:
8825:
8820:
8815:
8810:
8805:
8800:
8795:
8790:
8785:
8780:
8774:
8768:
8758:
8752:
8751:
8749:
8748:
8747:
8746:
8741:
8731:
8726:
8721:
8716:
8711:
8710:
8709:
8704:
8694:
8689:
8684:
8679:
8674:
8669:
8668:
8667:
8662:
8657:
8652:
8642:
8641:
8640:
8635:
8630:
8625:
8620:
8611:
8609:
8603:
8602:
8600:
8599:
8594:
8593:
8592:
8587:
8582:
8572:
8567:
8562:
8557:
8556:
8555:
8545:
8540:
8539:
8538:
8527:
8525:
8519:
8518:
8515:
8514:
8512:
8511:
8506:
8501:
8496:
8491:
8485:
8483:
8479:
8478:
8475:
8474:
8472:
8471:
8470:
8469:
8459:
8454:
8449:
8448:
8447:
8442:
8432:
8431:
8430:
8420:
8419:
8418:
8413:
8404:
8402:
8398:
8397:
8395:
8394:
8389:
8384:
8379:
8374:
8369:
8363:
8361:
8354:
8348:
8347:
8344:
8343:
8341:
8340:
8335:
8330:
8325:
8324:
8323:
8318:
8308:
8303:
8298:
8293:
8288:
8283:
8278:
8269:
8256:
8247:
8242:
8237:
8236:
8235:
8225:
8220:
8215:
8210:
8204:
8198:
8189:
8178:
8177:
8170:
8169:
8162:
8155:
8147:
8141:
8140:
8135:
8128:
8127:External links
8125:
8123:
8122:
8085:(2): 773â783.
8065:
8026:
7989:(2): 220â236.
7973:
7946:
7917:(4): 577â582.
7897:
7869:
7817:
7752:
7695:
7666:(6): 419â431.
7646:
7631:
7604:(6): 951â962.
7587:
7533:
7476:
7469:
7443:
7408:
7357:
7330:
7289:
7260:(3): 316â329.
7240:
7205:
7190:
7141:
7121:
7094:(4): 534â546.
7078:
7059:
7036:
7015:(4): 625â632.
6995:
6958:(6): 412â419.
6942:
6915:
6866:
6831:
6803:
6775:
6756:(3): 563â565.
6740:
6691:
6668:
6641:
6634:
6608:
6581:(6): 847â857.
6564:
6545:(1): 288â292.
6529:
6510:(3): 719â725.
6494:
6467:(6): 707â713.
6451:
6440:(4): 825â829.
6421:
6372:
6315:
6265:
6236:(5): 695â706.
6216:
6187:(3): 418â428.
6166:
6117:
6098:(2): 401â407.
6082:
6039:
5990:
5955:(4): 345â349.
5939:
5920:(3): 479â486.
5904:
5871:
5869:
5866:
5863:
5862:
5858:component free
5844:
5843:
5841:
5838:
5837:
5836:
5831:
5826:
5819:
5816:
5804:linear algebra
5799:
5796:
5757:
5754:
5753:
5752:
5743:
5737:
5719:
5718:
5695:
5672:
5625:magnetic field
5566:
5565:
5562:MELAS syndrome
5559:
5552:
5550:
5544:
5537:
5535:
5529:
5522:
5502:
5501:
5460:
5458:
5451:
5445:
5442:
5441:
5440:
5424:
5420:
5416:
5413:
5410:
5407:
5400:
5396:
5390:
5386:
5380:
5376:
5369:
5365:
5362:
5343:
5342:
5328:
5325:
5322:
5319:
5316:
5311:
5302:
5298:
5294:
5291:
5288:
5285:
5282:
5279:
5274:
5270:
5266:
5263:
5258:
5254:
5250:
5247:
5244:
5241:
5238:
5235:
5230:
5226:
5222:
5219:
5214:
5210:
5206:
5203:
5200:
5197:
5194:
5191:
5186:
5182:
5178:
5172:
5168:
5165:
5146:
5145:
5129:
5125:
5121:
5116:
5112:
5108:
5103:
5099:
5091:
5087:
5083:
5080:
5075:
5071:
5067:
5062:
5058:
5051:
5046:
5042:
5038:
5035:
5032:
5027:
5023:
5019:
5014:
5010:
5006:
5001:
4997:
4978:
4977:
4961:
4957:
4953:
4948:
4944:
4940:
4935:
4931:
4923:
4919:
4915:
4909:
4904:
4900:
4874:
4870:
4866:
4861:
4857:
4853:
4848:
4844:
4832:spherical case
4828:
4827:
4811:
4807:
4803:
4798:
4794:
4790:
4785:
4781:
4775:
4770:
4766:
4762:
4757:
4753:
4749:
4746:
4740:
4735:
4731:
4705:
4701:
4697:
4692:
4688:
4684:
4679:
4675:
4659:
4658:
4642:
4638:
4634:
4629:
4625:
4621:
4616:
4612:
4604:
4600:
4596:
4591:
4587:
4580:
4575:
4571:
