Tensor diffusion

Basser P J, Mattiello J and LeBihan D 1994 MR diffusion tensor speotrosoopy and imaging Biophys. J. 66 259-67... 

In order to overcome these problems, one approach, originally developed by Whitaker [420], Slattery [359], and Anderson and Jackson [17], involves the method of volume averaging. Using volume averaging theory, Whitaker and coworkers [193,264,268,337,436] found the effective diffusion tensor for a two-phase system to be given by... 

Trinh, S Locke, BR Arce, P, Effective Diffusivity Tensors of Point-Like Molecules in Anisohopic Media by Monte Carlo Simulation, Transport in Porous Media in review, 2001. 

This simple relaxation theory becomes invalid, however, if motional anisotropy, or internal motions, or both, are involved. Then, the rotational correlation-time in Eq. 30 is an effective correlation-time, containing contributions from reorientation about the principal axes of the rotational-diffusion tensor. In order to separate these contributions, a physical model to describe the manner by which a molecule tumbles is required. Complete expressions for intramolecular, dipolar relaxation-rates for the three classes of spherical, axially symmetric, and asymmetric top molecules have been evaluated by Werbelow and Grant, in order to incorporate into the relaxation theory the appropriate rotational-diffusion model developed by Woess-ner. Methyl internal motion has been treated in a few instances, by using the equations of Woessner and coworkers to describe internal rotation superimposed on the overall, molecular tumbling. Nevertheless, if motional anisotropy is present, it is wiser not to attempt a quantitative determination of interproton distances from measured, proton relaxation-rates, although semiquantitative conclusions are probably justified by neglecting motional anisotropy, as will be seen in the following Section. 

D. LeBihan, J. F. Mangin, C. Poupon, C.A. Clark, S. Pappata, N. Molko, H. Chabriat 2001, (Diffusion tensor imaging concepts and applications),/. Magn. Reson. Imag. 13, 534—546. 

Some micro- and mesoporous materials exhibit anisotropic pore structures, which may yield different values for the diffusivities in the three orthogonal spatial directions. In such systems, the self-diffusion should be described by a diffusion tensor rather than by a single scalar self-diffusion coefficient. By measuring over a... 

Fig. 3.1.4 Anisotropic self-diffusion of water in and filled symbols, respectively). The horizon-MCM-41 as studied by PFG NMR. (a) Depen- tal lines indicate the limiting values for the axial dence of the parallel (filled rectangles) and (full lines) and radial (dotted lines) compo-perpendicular (circles) components of the axi- nents of the mean square displacements for symmetrical self-diffusion tensor on the inverse restricted diffusion in cylindrical rods of length temperature at an observation time of 10 ms. / and diameter d. The oblique lines, which are The dotted lines can be used as a visual guide, plotted for short observation times only, repre-The full line represents the self-diffusion sent the calculated time dependences of the...

The spatial and temporal evolution of the concentration field is dependent on the velocity field vector v(r,t), the diffusion tensor D(r,t) and any reactions occurring in the system R(r,t). Non-dimensionalization of Eqn. (5.1.4) generates the Pedet... 

Key Words Carbon-13 spin relaxation, T, Measurements, Nuclear Overhauser effect, Rotation-diffusion tensor, HOESY experiments... 

If the considered molecule cannot be assimilated to a sphere, one has to take into account a rotational diffusion tensor, the principal axes of which coincide, to a first approximation, with the principal axes of the molecular inertial tensor. In that case, three different rotational diffusion coefficients are needed.14 They will be denoted as Dx, Dy, Dz and describe the reorientation about the principal axes of the rotational diffusion tensor. They lead to unwieldy expressions even for auto-correlation spectral densities, which can be somewhat simplified if the considered interaction can be approximated by a tensor of axial symmetry, allowing us to define two polar angles 6 and

symmetry axis of the considered interaction) in the (X, Y, Z) molecular frame (see Figure 5). As the tensor associated with dipolar interactions is necessarily of axial symmetry (the relaxation vector being... 

Fortunately, in the case of a rotational diffusion tensor with axial symmetry (such molecules are denoted "symmetric top"), some simplification occurs. Let us introduce new notations D// = Dz and D = Dx = Dy. Furthermore, we shall define effective correlation times ... 

It can be noticed that at least two independent relaxation parameters in the symmetric top case, and three in the case of fully anisotropic diffusion rotation are necessary for deriving the rotation-diffusion coefficients, provided that the relevant structural parameters are known and that the orientation of the rotational diffusion tensor has been deduced from symmetry considerations or from the inertial tensor. 

Figure 15 The model molecule used to demonstrate the possibilities of HOESY experiments in terms of carbon-proton distances and reorientational anisotropy. To a first approximation, the molecule is devoid of internal motions and its symmetry determines the principal axis of the rotation-diffusion tensor. Note that H, H,., H,-, H,/ are non-equivalent. The arrows indicate remote correlations.

Sullivan, E. V. and Pfefferbaum, A. diffusion tensor imaging in normal aging and neuropsychiatric disorders. Eur. J. Radiol. 45 244-255, 2003. 

Another rotational diffusion model known as the anisotropic viscosity model156,157 is very similar to the above model, and its main feature is to diagonalize the rotational diffusion tensor in the L frame defined by the director. A similar (but not the same) expression as Eq. (71) is J R(r)co)... 

How Can We Derive the Rotational Diffusion Tensor of a Molecule from Spin-Relaxation Data 293 ... 

Derivation of the Diffusion Tensor when Protein Structure is Known 295... 

The anisotropy of the overall tumbling will result in the dependence of spin-relaxation properties of a given 15N nucleus on the orientation of the NH-bond in the molecule. This orientational dependence is caused by differences in the apparent tumbling rates sensed by various internuclear vectors in an anisotropically tumbling molecule. Assume we have a molecule with the principal components of the overall rotational diffusion tensor Dx, Dy, and l)z (x, y, and z denote the principal axes of the diffusion tensor), and let Dx< Dy< Dz. 

Several approaches to determination of the overall rotational diffusion tensor from 15N relaxation data were suggested in the literature [15, 47, 49, 51-53]. The approach described here uses the orientational dependence of the ratio of spin-relaxation rates [49]... 

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