Apparent Diffusion Coefficients from High Angular Resolution Diffusion Images: Estimation and Applications
Résumé
High angular resolution diffusion imaging (HARDI) has recently been of great interest in characterizing non-Gaussian diffusion processes. In the white matter of the brain, non-Gaussian diffusion occurs when fiber bundles cross, kiss or diverge within the same voxel. One important goal in current research is to obtain more accurate fits of the apparent diffusion processes in these multiple fiber regions, thus overcoming the limitations of classical diffusion tensor imaging (DTI). This paper presents an extensive study of high order models for apparent diffusion coefficient estimation and illustrates some of their applications. In particular, we first develop the appropriate mathematical tools to work on noisy HARDI data. Using a meaningful modified spherical harmonics basis to capture the physical constraints of the problem, we propose a new regularization algorithm to estimate a diffusivity profile smoother and closer to the true diffusivities without noise. We define a smoothing term based on the Laplace-Beltrami operator for functions defined on the unit sphere. The properties of the spherical harmonics are then exploited to derive a closed form implementation of this term into the fitting procedure. We next derive the general linear transformation between the coefficients of a spherical harmonics series of order $\ell$ and the independent elements of the rank-$\ell$ high order diffusion tensor. An additional contribution of the paper is the careful study of the state of the art anisotropy measures for high order formulation models computed from spherical harmonics or tensor coefficients. Their ability to characterize the underlying diffusion process is analyzed. We are able to reproduce published results and also able to recover voxels with isotropic, single fiber anisotropic and multiple fiber anisotropic diffusion. We test and validate the different approaches on apparent diffusion coefficients from synthetic data, from a biological phantom and from a human brain dataset.
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