In the last decade diffusion MRI has become a powerful tool to non-invasively study white-matter integrity in the brain. Recently many research groups have focused their attention on multi-shell spherical acquisitions with the aim of effectively mapping the diffusion signal with a lower number of q-space samples, hence enabling a crucial reduction of acquisition time. One of the quantities commonly studied in this context is the so-called orientation distribution function (ODF). In this setting, the spherical harmonic (SH) transform has gained a great deal of popularity thanks to its ability to perform convolution operations efficiently and accurately, such as the Funk-Radon transform notably required for ODF computation from q-space data. However, if the q-space signal is described with an unsuitable angular resolution at any b-value probed, aliasing (or interpolation) artifacts are unavoidably created. So far this aspect has been tackled empirically and, to our knowledge, no study has addressed this problem in a quantitative approach. The aim of the present work is to study more theoretically the efficiency of multi-shell spherical sampling in diffusion MRI, in order to gain understanding in HYDI-like approaches, possibly paving the way to further optimization strategies.
Introduction. In the last decade diffusion MRI has become a powerful tool to non-invasively study white-matter integrity in the brain. Recently many research groups have focused their attention on multi-shell spherical acquisitions [1,2,3] with the aim of effectively mapping the diffusion signal with a lower number of q-space samples, hence enabling a crucial reduction of acquisition time. One of the quantities commonly studied in this context is the so-called orientation distribution function (ODF). In this setting, the spherical harmonic (SH) transform has gained a great deal of popularity thanks to its ability to perform convolution operations efficiently and accurately, such as the Funk-Radon transform notably required for ODF computation from q-space data. However, if the q-space signal is described with an unsuitable angular resolution at any b-value probed, aliasing (or interpolation) artifacts are unavoidably created. So far this aspect has been tackled empirically and, to our knowledge, no study has addressed this problem in a quantitative approach. The aim of the present work is to study more theoretically the efficiency of multi-shell spherical sampling in diffusion MRI, in order to gain understanding in HYDI-like approaches [4], possibly paving the way to further optimization strategies.
Theory. The ODF is a function on the sphere defined as the radial integration of the full diffusion propagator P: O( r ) = P( r, r) ∫ r 2 dr . The spherical harmonics Y lm ( r ), with l ∈ ℕ and m ≤ l , represent an orthonormal basis for functions on the sphere. We have proved the following original expression for the SH coefficients of the ODF as a function of the q-space signal E, the 3D Fourier transform of P (see [3] for a similar expression in function space):
where S lm represent the SH coefficients of the following radial integration of the q-space signal E, S( q) = E ( q,q)-1 q dq ∫ , with l < B 0 , where B 0 represents the angular band limit of the ODF. As acknowledged previously (see [5]), the Funk-Radon transform is represented efficiently in harmonic space by the simple factor P l (0). However, to our knowledge all research in diffusion MRI is based on spherical sampling distributions providing a non-theoretically exact SH transform. In this work, we advocate the use of equiangular grids, on which sampling theorems exist [6, 7] that allow the exact computation of the B 2 SH coefficients of a function on the sphere of band limit B on the basis of no more than exactly 2B 2 -3B+2 samples [7], and by means of fast algorithms. Methods. To simulate the diffusion process occurring in human brains, synthetic data were generated by means of the multi-tensor model [8] using diffusivity values typically observed in real cases (
A two-fiber model was considered, with different values of crossing angle α and volume fractions 1 and 2, with α ∈ {15°, 40°, 65°, 90°} and ρ 1 ∈ {1, 0.75, 0.5} , and for various fractional anisotropies (FA) (identical for the two fibers), with FA ∈ {0.8, 0.6, 0.4}. Firstly, the value of the angular band limits B of a spherical cut, at given b-value, of the signal E was studied. The discrepancy between the modeled signal E at each b-value and its forward-inverse SH transform Ẽ was evaluated in terms of the sum of the square values of E-Ẽ. Given the exact sampling theorem available on equiangular grids, such a measure quantifies the error arising from any underestimation of the band limit B and leading to angular aliasing. Secondly, in order to illustrate this angular aliasing effect, ODFs were reconstructed from simulated multi-shell data acquisitions with B on each sampled shell associated with either a low or high aliasing level. A 45° crossing angle was considered, with 1 = 0.5 and FA = 0.8, both without noise and with Rician noise at an SNR of 25. A multi-shell setting with 3 shells linearly sampled at b-values 1000, 3500 and 8000 s/mm 2 and a single-shell setting with 1 shell at b-value 3500 s/mm 2 were considered in order to illustrate the complementary radial aliasing induced by excessive reduction of the number of shells, in the fully model independent approach of interest. As a reference, in addition to the ground truth ODF computed analytically from the model, experiments were also simulated in a standard Diffusion Spectrum Imaging (DSI) setting using a state-of-the-art model independent approach.
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