4545:
4541:
4537:
4532:
4528:
4524:
4519:
4515:
4487:
4484:
4479:
4475:
4471:
4466:
4462:
4458:
4453:
4449:
4421:
4417:
4413:
4409:
4404:
4400:
4396:
4391:
4387:
4383:
4378:
4374:
4370:
4367:
4364:
4361:
4358:
4355:
4352:
4341:
4340:
4326:
4321:
4316:
4312:
4308:
4303:
4298:
4294:
4290:
4285:
4280:
4276:
4272:
4269:
4264:
4259:
4255:
4251:
4248:
4245:
4242:
4239:
4236:
4231:
4227:
4223:
4220:
4215:
4211:
4207:
4204:
4201:
4198:
4195:
4192:
4187:
4183:
4179:
4176:
4171:
4167:
4163:
4160:
4157:
4154:
4151:
4148:
4143:
4139:
4135:
4132:
4129:
4123:
4119:
4116:
4086:
4080:
4076:
4053:
4046:
4039:
4032:
4025:
4018:
4011:
4004:
3984:
3964:
3953:
3952:
3935:
3931:
3927:
3924:
3921:
3918:
3916:
3914:
3911:
3907:
3903:
3900:
3897:
3893:
3890:
3886:
3883:
3881:
3879:
3876:
3872:
3868:
3863:
3860:
3856:
3852:
3849:
3846:
3842:
3839:
3835:
3832:
3830:
3828:
3825:
3821:
3817:
3814:
3811:
3806:
3803:
3799:
3795:
3791:
3788:
3784:
3781:
3779:
3777:
3773:
3769:
3766:
3763:
3759:
3756:
3752:
3751:
3735:
3730:
3729:
3718:
3714:
3710:
3705:
3701:
3697:
3692:
3688:
3684:
3679:
3675:
3671:
3668:
3664:
3661:
3628:
3624:
3601:
3597:
3574:
3570:
3549:
3538:
3537:
3524:
3520:
3516:
3511:
3507:
3503:
3498:
3494:
3490:
3487:
3484:
3481:
3477:
3474:
3441:
3437:
3424:
3423:
3412:
3408:
3404:
3399:
3395:
3391:
3386:
3382:
3378:
3375:
3370:
3366:
3347:
3336:
3322:
3315:
3308:
3276:
3275:
3262:
3258:
3254:
3251:
3248:
3245:
3241:
3237:
3232:
3228:
3224:
3221:
3218:
3213:
3209:
3205:
3202:
3199:
3194:
3190:
3186:
3183:
3180:
3163:
3157:
3151:
3124:cell membranes
3084:
3081:
3003:
2999:
2995:
2973:
2969:
2965:
2943:
2939:
2935:
2863:
2860:
2857:
2854:
2851:
2848:
2845:
2842:
2839:
2836:
2833:
2830:
2827:
2824:
2819:
2815:
2811:
2808:
2803:
2799:
2795:
2792:
2787:
2783:
2779:
2732:
2729:
2724:
2720:
2716:
2713:
2708:
2704:
2700:
2697:
2692:
2688:
2684:
2672:
2669:
2663:ellipsoid. As
2588:
2585:
2580:
2579:
2566:
2557:
2554:
2549:
2545:
2541:
2538:
2534:
2530:
2526:
2523:
2519:
2515:
2514:
2509:
2506:
2502:
2498:
2493:
2490:
2485:
2481:
2477:
2474:
2470:
2466:
2465:
2460:
2457:
2453:
2449:
2445:
2442:
2438:
2434:
2429:
2426:
2421:
2417:
2416:
2414:
2409:
2403:
2400:
2365:
2362:
2341:
2338:
2305:
2302:
2295:
2292:
2277:
2273:
2250:
2246:
2223:
2219:
2196:
2193:
2189:
2177:
2176:
2161:
2158:
2154:
2148:
2145:
2141:
2135:
2132:
2129:
2125:
2121:
2117:
2113:
2110:
2083:
2069:
2068:
2053:
2049:
2045:
2042:
2038:
2034:
2029:
2024:
2020:
2014:
2010:
2004:
2000:
1994:
1991:
1986:
1982:
1971:
1966:
1962:
1947:
1946:
1935:
1930:
1924:
1921:
1919:
1916:
1914:
1911:
1910:
1907:
1904:
1902:
1899:
1897:
1894:
1893:
1890:
1887:
1885:
1882:
1880:
1877:
1876:
1874:
1869:
1866:
1863:
1857:
1854:
1848:
1845:
1842:
1836:
1833:
1801:
1798:
1784:
1783:
1769:
1766:
1760:
1754:
1751:
1745:
1742:
1739:
1732:
1728:
1720:
1717:
1711:
1706:
1702:
1698:
1693:
1689:
1685:
1679:
1672:
1668:
1660:
1657:
1649:
1645:
1641:
1635:
1632:
1624:
1620:
1613:
1607:
1604:
1598:
1592:
1589:
1583:
1580:
1574:
1571:
1563:
1560:
1554:
1522:
1499:
1498:
1483:
1479:
1471:
1468:
1462:
1457:
1453:
1449:
1444:
1440:
