diffusivity

Diffusivity (from the Latin diffundere " expand dispel; emanate leave ") is the property of a material to allow the spread of solutes. The propagation process itself is called diffusion.

The term stands for the diffusion coefficient on the one hand, but especially in the solid material in the figurative sense of the ability to allow diffusion. In the transferred meaning it is used especially in the neurosciences with respect to body tissue and tissue fluid contained in it, but also in hydrology (groundwater), solid-state physics and thermodynamics ( thermal diffusivity ).

In contrast to diffusion in liquid or gaseous media to a solute in the solid state can be (for example, due to crystal lattice structure ), but also liquid in the solid material is not ( for example, because the ground layers, nerve fibers ) is generally in all directions equally spread. Certain directions are preferred and there is anisotropy. In this case is at the first Fick law, instead of the diffusion coefficient of the diffusion tensor ( a 3 x 3 symmetric matrix with non-negative real eigenvalues ​​).

The direction-dependent propagation in tissue is called the apparent diffusion, the main diagonal elements of the diffusion tensor as the apparent diffusion coefficient ( ADC also for apparent diffusion coefficient ). For a quantitative description of the apparent diffusion and the associated anisotropy different metrics are used, which are formed among others from the eigenvalues ​​of the diffusion tensor. Thus, measures such as the fractional anisotropy ( FA), how much the eigenvalues ​​different from one another, ie, how much the diffusivity is different in different directions. The fractional anisotropy and relative anisotropy ( RA) treated with medium, axial and radial diffusivity below.

In the humanities, the term is transferred to the degree needed in the spread or something can be moved or applied to other areas.

Background

Diffusion of solutes is described by Fick's laws. The first of these two laws

Indicates that the particle flux is linearly dependent on the gradient of the concentration, the larger of two locations, the difference in concentration of the solute is, the more particles will flow from the higher to lower concentration. The coefficient is in this case called the diffusion coefficient or the diffusivity.

This description can be applied for various transport processes, ranging from chemical substances dissolved in liquids, via electrical line up to the heat conduction.

However, the simple description of the one-dimensional law above always breaks down if the transport can take place in certain directions preferred, but in others hindered. We then speak of directionality or anisotropy.

Possible reasons for an anisotropy can be that certain sites can accommodate larger concentrations of the solute in the solid state. Thus, in steel can accumulate at the crack tip dislocations higher concentrations of dissolved hydrogen. Even in the description of heat conduction play differences in the thermal capacity of the material involved, why and thermal conductivity must be made between the thermal diffusivity ( also called the thermal diffusivity ).

Much more dramatic, however, the directional dependence is found in body tissue in which impede the free flow of tissue fluid already in healthy tissue cell walls.

In case of anisotropy occurs in the place of the above one-dimensional Act

In the now

A 3 × 3 matrix, referred to as the diffusion tensor. This matrix is symmetric and therefore has only six independent components in a variant of Voigt 's notation as a vector

Can be written.

The eigenvalues ​​of the diffusion tensor with, and referred to, where λ1 ≥ λ2 ≥ λ3 ≥ 0. Plotting the apparent diffusion coefficient on the respective spatial directions, so you get a spheroid in general.

Measurement and Applications

The measurement of the diffusivity is effected by means of magnetic resonance imaging (MRI) at the so-called diffusion tensor imaging ( DTI), such that the brain ³ 0.5-8 mm voxel is located in each voxel and in at least six different directions, depending on the diffusion coefficient of self diffusion is measured by water in the tissue fluid. Such a diffusion coefficient as well as the apparent diffusion coefficient (also referred to ADC for apparent diffusion coefficient ). Due to this at least 6 values ​​per voxel, a diffusion tensor is calculated for each voxel. For this voxel-wise measures of the diffusivity or anisotropy are derived and plotted for B using tensor glyphs or color value images, or used for further analysis, such as the creation of models of fiber tracts ( tractography ).

Due to the diffusivity can be concluded that other properties of the fabric. Thus, conclusions about the gross structure (eg, fibrous nature ) and thus on age- or disease- related changes are drawn (changes in myelination of axons, for example ) of brain tissue. Since the movement of the tissue fluid in a fibrous tissue in the direction of the tissue fibers is substantially greater than transverse to the fiber direction can be made on the basis of models of Diffusivitätsmessungen fiber tracts in the brain ( tractography ). It can also be found at what point can flow freely due to the rupture of a membrane liquid.

Metrics

The diffusivity is described by the eigenvalues ​​of the diffusion tensor, in which case the mean diffusivity, axial diffusivity and the radial diffusivity are highlighted.

To describe the degree of anisotropy is formed from the eigenvalues ​​beyond the scale-invariant conditions

• Fractional anisotropy
• Relative anisotropy
• Volume ratio

Mean diffusivity

The mean diffusivity ( diffusion coefficient of the medium ) is defined as

It can theoretically be any size, which would mean according to arbitrarily large diffusivity. for the total absence of apparent diffusion ( no Brownian motion of molecules measured ).

Axial diffusivity

The axial diffusivity is the largest of the three eigenvalues.

Illustrative: The axial length of the diffusivity Diffusionstensorellipsoids and describes the magnitude of the apparent diffusion in the main direction, which is the direction of greatest flexibility. It is a marker of axonal integrity. The larger the axial diffusivity, the more intact the axons.

Radial diffusivity

The radial diffusivity is the average of the two eigenvalues ​​smaller, so more illustrative: The radial diffusivity? T ( t is transverse) is the average thickness of Diffusionstensorellipsoids in the longitudinal center and describes the average apparent diffusion in the plane perpendicular to the main direction. It is, for example, a marker for the integrity of the myelin. Demyelination increases the radial diffusivity.

Fractional anisotropy

The fractional anisotropy is defined as

Illustrative: The fractional anisotropy FA is the standard deviation of eigenvalues ​​divided by the mean value of the eigenvalue square. It is a measure of the directionality of the apparent diffusion. It's FA = 0 in the case of complete isotropy, ie λ1 = λ2 = λ3 > 0 At maximum anisotropy (full directionality of diffusion in just one direction) = λ3 are, so if λ1 > 0 and λ2 = 0, is FA = 1 for example, is a marker of the anatomy of the white matter of the brain, the greater the FA, the intact white matter.

Relative anisotropy

Anisotropy is defined as the relative

Illustrative: The relative anisotropy RA is the standard deviation of the eigenvalues ​​(calculated as the standard deviation of a population, that is, with n = 3 as the divisor ) divided by the mean of the eigenvalues. It is a measure of the directionality of the apparent diffusion. It is RA = 0 in the case of complete isotropy, ie λ1 = λ2 = λ3 > 0 At maximum anisotropy (full directionality of diffusion in just one direction) λ3 = 0, ie, if λ1 > 0 and λ2 = RA =.

Volume ratio

The volume ratio is defined as

Illustrative: The volume ratio VR is the product of the eigenvalues ​​divided by the third power of the average value of the eigenvalues. It is a measure of the directionality of the apparent diffusion. It is VR = 1 in the case of complete isotropy, ie λ1 = λ2 = λ3 > 0, if at least the smallest of the eigenvalues ​​is equal to 0, then VR = 0 This corresponds to the situation that the diffusion only in precisely one direction ( λ2 = λ3 = 0) or only in one plane ( λ3 = 0) takes place.