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326 PRICE D will affect the measured echo attenuation Ž 114, 116 . . The situation with anisotropic restricted diffusion is more complicated, and we will illustrate this with reference to diffusion in a cylinder with an arbitrary Ž polar. angle, , between the symmetry axis of the cylinder and the static magnetic field Ž which is also the direction of the gradient. Ž Fig. 9 . . Such a cylinder can be thought of, for example, as a simplistic model of a muscle-fiber cell. The SGP solution for this geometry is given by Ž 117 . 2 4 2 n 2 nm km 2K R Ž 2qR. sin Ž 2. 1 Ž 1. cosŽ 2 qL cos . EŽ q,. Ý Ý Ý 2 2 2 2 2 L nRL 2qR cos 2qR sin m n0 k1 m0 2 Ž . Ž . 2 2 Ž . 2 Ž 2 2. km km 2 n 2 2 km m ½ ž R / ž L / 5 J Ž 2qR sin . exp D 110 where L is the length of the cylinder, R is the radius of the cylinder, and km is the kth nonzero root of the equation J Ž . m km 0, where J is the Bessel function of the first kind, and the constant K nm depends on the values of the indexes n and m according to K nm 1 if nm0 Knm 1 if nm0orm0and n 0 Knm 1 if n, m0. 111 Now, the mathematical complexity is no concern to us here and the point that we wish to emphasize is the dependence; thus, in contradistinction to the cases of free diffusion and diffusion with a sphere, in an anisotropic system the spinecho attenuation is now a function of the direction of the gradient. In fact, if we had a less symmetric geometry Ž e.g., an elliptic cylinder . , then the equation for Eq, Ž . should also be dependent on the azimuthal angle, . If we set 0, then the solution given by Eq. 110 reduces, as expected, to the solution for diffusion between planes Ži.e., Eq. 92 and noting that L2R . . Similarly, if 2, Eq. 110 reduces to the solution for diffusion in a cylinder Ž 110 . , EŽ q,. 2 22qR Ž . Ý Ý k1 m0 2 2 2 K J Ž 2qR. exp Ž R. 0m km m km D4 2 2 2 2 2 km km Ž 2qR. Ž m . . 112 The echo-attenuation curves for diffusion in a cylinder versus are plotted for three different values of in Fig. 10. The long time limiting formula for the cylinder is given by Ž 117 . Ž . E q, 2 8R 1cosŽ 2 qL cos .J Ž 2 qR sin . 1 . 4 2 2 Ž 2qR. L Ž cos sin . 113 When 0, this, of course, reduces to the long time for diffusion between planes as given in Eq. 93 Ž n.b. L 2 R . , and when 2, this reduces to Ž 117 . 2J Ž 2qR. 1 EŽ q,. . 114 2 Ž 2qR. The echo-attenuation curves for diffusion in a cylinder versus qR are plotted for three different values of in Fig. 11. Clearly, the dependence on the attenuation curves and the diffraction patternsor alternately, we can think of this as the orientation of D with respect to the gradientprovides an additional structural probe. If the restricted diffusion effects are not accounted for and free diffusion is assumed, and Eq. 104 , which is valid only for free diffusion, is used to analyze the attenuation data, then D is really an apparent diffusion ten- 2 2
sor, D app. When a single effective diffusion tensor is estimated for the entire sample, it is sometimes referred to as ‘‘diffusion tensor MR spectroscopy,’’ and when, as is commonly the case in imaging studies, the estimation is performed for each voxel, it is referred to as ‘‘diffusion tensor MR imaging.’’ From the discussion on restricted diffusion above it should be clear that D app is per- haps better written as D Ž . app , since it will be observation time dependent. For sufficiently short Figure 9 Schematic diagram of diffusion in a cylinder. The cylinder is of length L and radius R with the symmetry axis of the cylinder Ž . subtends an angle with the direction of the gradient and static field, which is taken to be z in the present case. The laboratory or gradient frame is given by Ž x, y, z . , where z coincides with the gradient direction. The director frame for the cylinder is given by Ž x, x, z . , where z coincides with the symmetry axis of the cylinder. If this were an elliptic cylinder for example, the director frame would be uniquely determined. Clearly, if 0, the two reference frames coincide. If 0 and a PFG diffusion measurement is performed, the spin-echo attenuation will be described by diffusion between planar boundaries Ži.e., Eq. 92 . . Conversely, if 2, the spin-echo attenuation will be described by diffusion within a cylinder ŽEq. 112 . . PULSED-FIELD GRADIENT NMR 327 values of such that the diffusion of the probe species is unaffected by the boundaries, D Ž . app would be isotropic, whereas for larger , itbecomes increasingly anisotropic. This can be easily visualized from the divergence of the echo-signal attenuation plots for different values of versus given in Fig. 10. For the case of diffusion within a cylinder Ž Fig. 9 . , D would be given by app xx 0 yy 0 Dzz D 0 0 D 0 D 0 115 app where D D xx yy owing to the axial symmetry of the cylinder. We remark that D , D , and D xx yy zz are, of course, eigenvalues of the matrix D app and are often termed the ‘‘principal diffusivities.’’ Ideally, it is possible to determine the dimensions of the restricting geometry from the restricted displacementfor example, in the case of the cylinder, the long axis of the cylinder gives the largest apparent diffusion coefficientbut this requires prior knowledge of the sample orientation so that it is possible to align the director frame of reference coincident with the gradient frame of reference in the PFG experiment. Sometimes it is useful to represent D app graphically by a diffusion ellipsoid Ž 45. Ž Fig. 12 . , which can be constructed using ž / ž / ' ' ž ' / xx yy zz 2 2 2 x y z 1. 2D 2D 2D 116 Equation 116 can be derived starting from Eq. 27 , but substituting the Dapp for the scalar D. We note from Eq. 33 that the major axes of the ellipsoids in Eq. 116 are the mean diffusion 2 distances Ž i.e., '² x : '2 D , etc. . xx . As Dapp becomes more anisotropic the ellipsoid becomes more prolate. For example, in muscle fiber, the effective diffusion ellipsoid reflects the fiber orientation and the mean diffusion distances. Generally, the relative alignment between the gradient and director frames is not known, the diffusion measurement returns a mixture of the different elements of the diffusion tensor, and the orientational dependence becomes a problem.
Page 1 and 2: Pulsed-Field Gradient Nuclear Magne
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Page 5 and 6: the gradient oriented along the z-a
Page 7 and 8: ent pulses cancel and all spins ref
Page 9 and 10: ent echo, on the other hand, refocu
Page 11 and 12: For the case of Ž three-dimensiona
Page 15 and 16: and S corresponds to the value of
Page 17 and 18: every time a pulse is applied. Thu
Page 19 and 20: Finally, noting the standard integr
Page 21 and 22: Now, if we integrate Eq. 83 over al
Page 23 and 24: In the long time limit 1 Ž i.e.,
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Page 33 and 34: Experimental aspects of PFG NMR wil
Page 35 and 36: 32. P. C. M. Van Zijl and C. T. W.
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