1436:
1430:
1423:
1419:
1411:
1408:
1400:
1396:
1392:
1386:
1383:
1375:
1371:
1364:
1358:
1355:
1349:
1343:
1340:
1334:
1331:
1325:
1322:
1314:
1311:
1305:
1289:Bloch equation
1280:
1277:
1276:
1275:
1264:
1261:
1258:
1255:
1252:
1249:
1246:
1241:
1237:
1233:
1230:
1224:
1221:
1216:
1213:
1210:
1207:
1204:
1201:
1198:
1177:
1176:
1165:
1162:
1159:
1156:
1153:
1150:
1147:
1144:
1141:
1138:
1132:
1129:
1124:
1121:
1118:
1115:
1112:
1109:
1106:
1077:
1076:
1065:
1062:
1059:
1056:
1053:
1050:
1047:
1044:
1041:
1038:
1035:
1032:
1029:
1026:
1023:
1020:
989:
969:
957:
954:
949:
948:
937:
932:
928:
924:
919:
915:
911:
907:
903:
900:
897:
894:
891:
888:
885:
882:
877:
873:
868:
864:
861:
858:
855:
852:
849:
846:
841:
837:
833:
830:
827:
824:
821:
818:
815:
812:
809:
806:
803:
799:
796:
793:
767:
763:
740:
736:
715:
712:
709:
689:
678:
677:
666:
663:
660:
657:
654:
651:
648:
645:
642:
639:
636:
629:
625:
620:
617:
614:
611:
608:
578:
558:
538:
518:
494:
474:
452:
448:
436:
435:
423:
419:
415:
409:
406:
401:
398:
394:
388:
384:
378:
374:
368:
364:
360:
356:
352:
349:
346:
339:
335:
330:
327:
324:
321:
318:
286:
282:
259:
255:
233:
230:
113:
110:
108:in the brain.
89:, fibers, and
87:macromolecules
50:
49:
42:
36:
35:
32:
24:
23:
15:
9:
6:
4:
3:
2:
9116:
9105:
9102:
9100:
9097:
9095:
9092:
9090:
9087:
9086:
9084:
9069:
9061:
9060:
9057:
9051:
9048:
9046:
9043:
9042:
9040:
9036:
9030:
9027:
9025:
9022:
9020:
9017:
9016:
9014:
9012:
9008:
9002:
8999:
8997:
8994:
8992:
8989:
8985:
8982:
8981:
8980:
8977:
8976:
8974:
8972:
8968:
8964:
8950:
8947:
8945:
8942:
8940:
8937:
8935:
8932:
8930:
8927:
8926:
8924:
8921:
8917:
8913:
8907:
8904:
8903:
8901:
8898:
8894:
8890:
8887:
8885:
8880:
8870:
8867:
8865:
8864:Ga-68-DOTATOC
8862:
8860:
8857:
8855:
8852:
8851:
8849:
8845:
8839:
8836:
8834:
8831:
8829:
8826:
8824:
8821:
8819:
8816:
8814:
8811:
8809:
8806:
8804:
8801:
8799:
8796:
8794:
8791:
8789:
8786:
8784:
8781:
8779:
8776:
8775:
8772:
8769:
8767:
8762:
8759:
8757:
8753:
8745:
8742:
8740:
8737:
8736:
8735:
8732:
8730:
8727:
8725:
8722:
8720:
8717:
8715:
8712:
8708:
8705:
8703:
8700:
8699:
8698:
8695:
8693:
8690:
8688:
8685:
8683:
8680:
8678:
8677:Intravascular
8675:
8673:
8670:
8666:
8663:
8661:
8658:
8656:
8653:
8651:
8648:
8647:
8646:
8643:
8639:
8636:
8634:
8631:
8629:
8626:
8624:
8621:
8619:
8616:
8615:
8613:
8612:
8610:
8608:
8604:
8598:
8597:Synthetic MRI
8595:
8591:
8588:
8586:
8583:
8581:
8578:
8577:
8576:
8573:
8571:
8568:
8566:
8563:
8561:
8558:
8554:
8551:
8550:
8549:
8546:
8544:
8541:
8537:
8534:
8533:
8532:
8529:
8528:
8526:
8524:
8520:
8510:
8507:
8505:
8502:
8500:
8497:
8495:
8492:
8490:
8487:
8486:
8484:
8480:
8468:
8465:
8464:
8463:
8460:
8458:
8455:
8453:
8450:
8446:
8443:
8441:
8438:
8437:
8436:
8433:
8429:
8426:
8425:
8424:
8421:
8417:
8414:
8412:
8409:
8408:
8406:
8405:
8403:
8399:
8393:
8390:
8388:
8387:Electron beam
8385:
8383:
8380:
8378:
8375:
8373:
8370:
8368:
8365:
8364:
8362:
8358:
8355:
8353:
8349:
8339:
8338:Orbital x-ray
8336:
8334:
8331:
8329:
8326:
8322:
8319:
8317:
8314:
8313:
8312:
8309:
8307:
8304:
8302:
8299:
8297:
