Figure 1.
Illustration of the signal decrease when the echo time and the b-value increase in DWI.
Diffusion-weighted acquisition with (a),
(b),
(c) and
(d). Comparison for
ms (first line) obtained when using our CUSP sequence with
, and
ms (second line) obtained when using multi-shell HARDI sequence with
. It shows how the signal amplitude decreases (and so does the signal-to-noise ratio) when the b-value and the TE increase (first line versus second line). Acquisitions with a short TE should be favored, particularly when imaging at high b-value.
Figure 2.
Intra-voxel orientation heterogeniety and partial volume averaging leads to a non-monoexponential decay in a voxel.
(a): Illustration of the monoexponential decay arising from a single tensor (FA = , diffusivity =
) as shown by the linearity of
in both the parallel and perpendicular directions with respect to the tensor orientation (noise-free case). (b): Illustration that mixing of an isotropic compartment (
,
) and two crossing fascicles represented by two single tensors (
, FA =
, diffusivity =
, crossing angle =
) using Equation 2 leads to a non-monoexponential decay in the voxel, even for b-values below
. This illustrates that a non-monoexponential decay in a voxel may arise from a sum of mono-exponential behaviors.
Figure 3.
CUbe and SPhere (CUSP) imaging can be constructed as a truncated or a projected multi-shell HARDI.
(a): In CUSP-T (Truncated), we consider a multi-shell HARDI with uniformly spaced radius (blue, green, red) and truncate those parts of the shells that project outside of the cube of constant TE of the inner shell. (b): CUSP-xT (eXponential Truncated) employs portion of multiple shells with exponentially spaced radius to achieve an improved uniformity of SNR. (c): In CUSP-P (Projected), we consider an inner shell at (blue) and an outer shell at
(red). The gradients of the outer shell are projected to the cube of constant TE (grey) to avoid any increase in TE. In these figures, the spherical and cubic sampling were shown in different partitions of q-space for visualization purpose.
Figure 4.
Qualitative evaluation of CUSP-MFM.
(a) One hundred synthetic tensors crossing at in various configurations. (b) Estimated tensors with a CUSP35 gradient encoding scheme (SNR =
dB) superimposed on the first tensor's fraction
(window:
; level
). (c) Estimated tensors with the HARDI35 gradient encoding scheme. With a single non-zero b-value (Fig. c), the tensor eigenvalues and the fractions are collinear, leading to a poor multi-fascicle estimate. When using CUSP35 (Fig. b), the system is better determined, leading to a better estimate. Both the tensors and the fraction
are more uniform when using CUSP35-MFM compared to HARDI35-MFM.
Figure 5.
Quantitative evaluation of the CUSP-MFM estimation accuracy.
Quantitative evaluation of the estimation accuracy for the fractions (first line, fAAD metric) and the tensors (second line, tALED metric). Each plot shows the quality metric (fAAD, tALED) in function of the crossing angle for various gradient encoding scheme and various signal-to-noise ratios. It shows that employing a large number of directions (HARDI256) does not dramatically improve the results whereas introducing multiple non-zero b-values does (CUSP35). CUSP35-MFM consistently provides the best estimation accuracy.
Figure 6.
Quantitative evaluation of the angle detection accuracy.
Evaluation of the angle detection accuracy in term of average minimum angle error (tAMA) and comparison with the ball-and-stick model of FSL. CUSP35-MFM provides on average the best angular resolution, particularly for angles lower than degrees, while it provides more information for clinical studies by estimating the full tensors: diffusion parameters such as the fractional anisotropy or the radial diffusivity can be computed for each fascicle independently.
Figure 7.
Angular dependency of the fractional anisotropy with CUSP and a multi-shell HARDI.
The DW-images for a single tensor with constant FA (FA = ) were simulated one hundred times for various tensor orientations (
–
) and for both CUSP and a multi-shell HARDI, corrupted by Rician noise (SNR on the
:
dB). We report the mean of the estimated FA for each angle, for CUSP-35 (a) and CUSP-65 (b). The average FA is compared to the average FA obtained when using MSSHELL-35 (MS35: 5
and three shells of
gradients each at
,
,
) and MSSHELL-65 (MS65). The angular dependency of CUSP and of a multi-shell HARDI is similar.
