Computation of the fractional anisotropy and mean diffusivity maps without tensor decoding and diagonalization: Theoretical analysis and validation

Abstract

Diffusion tensor MRI (DT-MRIKey words: magnetic resonance imaging; diffusion tensor encoding; fractional anisotropy; relative anisotropy; mean diffusivity; icosahedral encoding; Monte Carlo simulations Diffusion tensor MRI (DT-MRI) has become an increasingly important modality for understanding the organization of normal brain structures (1-4) and the evolution of neurological and psychiatric disorders (1,5-8). Currently, there are different methods for acquiring, processing, and modeling the measured DT-MRI data (1,9 -14). However, there are many technical and scientific challenges limiting the interpretation, validation, and assignment of the true contributors to the DT-MRI signal in complex biological systems (1,(7)(8)(9)(10)(11)(12)(13)(14)(15)(16)(17). In general, the DT-MRI derived and rotationally invariant maps, such as (D), RA, and FA (6,18,19), are computed offline because of the intense computational nature of the analysis. The availability of online DW procedures to compute these measures would extend the utility of DT-MRI to clinical applications for acute diseases, where immediate feedback to the attending clinicians is important.It was recently shown that useful information can be gained by applying spatially independent component analysis (offline and computationally intensive) and higher-moment statistics to the diffusion data (12). Additional studies (20) have suggested that anisotropy measures can be used to investigate the complex fiber structures within a voxel without certain model assumptions about the signal sources. The relationship between DW-based invariants and those obtained from the single tensor model (when it is an operationally acceptable working model) has not been formally explored. The influence of bias introduced by the encoding scheme and the SNR has also not been addressed.There is considerable evidence that icosahedral encoding sets are the least biased and the most rotationally invariant sets (9,21-26) suitable for acquiring DW data. This work shows that by employing the uniformly distributed principal icosahedron sampling scheme, a diffusion anisotropy measurement analogous to FA can be directly obtained from the DW data. The effects of SNR, encoding scheme, and diffusion sensitization on the accuracy of the method are also investigated using Monte Carlo simulations and normal brain DT-MRI measurements. THEORYBefore the algorithm and the requirements to obtain the rotationally invariant (D), RA, and FA maps directly from the DW data are presented, the basic mathematical underpinnings of the DT-MRI tensor estimation and encoding theory, using matrix algebra, are briefly summarized (1, 24 -27). Self-DT Encoding TheorySince the DT is symmetric, a minimum of six noncollinear encoding directions are needed to obtain the six independent elements of the DT, D, represented as a 3 ϫ 3 matrix [1]The unique tensor elements can be represented by the column vector (24):

Diffusion tensor imaging-based tissue segmentation: Validation and application to the developing child and adolescent brain

“…al., 1995;Hasan and Narayana, 2003;Le Bihan, 1995;Pierpaoli et al, 1996). This problem is mitigated by the use of rotationally-invariant encoding schemes at high SNR as implemented in this paper.…”

“…For example, the encoding scheme adopted in our work (Icosa21b) can provide independent rotationally-invariant data sets (Icosa6 and Icosa15) at different SNR values using the same data (Hasan and Narayana, 2003), and hence some sensitivity analysis can be conducted to show the SNR performance of this approach both regionally and globally. These studies will be reported in a future communication.…”

“…The diffusion-weighted data were acquired using a single-shot spin echo diffusion sensitized echo-planar imaging (EPI) sequence with the balanced Icosa21 encoding scheme (Hasan, 2006;Hasan and Narayana 2003;, a diffusion sensitization of b=1000 sec.mm −2 , a repetition and echo times of T R =6.1 s, T E = 84 ms, respectively. EPI image distortion artifacts were reduced by using a SENSE acceleration factor or k-space undersampling of R of two (Bammer et.…”

Diffusion MR imaging characteristics of the developing primate brain

“…Due to the large number of diffusion directions evaluated, this sampling scheme was found to provide precise, rotationally-invariant parameter estimates when evaluated using simulated data (data not shown). For example, comparisons of the random direction sampling scheme with a 45 direction scheme based on icosahedral sampling (Batchelor et al, 2003;Hasan and Narayana, 2003) yielded extremely small differences in parameter estimates from simulated data (approximately one order of magnitude less than the uncertainty in the estimates). The diffusion-sensitizing gradient pulse length, d, was 5 ms.…”

A framework for quality control and parameter optimization in diffusion tensor imaging: theoretical analysis and validation

“…The relationship between the measured diffusion signals using uniformly-distributed icosahedral tensor encoding schemes and the mean diffusivity can be expressed as [16][17][18]:…”