8294:
8292:
8289:
8287:
8284:
8282:
8279:
8277:
8273:
8270:
8268:
8264:
8260:
8257:
8255:
8251:
8248:
8246:
8243:
8241:
8238:
8234:
8233:Bronchography
8231:
8230:
8229:
8226:
8224:
8221:
8219:
8216:
8214:
8211:
8209:
8206:
8205:
8202:
8199:
8197:
8193:
8190:
8188:
8183:
8179:
8175:
8168:
8163:
8161:
8156:
8154:
8149:
8148:
8145:
8139:
8136:
8134:
8131:
8130:
8118:
8114:
8110:
8106:
8102:
8098:
8093:
8088:
8084:
8080:
8076:
8069:
8061:
8057:
8053:
8049:
8045:
8041:
8037:
8030:
8022:
8018:
8014:
8010:
8005:
8000:
7996:
7992:
7988:
7984:
7977:
7969:
7965:
7961:
7957:
7950:
7942:
7938:
7934:
7930:
7925:
7920:
7916:
7912:
7908:
7901:
7893:
7889:
7882:
7881:
7873:
7865:
7861:
7857:
7853:
7848:
7843:
7839:
7835:
7831:
7824:
7822:
7813:
7809:
7804:
7799:
7795:
7791:
7787:
7783:
7779:
7775:
7771:
7767:
7763:
7756:
7748:
7744:
7739:
7734:
7730:
7726:
7722:
7718:
7714:
7710:
7706:
7699:
7691:
7687:
7682:
7677:
7673:
7669:
7665:
7661:
7657:
7650:
7642:
7635:
7627:
7623:
7619:
7615:
7611:
7607:
7603:
7599:
7591:
7583:
7579:
7574:
7569:
7564:
7559:
7555:
7551:
7547:
7540:
7538:
7529:
7525:
7520:
7515:
7511:
7507:
7503:
7499:
7495:
7491:
7487:
7480:
7472:
7466:
7462:
7458:
7454:
7447:
7439:
7435:
7431:
7427:
7423:
7419:
7412:
7404:
7400:
7395:
7390:
7385:
7380:
7376:
7372:
7368:
7361:
7347:
7346:
7341:
7334:
7326:
7322:
7317:
7312:
7308:
7304:
7300:
7293:
7285:
7281:
7276:
7271:
7267:
7263:
7259:
7255:
7251:
7244:
7236:
7232:
7228:
7224:
7221:(2): 93â108.
7220:
7216:
7209:
7201:
7194:
7186:
7182:
7177:
7172:
7168:
7164:
7160:
7156:
7152:
7145:
7134:
7133:
7125:
7117:
7113:
7109:
7105:
7101:
7097:
7093:
7089:
7082:
7074:
7071:(in French).
7070:
7063:
7055:
7051:
7047:
7040:
7032:
7028:
7023:
7018:
7014:
7010:
7006:
6999:
6991:
6987:
6983:
6979:
6974:
6969:
6965:
6961:
6957:
6953:
6946:
6932:
6931:
6926:
6919:
6911:
6907:
6902:
6897:
6893:
6889:
6885:
6881:
6877:
6870:
6862:
6858:
6854:
6850:
6847:(2): 318â22.
6846:
6842:
6835:
6821:
6817:
6810:
6808:
6793:
6789:
6782:
6780:
6771:
6767:
6763:
6759:
6755:
6751:
6744:
6736:
6732:
6728:
6724:
6719:
6714:
6710:
6706:
6702:
6695:
6687:
6683:
6679:
6672:
6664:
6660:
6656:
6652:
6645:
6637:
6631:
6627:
6623:
6619:
6612:
6604:
6600:
6596:
6592:
6588:
6584:
6580:
6576:
6568:
6560:
6556:
6552:
6548:
6544:
6540:
6533:
6525:
6521:
6517:
6513:
6509:
6505:
6498:
6490:
6486:
6482:
6478:
6474:
6470:
6466:
6462:
6455:
6447:
6443:
6439:
6435:
6428:
6426:
6417:
6413:
6408:
6403:
6399:
6395:
6391:
6387:
6383:
6376:
6368:
6364:
6359:
6354:
6350:
6346:
6342:
6338:
6334:
6330:
6326:
6319:
6311:
6307:
6302:
6297:
6293:
6289:
6285:
6281:
6277:
6269:
6261:
6257:
6252:
6247:
6243:
6239:
6235:
6231:
6227:
6220:
6212:
6208:
6203:
6198:
6194:
6190:
6186:
6182:
6178:
6170:
6162:
6158:
6153:
6148:
6144:
6140:
6136:
6132:
6128:
6121:
6113:
6109:
6105:
6101:
6097:
6093:
6086:
6078:
6074:
6070:
6066:
6062:
6058:
6054:
6050:
6043:
6035:
6031:
6027:
6023:
6018:
6013:
6009:
6005:
6001:
5994:
5986:
5982:
5978:
5974:
5970:
5966:
5962:
5958:
5954:
5950:
5943:
5935:
5931:
5927:
5923:
5919:
5915:
5908:
5900:
5894:
5890:
5887:(in French).
