High angular resolution diffusion imaging (HARDI) was introduced to overcome a major limitation of conventional diffusion tensor imaging (DTI). Methods that describe the apparent diffusion coefficient (ADC) profile using high-order tensors is one of the estimation methods associated with HARDI data [1]. Several anisotropy indices such as generalized anisotropy (GA) and generalized fractional anisotropy (GFA) for HARDI profiles have been reported in the literature [2, 3, and 4]. Since GFA depends on the number of tessellation of the sphere, it is not well-defined and GA overestimates the white matter anisotropy. We propose here a well-defined and rotationally invariant anisotropy measure for HARDI data using the L²-norm of the diffusivity function of the higher order tensor. This is an extension of the conventional fractional anisotropy measure (FA). The new proposed high order tensor fractional anisotropy measure (HFA) was compared with GA and GFA using the contrast-to-noise ratio (CNR)[5] and coefficient of variation (CV) as the metrics. The results of different anisotropic measures were compared for single, two, and three crossing fiber regions in human brain.

The HARDI data was acquired on a Philips 3T scanner on five males with no history of neurological or neuropsychiatric disorders. A32-channel head coil with a SENSE factor of 2 was used for data acquisition. Multi-slice, diffusion-weighted images were acquired using a single shot spin echo EPI sequence with 81 diffusion encoding directions. The sequence parameters were: FOV = 256x256 mm2, 44 contiguous slices with 3 mm slice thickness, TR/TE = 8235/72 ms, b-value = 1600 s/mm2. The new anisotropy measure, HFA, is defined based on L² distance of square-integrable functions over a sphere. The L²-distance between two square-integrable functions, f and g, over a sphere is given by [1]

$$\parallel f-g \parallel_2=\sqrt{\frac{1}{4\pi}\int_{S^2}(f-g)^2d\sigma}$$

The exact formula for the integration of a monomial $$P(X)=x^{\alpha_1}y^{\alpha_2}z^{\alpha_3}$$ is given by [6]

$$\int_{S^2}Pd\sigma = \begin{cases}0 & if&some& \alpha_j &are& odd\\\frac{2\Gamma(\beta_1)\Gamma(\beta_2)\Gamma(\beta_3)}{\Gamma(\beta_1+\beta_2+\beta_3)} & if &all &\alpha_j &are& even\end{cases}$$

where α_1,α_{2, and} α₃ are nonnegative integers , β_j = (α_j+1)/2, Γ is the gamma function and σ is a measure on the surface of a sphere .

Similar to the FA definition $$FA=\sqrt{\frac{3}{2}\frac{\sum_1^3(\lambda_i-\lambda)^2}{\sum_1^3\lambda_i^2}}=\sqrt{\frac{3}{2}}\frac{\parallel D-<D>\parallel_2}{\parallel D\parallel_2}$$

based on the L²-distance HFA can be expressed as

$$HFA=\frac{\parallel D- <D>\parallel_2}{\parallel D\parallel_2}$$

where D is diffusivity function and <D> is the mean diffusivity value as defined in [4] and I^S is the totally symmetric identity tensor [3, 7]. For fourth order tensors, I^S is expressed as in [7]. To quantitatively compare various 4^th and 6^th order tensor anisotropy measures (HFA, GFA and GA) we used the contrast-to- noise ratio (CNR) which is defined in [5] and coefficient of variation.

The various anisotropy measures were compared for three regions: single, two and three fiber crossings (Fig 1). From Fig.2 we can see that HFA, GFA and GA values derived from the 6^th order are similar to those based on the 4^th order tensor for all three different fiber crossing regions (Fig. 1). For all the three anisotropic measures based on the HARDI data the CNR between the gray and white matter regions for order 4 was found to be superior or equal to that of order 6 (Fig. 3). The FA derived from conventional tensor model of HARDI data has better CNR than the others for single fiber bundle (splenium) whereas GA had a better CNR than others in two and three crossing regions (Fig.3). This might be because of the over estimation of GA in white matter regions. GA has the lowest CV in all three white matter regions and FA has the highest CV in two and three crossing fiber bundle regions (Fig.4). Landgraf et al [8] have shown that the anisotropic measure defined using the L²-norm of ODF function is the limiting value of the generalized fractional anisotropy (GFA) based on the ODF values. For most of the comparisons we performed, HFA and GFA behaved very similarly. Unlike GFA, HFA that we introduced in this work is well-defined as it only depends on the independent tensor elements and it is also rotationally invariant. For all the three anisotropic measures of all three different white matter regions the CNR of the 4^th order is greater than that of the 6th order. HFA and GFA have shown better CNR than FA and GA in two and three crossing regions. The CV and CNR of HFA and GFA are very similar. The results described above were very similar across all the five subjects, suggesting the robustness of the proposed methodology.