904:
880:
892:
128:
359:
673:
856:
constraints switch direction on a smaller scale than the measured one, then the measured FA will be attenuated. For example, the brain can be thought of as a fluid permeated by many fibers (nerve axons). However, in most parts the fibers go in all directions, and thus although they constrain the diffusion the FA is
138:
847:
since the diffusion is isotropic, and there is equal probability of diffusion in all directions. The eigenvectors and eigenvalues of the
Diffusion Tensor give a complete representation of the diffusion process. FA quantifies the pointedness of the ellipsoid, but does not give information about which
918:
diffusion processes, which has been found to be inadequate in accurately representing the true diffusion process in the human brain. Due to this, higher order models using spherical harmonics and
Orientation Distribution Functions (ODF) have been used to define newer and richer estimates of the
855:
unless the diffusion process is being constrained by structures such as network of fibers. The measured FA may depend on the effective length scale of the diffusion measurement. If the diffusion process is not constrained on the scale being measured (the constraints are too far apart) or the
455:
31:
process. A value of zero means that diffusion is isotropic, i.e. it is unrestricted (or equally restricted) in all directions. A value of one means that diffusion occurs only along one axis and is fully restricted along all other directions. FA is a measure often used in
766:
872:
can also exhibit anisotropic diffusion because the needle or plate-like shapes of their molecules affect how they slide over one another. When the FA is 0 the tensor nature of D is often ignored, and it is called the diffusion constant.
354:{\displaystyle {\text{FA}}={\sqrt {{\frac {3}{2}}\left({\frac {(\lambda _{1}-{\hat {\lambda }})^{2}+(\lambda _{2}-{\hat {\lambda }})^{2}+(\lambda _{3}-{\hat {\lambda }})^{2}}{\lambda _{1}^{2}+\lambda _{2}^{2}+\lambda _{3}^{2}}}\right)}}}
919:
anisotropy, called
Generalized Fractional Anisotropy. GFA computations use samples of the ODF to evaluate the anisotropy in diffusion. They can also be easily calculated by using the Spherical Harmonic coefficients of the ODF model.
835:(this rarely happens in real data), in which case D has only one nonzero eigenvalue and the ellipsoid reduces to a line in the direction of that eigenvector. This means that the diffusion is confined to that direction alone.
668:{\displaystyle {\text{FA}}={\sqrt {{\frac {1}{2}}\left({\frac {(\lambda _{1}-\lambda _{2})^{2}+(\lambda _{2}-\lambda _{3})^{2}+(\lambda _{3}-\lambda _{1})^{2}}{\lambda _{1}^{2}+\lambda _{2}^{2}+\lambda _{3}^{2}}}\right)}},}
444:
822:
118:
684:
124:. The eigenvectors give the directions in which the ellipsoid has major axes, and the corresponding eigenvalues give the magnitude of the peak in each eigenvector direction.
969:
J. Cohen-Adad, M. Descoteaux, S. Rossignol, RD Hoge, R. Deriche, and H. Benali (2008). "Detection of multiple pathways in the spinal cord using q-ball imaging".
935:
Basser, P.J. & Pierpaoli, C. (1996). "Microstructural and physiological features of tissues elucidated by quantitative-diffusion-tensor MRI".
367:
37:
864:
the fibers are aligned over a large enough scale (on the order of a mm) for their directions to mostly agree within the resolution element of a
777:
952:Özarslan, E. Vemuri, B.C. & Mareci, T. H. (2005). "Generalized scalar measures for diffusion MRI using trace, variance, and entropy".
903:
70:
843:
This can be visualized with an ellipsoid, which is defined by the eigenvectors and eigenvalues of D. The FA of a sphere is
827:
Note that if all the eigenvalues are equal, which happens for isotropic (spherical) diffusion, as in free water, the FA is
1023:
879:
891:
761:{\displaystyle {\text{FA}}={\sqrt {{\frac {1}{2}}\left(3-{\frac {1}{{\text{trace}}({\text{R}}^{2})}}\right)}}}
993:
53:
64:
A Diffusion
Ellipsoid is completely represented by the Diffusion Tensor, D. FA is calculated from the
1018:
865:
1013:
20:
8:
998:
33:
1008:
1003:
869:
861:
987:
439:{\displaystyle {\hat {\lambda }}=(\lambda _{1}+\lambda _{2}+\lambda _{3})/3}
914:
One drawback of the
Diffusion Tensor model is that it can account only for
49:
817:{\displaystyle {\text{R}}={\frac {\text{D}}{{\text{trace}}({\text{D}})}}}
45:
65:
24:
28:
915:
121:
127:
113:{\displaystyle \lambda _{1}\geq \lambda _{2}\geq \lambda _{3}}
41:
868:, and it is these regions that stand out in an FA image.