5886:
5879:
5877:
5872:
5859:
5855:
5849:
5845:
5835:
5832:
5830:
5827:
5825:
5822:
5821:
5815:
5812:
5807:
5805:
5795:
5791:
5789:
5785:
5779:
5776:
5775:tessellations
5772:
5767:
5762:
5751:
5747:
5744:
5741:
5738:
5735:
5731:
5727:
5724:
5723:
5722:
5716:
5712:
5709:
5705:
5701:
5696:
5693:
5689:
5685:
5681:
5677:
5673:
5670:
5666:
5662:
5658:
5654:
5650:
5649:
5648:
5645:
5643:
5639:
5635:
5631:
5626:
5617:
5609:
5605:
5603:
5599:
5595:
5591:
5587:
5583:
5578:
5575:
5571:
5563:
5556:
5551:
5548:
5541:
5536:
5533:
5526:
5521:
5520:
5519:
5517:
5513:
5512:perfusion MRI
5509:
5498:
5495:
5487:
5484:December 2013
5477:
5473:
5467:
5466:
5461:This section
5459:
5455:
5450:
5449:
5422:
5414:
5408:
5398:
5394:
5388:
5384:
5378:
5374:
5367:
5352:
5351:
5350:
5348:
5323:
5317:
5309:
5300:
5289:
5283:
5277:
5272:
5268:
5261:
5256:
5245:
5239:
5233:
5228:
5224:
5217:
5212:
5201:
5195:
5189:
5184:
5180:
5170:
5155:
5154:
5153:
5151:
5127:
5123:
5119:
5114:
5110:
5106:
5101:
5097:
5089:
5085:
5081:
5078:
5073:
5069:
5065:
5060:
5056:
5049:
5044:
5040:
5036:
5033:
5030:
5025:
5021:
5017:
5012:
5008:
5004:
4999:
4995:
4987:
4986:
4985:
4983:
4959:
4955:
4951:
4946:
4942:
4938:
4933:
4929:
4921:
4917:
4913:
4907:
4902:
4898:
4890:
4889:
4888:
4872:
4868:
4864:
4859:
4855:
4851:
4846:
4842:
4833:
4809:
4805:
4801:
4796:
4792:
4788:
4783:
4779:
4768:
4764:
4760:
4755:
4751:
4744:
4738:
4733:
4729:
4721:
4720:
4719:
4703:
4699:
4695:
4690:
4686:
4682:
4677:
4673:
4664:
4640:
4636:
4632:
4627:
4623:
4619:
4614:
4610:
4602:
4598:
4594:
4589:
4585:
4578:
4573:
4569:
4561:
4560:
4559:
4543:
4539:
4535:
4530:
4526:
4522:
4517:
4513:
4504:
4499:
4485:
4482:
4477:
4473:
4469:
4464:
4460:
4456:
4451:
4447:
4438:
4432:
4419:
4415:
4411:
4402:
4398:
4394:
4389:
4385:
4381:
4376:
4372:
4365:
4359:
4353:
4319:
4314:
4310:
4306:
4301:
4296:
4292:
4288:
4283:
4278:
4274:
4267:
4257:
4246:
4240:
4234:
4229:
4225:
4218:
4213:
4202:
4196:
4190:
4185:
4181:
4174:
4169:
4158:
4152:
4146:
4141:
4137:
4127:
4121:
4106:
4105:
4104:
4102:
4084:
4078:
4074:
4065:
4061:
4060:
4052:
4049: â
4045:
4038:
4035: â
4031:
4024:
4021: â
4017:
4010:
4003:
3999:
3982:
3962:
3933:
3929:
3925:
3922:
3919:
3917:
3909:
3905:
3898:
3884:
3882:
3874:
3870:
3861:
3858:
3854:
3847:
3833:
3831:
3823:
3819:
3809:
3804:
3801:
3797:
3782:
3780:
3775:
3771:
3742:
3741:
3740:
3738:
3732:which equals
3716:
3712:
3703:
3699:
3695:
3690:
3686:
3682:
3677:
3673:
3666:
3651:
3650:
3649:
3647:
3642:
3626:
3622:
3599:
3595:
3572:
3568:
3522:
3518:
3514:
3509:
3505:
3501:
3496:
3492:
3488:
3464:
3463:
3462:
3460:
3455:
3439:
3435:
3410:
3406:
3397:
3393:
3389:
3384:
3380:
3373:
3368:
3364:
3356:
3355:
3354:
3353:
3346:
3342:
3335:
3330:
3328:
3321:
3314:
3307:
3303:
3302:
3297:
3296:
3291:
3290:
3283:
3281:
3260:
3256:
3252:
3249:
3246:
3243:
3239:
3230:
3226:
3222:
3219:
3216:
3211:
3207:
3203:
3200:
3197:
3192:
3188:
3184:
3181:
3171:
3170:
3169:
3166:
3160:
3154:
3149:
3143:
3141:
3137:
3133:
3129:
3125:
3121:
3118:
3114:
3110:
3105:
3102:
3098:
3089:
3080:
3079:information.
