A novel tensor distribution model for the diffusion-weighted MR signal

Abstract

Diffusion MRI is a non-invasive imaging technique that allows the measurement of water molecule diffusion through tissue in vivo. The directional features of water diffusion allow one to infer the connectivity patterns prevalent in tissue and possibly track changes in this connectivity over time for various clinical applications. In this paper, we present a novel statistical model for diffusion-weighted MR signal attenuation which postulates that the water molecule diffusion can be characterized by a continuous mixture of diffusion tensors. An interesting observation is that this continuous mixture and the MR signal attenuation are related through the Laplace transform of a probability distribution over symmetric positive definite matrices. We then show that when the mixing distribution is a Wishart distribution, the resulting closed form of the Laplace transform leads to a Rigaut-type asymptotic fractal expression, which has been phenomenologically used in the past to explain the MR signal decay but never with a rigorous mathematical justification until now. Our model not only includes the traditional diffusion tensor model as a special instance in the limiting case, but also can be adjusted to describe complex tissue structure involving multiple fiber populations. Using this new model in conjunction with a spherical deconvolution approach, we present an efficient scheme for estimating the water molecule displacement probability functions on a voxel-by-voxel basis. Experimental results on both simulations and real data are presented to demonstrate the robustness and accuracy of the proposed algorithms.

Fig. 1

The Wishart distributed tensors lead to a Rigaut-type signal decay.

Fig. 2

Diffusion tensor fitting of a simulated data set. (a) Visualization of the noiseless tensor field. (b) Comparison of the accuracy of the estimated dominant eigenvectors using different methods under different noise levels.

Fig. 3

Probability maps from a simulated image of two crossing fiber bundles computed using (a) DOT and (b) proposed method.

Fig. 4

Simulations of 1, 2 and 3 fibers (b = 1500 s/mm²). Orientations: azimuthal angles φ₁ = 30, φ₂ = {20, 100}, φ₃ = {20, 75, 135}; polar angles were all 90°. Top: Q-ball ODF surfaces; Bottom: Probability surfaces computed using proposed method.

Fig. 5

Resistance to noise (2 fibers, σ = 0.08): (a) ODF from QBI; (b) Proposed method.

Fig. 6

Resistance to noise (3 fibers, σ = 0.04): (a) ODF from QBI; (b) Proposed method.

Fig. 7

Probability maps computed from a rat optic chiasm data set overlaid on axially oriented GA maps. The decussations of myelinated axons from the two optic nerves at the center of the optic chiasm are readily apparent. Decussating fibers carry information from the temporal visual fields to the geniculate body. Upper left corner shows the corresponding reference (S₀) image.

Fig. 8

Probability maps of coronally oriented GA images of a control and an epileptic hippocampus. Upper left corner shows the corresponding reference (S₀) images where the rectangle regions enclose the hippocampi. In the control hippocampus, the molecular layer and stratum radiatum fiber orientations paralleled the apical dendrites of granule cells and pyramidal neurons respectively, whereas in the stratum lacunosum, moleculare orientations paralleled Schaffer collaterals from CA1 neurons. In the epileptic hippocampus, the overall architecture is notably altered; the CA1 subfield is lost, while an increase in crossing fibers can be seen in the hilus and dentate gyrus (dg). Increased crossing fibers can also be seen in the entorhinal cortex (ec). Fiber density within the statum lacunosum moleculare and statum radiale is also notably reduced, although fiber orientation remains unaltered.