Figure 8.
Two uniform crossing fascicles have uniform characteristics (FA) with CUSP-MFM.
Estimation of two synthetic crossing fascicles (angle = , SNR =
dB) with HARDI35-MFM (first line) and with CUSP35-MFM (second line). (a) Estimated tensors and (b) fraction of the isotropic water compartment. (c) Illustration of the fractional anisotropy uniformity for each fascicle. In this image, the z-coordinate of the tract streamlines encode for the fractional anisotropy along the tracts (red:
; green:
). It shows the FA of two uniform fascicles to be qualitatively more uniform with CUSP35-MFM than with HARDI35-MFM.
Figure 9.
Quantitative evaluation of the fascicle characteristics along a uniform fascicle.
Quantitative evaluation of diffusion parameters along the horizontal synthetic tract of Fig. 8 for SNR of dB and
dB and for HARDI35-MFM and CUSP35-MFM. (a) Fractional anisotropy assessment. (b) Radial diffusivity assessment. It shows the FA and the RD of two uniform fascicles to be almost uniform with CUSP35-MFM, which is clinically relevant to assess individual fascicle characteristics when studying white matter development or degeneration.
Figure 10.
Evaluation with in vivo short duration DWI acquisitions with , without any model selection.
Comparison of the HARDI35 (first column) and CUSP35 (second column) acquisitions. Fig.a and Fig.b: in contrast to CUSP35-MFM (b), HARDI35-MFM (a) leads to degenerate tensors (area 1) and confounds CSF contamination and fascicles (area 2). Fig.c and Fig.d: when ignoring the estimation of the isotropic compartment, the performance of CUSP35-MFM (d) are strongly affected. The diffusion of unrestricted water cannot be ignored when using a multiple b-values acquisitions. Fig.e and Fig.f: FSL estimates sticks with noisy orientation (area 4), and leads to non-aligned sticks in a single fascicle region of the corpus callosum (area 3). Fig.g and Fig.h: FSL estimation after denoising the DW images (dHARDI35 and dCUSP35).
Figure 11.
Evaluation with in vivo short duration DWI acquisitions with fascicles, without any model selection.
Estimation of fascicles with HARDI35 (a) and with CUSP35 (b), which contain only
images. The tensors with fraction of occupancy smaller than
were not visualized. CUSP35-MFM results in the estimation of three fascicles in a region (see outlined voxels) that matches the known anatomy, the centrum semiovale.
Figure 12.
Estimation of fascicles with F-test model order selection and CUSP-65.
(a) Estimated MFM superimposed on the T1-weighted anatomical image. Particularly, three tensor were correctly estimated in the centrum semiovale, which is a known brain region in which three fascicles are crossing. (b) Illustration of the tractography streamlines passing through the voxel encircled in yellow in (a), showing the three crossing fascicles.
Figure 13.
CUSP-MFM enables the estimation of diffusion tensor parameters which do not vary with the partial volume effect.
We computed the FA along a same tract (Fig.a) for various artificial rotations of the diffusion-weighted images. For each streamline point, the most aligned anisotropic tensor with the streamline orientation was selected and its FA assessed. Fig.b shows the variance of the FA along the tract across the rotations, when using the CUSP or the HARDI acquisition and the MFM estimator without regularization and with the same parameters. HARDI has dramatically increased variance, as it conflates tensor size with partial voluming. CUSP does not. Fig.c shows the FA variance when adding the regularization to the estimation with both CUSP and HARDI. Fig.d shows the corresponding value of the FA along the tract. It shows that CUSP-MFM enables estimation of diffusion tensor parameters which do not vary with the partial volume fractions nor the regularization.
Figure 14.
Comparison of CUSP and multi-shell HARDI via residual bootstrapping.
(a) T1-weighted image showing the anatomy. (b) Standard deviation of the maximum FA when using MSHARDI-65-MFM. (c) Standard deviation of the maximum FA when using CUSP-65-MFM. The standard deviation of the maximum FA is significantly lower when using CUSP, showing a lower uncertainty in the MFM estimates.