3078:
3074:
3070:
3066:
3062:
3058:
3054:
3050:
3046:
3044:
3040:
3036:
3032:
3027:
3023:
3019:
3001:
2997:
2993:
2971:
2967:
2963:
2941:
2937:
2933:
2924:
2920:
2916:
2911:
2909:
2905:
2901:
2897:
2893:
2889:
2885:
2881:
2877:
2861:
2858:
2855:
2852:
2849:
2846:
2843:
2840:
2837:
2834:
2831:
2828:
2825:
2822:
2817:
2813:
2809:
2806:
2801:
2797:
2793:
2790:
2785:
2781:
2777:
2768:
2766:
2762:
2758:
2754:
2750:
2746:
2730:
2727:
2722:
2718:
2714:
2711:
2706:
2702:
2698:
2695:
2690:
2686:
2682:
2668:
2666:
2661:
2656:
2652:
2650:
2646:
2642:
2638:
2634:
2630:
2626:
2622:
2618:
2612:
2610:
2606:
2603:
2599:
2595:
2584:
2564:
2555:
2552:
2547:
2539:
2536:
2532:
2524:
2521:
2517:
2507:
2504:
2500:
2491:
2488:
2483:
2475:
2472:
2468:
2458:
2455:
2451:
2443:
2440:
2436:
2427:
2424:
2419:
2412:
2407:
2398:
2388:
2387:
2386:
2383:
2381:
2377:
2371:
2361:
2359:
2354:
2352:
2346:
2337:
2334:
2330:
2327:
2323:
2319:
2310:
2301:
2299:
2291:
2275:
2271:
2248:
2244:
2221:
2217:
2194:
2191:
2187:
2159:
2156:
2152:
2146:
2143:
2139:
2133:
2130:
2127:
2123:
2119:
2115:
2111:
2108:
2101:
2100:
2099:
2097:
2081:
2074:
2051:
2047:
2043:
2040:
2036:
2032:
2027:
2022:
2018:
2012:
2008:
2002:
1998:
1992:
1989:
1984:
1980:
1969:
1964:
1960:
1952:
1951:
1950:
1933:
1928:
1922:
1917:
1912:
1905:
1900:
1895:
1888:
1883:
1878:
1872:
1867:
1864:
1861:
1852:
1846:
1843:
1840:
1831:
1821:
1820:
1819:
1816:
1796:
1764:
1749:
1743:
1737:
1730:
1726:
1715:
1704:
1700:
1696:
1691:
1687:
1677:
1670:
1666:
1655:
1647:
1643:
1639:
1630:
1622:
1618:
1611:
1602:
1596:
1587:
1581:
1578:
1572:
1569:
1558:
1552:
1542:
1541:
1540:
1538:
1537:
1520:
1511:
1507:
1502:
1481:
1477:
1466:
1455:
1451:
1447:
1442:
1438:
1428:
1421:
1417:
1406:
1398:
1394:
1390:
1381:
1373:
1369:
1362:
1353:
1347:
1338:
1332:
1329:
1323:
1320:
1309:
1303:
1293:
1292:
1291:
1290:
1286:
1285:magnetization
1262:
1256:
1253:
1250:
1244:
1239:
1231:
1228:
1222:
1211:
1208:
1205:
1199:
1186:
1185:
1184:
1182:
1160:
1157:
1154:
1148:
1145:
1139:
1136:
1130:
1119:
1116:
1113:
1107:
1094:
1093:
1092:
1091:of the flux:
1090:
1086:
1082:
1060:
1057:
1054:
1048:
1042:
1039:
1036:
1030:
1027:
1024:
1018:
1011:
1010:
1009:
1007:
1003:
987:
967:
953:
930:
926:
922:
917:
913:
905:
895:
892:
889:
886:
883:
875:
871:
866:
859:
856:
853:
850:
847:
839:
835:
828:
825:
822:
816:
813:
810:
807:
804:
783:
782:
781:
765:
761:
738:
734:
713:
710:
707:
687:
661:
658:
655:
652:
649:
646:
640:
637:
634:
627:
623:
615:
612:
606:
596:
595:
594:
590:
576:
536:
516:
508:
492:
472:
450:
446:
421:
417:
413:
407:
404:
399:
392:
386:
382:
376:
372:
366:
362:
358:
354:
350:
347:
344:
337:
333:
325:
322:
316:
306:
305:
304:
300:
284:
280:
257:
253:
243:
239:
229:
225:
223:
219:
215:
209:
207:
203:
198:
196:
192:
188:
184:
181:
177:
173:
169:
165:
162:
158:
154:
150:
146:
141:
139:
135:
131:
130:T2 relaxation
127:
123:
119:
109:
107:
104:
100:
96:
92:
88:
84:
80:
76:
72:
68:
67:MRI sequences
64:
60:
56:
47:
43:
41:
37:
33:DTI Color Map
30:
25:
22:Diffusion MRI
20:
9099:Neuroimaging
9045:Acute stroke
9011:Thermography
8766:scintigraphy
8756:Radionuclide
8744:pre-hospital
8590:Tractography
8579:
8509:Radiodensity
8411:calcium scan
8372:Quantitative
8082:
8078:
8068:
8043:
8039:
8029:
7986:
7982:
7976:
7959:
7955:
7949:
7914:
7910:
7900:
7879:
7872:
7837:
7833:
7772:(1): 20935.
