The fractional anisotropy (FA) value for Diffusion Tensor Imaging is widely used to determine the anisotropy of diffusion in a given voxel. As the FA value is based on the tensor's eigenvectors it is not possible to calculate this quantity for HARDI diffusion profiles. In this paper we introduce an anisotropy index for HARDI data that utilizes the L2-norm as the most natural notion of distance for square-integrable functions on the two-sphere such as HARDI diffusion profiles and show that it is the limit of the generalized fractional anisotropy (GFA) index. Our index is well-defined and rotationally invariant and thus resolves the unsatisfactory issues with the GFA index.
The gray matter in the human brain comprises about 10 billion neuronal cells that use their axons to transmit information. These axons form bundles called neuronal fiber tracts that connect functional areas.
The localization of white matter tracts is of great interest for neurological research about brain structure and function, as well as for neurosurgery to preserve important tract systems during surgical intervention.
The noninvasive and in vivo reconstruction of fiber tracts is possible with Diffusion MRI. This is a medical imaging modality that is sensitive to the random thermal movement of water molecules. This information allows for inference about the structure of the tissue in the human brain because diffusion tends to be hampered in directions orthogonal to fiber bundles.
A widely used model to describe the measured diffusion is the diffusion
tensor [
The fractional anisotropy value (FA) is a scalar measure derived from the
diffusion tensor and quantifies the anisotropy of diffusion. As diffusion tends
to be more anisotropic in white matter this value can be applied to distinguish
white matter from gray matter within the brain. Tracking algorithms therefore
often operate with an FA threshold to prevent fiber tracking algorithms from
leaving regions of white matter as in [
However, the diffusion tensor can only resolve a single diffusion direction
per voxel. To overcome this deficit high angular resolution diffusion imaging
(HARDI) techniques such as Q-Ball imaging [
The FA value is based on the eigenvalues of the diffusion tensor. Therefore,
it is not possible to calculate the FA value of a general ADC profile due to its
lack of eigenvalues. Several anisotropy indices for HARDI diffusion profiles can
be found in the literature, such as the indices of Frank [
In this paper we generalize the popular GFA index [
This section starts with the definition of the GFA index and its drawbacks. Thereafter the L-index is defined. 2.1
The GFA index
The GFA index defined by Tuch [
i=1(f (xi) − ⟨f ⟩n)2 (n − 1) ∑n i=1 f (xi)2 (1) where ⟨f ⟩n = n1 ∑n
i=1 f (xi) is the mean value.
This index is automatically scaled to the unit interval and maps isotropic ODFs to zero. In the special case of diffusion tensors, GFA reduces to the FA value if the chosen directions happen to be the eigenvectors of the tensor.
The GFA index has, however, some severe drawbacks. First of all, it is not well-defined as it strongly depends on the number and choice of the directions in which the function is evaluated. Secondly, GFA(f ) is not rotationally invariant as one would expect from an anisotropy index. 2.2
The L-Index The basic idea of our anisotropy index is to measure how spherical the ADCprofile is. The ODF is expressed as a linear combination of spherical harmonics that are an orthonormal basis of the Hilbert space L2 of square-integrable functions. The natural measure of distance in this space is the L2-norm ∥·∥2.
If the mean value ⟨f ⟩ of f defined by ⟨f ⟩ = 2π2 1 ∫ and the corresponding constant function (which we also denote by ⟨f ⟩) is taken as the closest approximation of f by a sphere, the L2 distance ∥f − ⟨f ⟩∥2 between f and ⟨f ⟩ contains the information about how close f resembles a sphere. Normalizing with the L2 norm of f yields the L-index
The main result of this paper is that the L-index is the limit of the GFAn index as n increases if one chooses the xi as rectilinear grid points in the [0, π) × [0, 2π) domain.
Theorem 1. limn!1 GFAn(f ) = L(f ) Proof. Let f : [0, a] × [0, b] → R be a real valued function on a rectangle. Define grid points xi(jn) := (i na , j nb ), i, j = 0, . . . , n for each n ∈ N Furthermore let n n Φ(f, n) = ∑ ∑ nab2 f (xi(jn))2 Now consider the characteristic functions χAi(jn) : [0, a] × [0, b] → {0, 1} for the sets Ai(jn) = [(i − 1) na , i a ) × [(j − 1) nb , j nb ). The characteristic function χA of a n set A is defined by χA(x) = 1 if x ∈ A and χA(x) = 0 otherwise. Then we have Φ(f, n) = ∫ a ∫ b n n
∑ ∑ f (xi(jn))2χAi(jn) (x, y) dy dx 0 0 i=1 j=1 (3) (4) (5) 20 40 60 80 100 120 20 40 60 80 100 120 20 40 60 80 100 120 20 40 60 80 100 120 which converges to ∥f ∥22 = ∫0a ∫0b f (x, y)2dydx as the sum describes a step function that converges to f 2.
A similar argument shows that limn!1⟨f ⟩n = ⟨f ⟩. Together this yields the
convergence because
lim GFAn(f ) = lim
n!1 n!1
√ n
n − 1
√Φ(f − ⟨f ⟩n, n) = ∥f − ⟨f ⟩∥2
√Φ(f, n) ∥f ∥2
(6)
So basically GFAn(f ) is an approximation to the true anisotropy value L(f )
using the rectangle rule for numerical integration. Note that GFAn(f ) ∈ [
Application to human brain data The Diffusion MRI dataset used in this work was provided for the IEEE Visualisation Contest 2010. It is courtesy of Prof. B. Terwey, Klinikum Mitte, Bremen, Germany and was acquired on a Siemens 3T Verio MR scanner. 30 gradient directions and two averages per gradient were acquired with b = 1000 s/mm2.
ADC-profiles were reconstructed with maximal order M = 6 as in [
Figure 1 shows the FA values (a) and L-index values (b) of a single slice in the dataset. Note that due to the fact that the distribution of the L-index is relatively narrow the values have been thresholded at 0.5 for contrast enhancement. From the histograms over the whole dataset for FA values and the L-index, we conclude that 0.5 is a reasonable threshold (Fig. 2).
7000 6000 5000 4000 3000 2000 1000 We showed that the L-index resolves the unsatisfactory issues with the GFA index not being well-defined nor rotationally invariant. Moreover the GFA index can be interpreted as approximation of the L-index by numerical integration.
Due to the strong linear correlation between the FA value and the L-index the latter can be used to segment white matter in the brain or serve as a stopping criterion for tracking algorithms based on HARDI data.