23:
value between zero and one that describes the degree of
851:
Note that the FA of most liquids, including water, is
780:
687:
458:
370:
141:
73:
816:
760:
667:
438:
353:
112:
985:
909:FA value of 0.6030, the DT matrix is diagonal()
885:FA value of 0.7698, the DT matrix is diagonal()
56:in 3 dimensions, normalized to the unit range.
771:where R is the "normalized" diffusion tensor:
946:
52:. The FA is an extension of the concept of
897:FA value of 0, the DT matrix is diagonal()
929:
446:being the mean value of the eigenvalues.
963:
831:. The FA can reach a maximum value of
126:
937:Journal of Magnetic Resonance, Series B
986:
13:
14:
1035:
954:Magnetic Resonance in Medicine,
902:
890:
878:
678:which is further equivalent to:
449:An equivalent formula for FA is
860:. In some regions, such as the
36:where it is thought to reflect
808:
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730:
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552:
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486:
425:
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377:
280:
273:
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54:eccentricity of conic sections
1:
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848:direction it is pointing to.
59:
7:
10:
1040:
1024:Magnetic resonance imaging
838:
131:Diffusion Tensor Schematic
17:Fractional anisotropy (FA)
866:magnetic resonance image
818:
762:
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440:
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132:
114:
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670:
441:
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130:
115:
778:
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139:
71:
994:Transport phenomena
652:
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814:
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120:of the diffusion
34:diffusion imaging
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1019:Medical imaging
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870:Liquid crystals
862:corpus callosum
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61:
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44:diameter, and
9:
6:
4:
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2:
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43:
39:
38:fiber density
35:
30:
26:
22:
18:
1014:Neuroimaging
974:
970:
965:
957:
953:
948:
940:
936:
931:
913:
850:
844:
842:
826:
770:
677:
448:
363:
63:
50:white matter
16:
15:
66:eigenvalues
46:myelination
988:Categories
977:, 739-749.
971:NeuroImage
960:, 866-876.
943:, 209-219.
923:References
60:Definition
25:anisotropy
999:Diffusion
717:−
640:λ
622:λ
604:λ
582:λ
578:−
569:λ
543:λ
539:−
530:λ
504:λ
500:−
491:λ
417:λ
404:λ
391:λ
378:^
375:λ
329:λ
311:λ
293:λ
274:^
271:λ
265:−
256:λ
233:^
230:λ
224:−
215:λ
192:^
189:λ
183:−
174:λ
102:λ
98:≥
89:λ
85:≥
76:λ
29:diffusion
916:Gaussian
1009:Tensors
1004:Imaging
839:Details
122:tensor
42:axonal
21:scalar
797:trace
727:trace
364:with
27:of a
19:is a
941:111
48:in
990::
975:42
973:,
958:53
956:,
939:,
690:FA
461:FA
144:FA
40:,
858:0
853:0
845:0
833:1
829:0
809:)
805:D
801:(
792:D
787:=
783:R
753:)
746:)
741:2
736:R
731:(
722:1
714:3
710:(
704:2
701:1
694:=
663:,
657:)
649:2
644:3
636:+
631:2
626:2
618:+
613:2
608:1
596:2
592:)
586:1
573:3
565:(
562:+
557:2
553:)
547:3
534:2
526:(
523:+
518:2
514:)
508:2
495:1
487:(
481:(
475:2
472:1
465:=
434:3
430:/
426:)
421:3
413:+
408:2
400:+
395:1
387:(
384:=
346:)
338:2
333:3
325:+
320:2
315:2
307:+
302:2
297:1
285:2
281:)
260:3
252:(
249:+
244:2
240:)
219:2
211:(
208:+
203:2
199:)
178:1
170:(
164:(
158:2
155:3
148:=
106:3
93:2
80:1
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