7769:
7765:
7755:
7715:(1): 14982.
7712:
7708:
7698:
7663:
7659:
7649:
7634:
7601:
7597:
7590:
7553:
7549:
7496:(1): 10742.
7493:
7489:
7479:
7452:
7446:
7421:
7417:
7411:
7377:(3): 63â74.
7374:
7370:
7360:
7349:. Retrieved
7343:
7333:
7306:
7302:
7292:
7257:
7253:
7243:
7218:
7214:
7208:
7199:
7193:
7158:
7154:
7144:
7131:
7124:
7091:
7087:
7081:
7072:
7068:
7062:
7045:
7039:
7012:
7008:
6998:
6955:
6951:
6945:
6934:. Retrieved
6928:
6918:
6883:
6879:
6869:
6844:
6840:
6834:
6823:. Retrieved
6819:
6795:. Retrieved
6791:
6753:
6749:
6743:
6708:
6704:
6694:
6677:
6671:
6654:
6650:
6644:
6617:
6611:
6578:
6574:
6567:
6542:
6538:
6532:
6507:
6503:
6497:
6464:
6460:
6454:
6437:
6433:
6389:
6385:
6375:
6332:
6328:
6318:
6283:
6279:
6268:
6233:
6229:
6219:
6184:
6180:
6169:
6134:
6130:
6120:
6095:
6091:
6085:
6052:
6048:
6042:
6007:
6003:
5993:
5952:
5948:
5942:
5917:
5913:
5907:
5888:
5884:
5848:
5834:Tractography
5824:Connectogram
5811:tractography
5808:
5801:
5792:
5780:
5770:
5763:
5759:
5720:
5715:connectomics
5684:motor cortex
5676:tractography
5671:and a tumor.
5665:brain tumors
5646:
5622:
5601:
5582:white matter
5579:
5567:
5505:
5490:
5481:
5470:Please help
5465:verification
5462:
5444:Applications
5347:volume ratio
5346:
5344:
5149:
5147:
4981:
4979:
4831:
4829:
4662:
4660:
4502:
4500:
4433:
4342:
4100:
4063:
4057:
4050:
4043:
4036:
4029:
4022:
4015:
4008:
4001:
3954:
3733:
3731:
3645:
3643:
3539:
3458:
3456:
3425:
3351:
3344:
3340:
3333:
3331:
3326:
3319:
3312:
3305:
3299:
3293:
3289:eigenvectors
3287:
3284:
3279:
3277:
3164:
3158:
3152:
3144:
3131:
3128:microtubules
3119:
3106:
3094:
3072:
3068:
3064:
3060:
3056:
3052:
3047:
3043:eigenvectors
3038:
3034:
3030:
3025:
3021:
3017:
2912:
2907:
2903:
2899:
2895:
2891:
2887:
2883:
2879:
2875:
2769:
2756:
2752:
2748:
2674:
2657:
2653:
2648:
2644:
2640:
2636:
2632:
2628:
2624:
2620:
2616:
2613:
2608:
2604:
2601:
2590:
2581:
2384:
2373:
2357:
2355:
2351:tractography
2347:
2343:
2331:
2326:T2 weighting
2321:
2317:
2315:
2300:
2297:
2178:
2095:
2072:
2070:
1948:
1817:
1785:
1534:
1503:
1500:
1282:
1178:
1078:
959:
950:
679:
591:
437:
301:
235:
226:
222:tractography
210:
199:
191:brain tumors
186:
185:
176:white matter
167:
166:
153:white matter
142:
133:
117:
115:
112:Introduction
106:tractography
103:white matter
98:
94:
62:
58:
54:
53:
8934:Cardiac PET
8707:renal tract
8682:Gynecologic
8614:Techniques
8585:restriction
8560:Angiography
8543:Neurography
8489:Fluoroscopy
8435:Angiography
8416:angiography
8360:Techniques:
8321:Aortography
8311:Angiography
8291:Cystography
8281:Mammography
8223:Myelography
8218:Sialography
8187:radiography
8046:: 129â135.
7892:10027851300
7345:Radiopaedia
6930:Radiopaedia
6820:XRayPhysics
6686:10018514722
6335:(1): 4749.
5766:icosahedron
5688:spinal cord
5642:eigenvector
4663:planar case
4503:linear case
3295:eigenvalues
3278:We use the
3113:attenuation
3077:eigenvector
2919:diagonalize
2765:hyperboloid
2073:attenuation
1506:H.C. Torrey
157:grey matter
9083:Categories
8847:Full body:
8633:endoscopic
8607:Ultrasound
8536:functional
8333:Lymphogram
8328:Venography
8296:Arthrogram
7956:Biometrika
7351:2017-10-15
7075:: 459â461.
6973:2108/34611
6936:2017-10-15
6825:2017-10-15
6814:Hammer M.
6797:2018-03-15
6049:NeuroImage
5868:References
5829:Connectome
5784:tomography
5586:anisotropy
2665:Adolf Fick
2368:See also:
2179:where the
1089:divergence
180:anisotropy
9050:Pregnancy
8929:Brain PET
8897:gamma ray
8833:DMSA scan
8687:Obstetric
8580:diffusion
8575:Sequences
8553:perfusion
8445:Pulmonary
8392:Cone beam
8286:Pyelogram
8117:264099038
8101:0740-3194
8060:121103537
8040:Math. Ann
7999:CiteSeerX
7794:2045-2322
7729:2045-2322
7155:Neurology
7054:576214706
6923:Bhuta S.
6841:Radiology
6504:Radiology
6092:Radiology
5985:250787827
5854:classical
5598:diffusion
5415:λ
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5385:λ
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3148:Cartesian
3101:ellipsoid
3097:diffusion
2923:Cartesian
2761:ellipsoid
2658:In 1848,
2402:¯
2304:ADC image
2294:Grayscale
2124:∑
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7812:34686721
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5756:Research
5704:dementia
5570:oncology
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242:moieties
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9094:Tensors
9089:Imaging
8967:Optical
8949:PET-MRI
8729:Carotid
8724:Scrotal
8618:doppler
8548:Cardiac
8457:Thyroid
8401:Targets
8352:CT scan
8021:6080176
7991:Bibcode
7941:2048187
7864:1368461
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7774:Bibcode
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6407:8928440
6386:Obesity
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6337:Bibcode
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5922:Bibcode
5899:8814916
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2810:c
2807:+
2802:2
2798:y
2794:b
2791:+
2786:2
2782:x
2778:a
2757:c
2753:b
2749:a
2731:1
2728:=
2723:2
2719:z
2715:c
2712:+
2707:2
2703:y
2699:b
2696:+
2691:2
2687:x
2683:a
2609:z
2605:y
2565:|
2556:z
2553:z
2548:D
2540:z
2537:y
2533:D
2525:z
2522:x
2518:D
2508:z
2505:y
2501:D
2492:y
2489:y
2484:D
2476:y
2473:x
2469:D
2459:z
2456:x
2452:D
2444:y
2441:x
2437:D
2428:x
2425:x
2420:D
2413:|
2408:=
2399:D
2358:b
2276:z
2272:G
2249:y
2245:G
2222:x
2218:G
2195:j
2192:i
2188:b
2160:j
2157:i
2153:D
2147:j
2144:i
2140:b
2134:j
2131:,
2128:i
2116:e
2112:=
2109:A
2082:A
2052:0
2048:D
2044:b
2037:e
2028:D
2023:3
2019:t
2013:2
2009:G
2003:2
1993:3
1990:1
1981:e
1970:M
1965:=
1961:M
1934:,
1929:]
1923:1
1918:0
1913:0
1906:0
1901:1
1896:0
1889:0
1884:0
1879:1
1873:[
1865:D
1862:=
1853:I
1844:D
1841:=
1832:D
1797:D
1765:M
1750:D
1738:+
1731:1
1727:T
1716:k
1710:)
1705:0
1701:M
1692:z
1688:M
1684:(
1671:2
1667:T
1656:j
1648:y
1644:M
1640:+
1631:i
1623:x
1619:M
1603:B
1588:M
1579:=
1573:t
1570:d
1559:M
1553:d
1521:M
1482:1
1478:T
1467:k
1461:)
1456:0
1452:M
1443:z
1439:M
1435:(
1422:2
1418:T
1407:j
1399:y
1395:M
1391:+
1382:i
1374:x
1370:M
1354:B
1339:M
1330:=
1324:t
1321:d
1310:M
1304:d
1263:.
1260:)
1257:t
1254:,
1251:x
1248:(
1240:2
1232:D
1229:=
1223:t
1215:)
1212:t
1209:,
1206:x
1203:(
1164:)
1161:t
1158:,
1155:x
1152:(
1149:J
1137:=
1131:t
1123:)
1120:t
1117:,
1114:x
1111:(
1064:)
1061:t
1058:,
1055:x
1052:(
1043:D
1037:=
1034:)
1031:t
1028:,
1025:x
1022:(
1019:J
988:J
936:)
931:2
927:b
918:1
914:b
910:(
906:/
902:]
899:)
896:z
893:,
890:y
887:,
884:x
881:(
876:1
872:S
867:/
863:)
860:z
857:,
854:y
851:,
848:x
845:(
840:2
836:S
832:[
823:=
820:)
817:z
814:,
811:y
808:,
805:x
802:(
798:C
795:D
792:A
766:2
762:b
739:1
735:b
714:C
711:D
708:A
688:D
665:)
662:C
659:D
656:A
650:b
644:(
635:=
628:0
624:S
619:)
616:E
613:T
610:(
607:S
577:D
517:G
473:S
451:0
447:S
422:]
418:D
414:)
408:3
393:(
387:2
377:2
373:G
367:2
355:[
345:=
338:0
334:S
329:)
326:E
323:T
320:(
317:S
285:2
281:T
258:1
254:T
97:(
57:(
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