Synopsis

Keywords: Data Processing, Diffusion/other diffusion imaging techniques

Gradient nonlinearity correction is well-established but not straightforward to implement in existing diffusion software packages due to it producing gradient tables that vary by voxel. We propose a simple, practical approach that approximates full correction by: (1) scaling the diffusion signal and (2) resampling the gradient orientations. Our approach results in uniform gradients across the corrected image and provides the key advantage of seamless integration into current diffusion pipelines. The proposed method resulted in negligible differences in multi- compartment indices from the standard voxel-wise empirical correction.

INTRODUCTION

In practice, magnetic field gradients are nonlinear given gradient coil designs and engineering constraints. These nonlinearities in magnetic field gradients give rise to spatial image warping¹. In diffusion weighted magnetic resonance imaging (DW-MRI), these systematic spatial distortions result in spatially dependent biases in the magnitude and direction of diffusion gradients^2-4. Thus, the achieved diffusion weighted (DW) encoding scheme varies from the one given to the scanner as input. Neglecting correction of nonlinearities can impact diffusion scalar metrics⁵, tractography⁶, group-wise studies⁵, and the reproducibility of apparent diffusion coefficient⁷. Hence, a correction step is important⁵. However, the state-of-the-art correction technique proposed by Bammer et al.⁸ has not become standard practice. We observe that integration into pipelines and models is not straightforward due to the voxel-wise varying gradient table that results (Figure 1). Our work bridges this gap with simple and efficient approximation that estimates the desired diffusion signal from the standard voxel-wise gradient table correction. We validate the proposed technique with a multi-compartment neurite orientation dispersion and density imaging (NODDI) model. Our method allows nonlinearity-corrected DW-MRI to be seamlessly integrated in diffusion workflows.

METHODS

Data: For this study, five de-identified subjects from the MASiVar dataset⁹ approved by the Institutional Review Board were used. The subject scans were acquired at b-value of 1000 s/mm² for 40 directions and b-value of 2000 s/mm² for 56 directions. They were scanned on a 3T Philips Achieva scanner with gradient strength of 80 mT/m and 200 T/m/s slew-rate. The empirical field maps were obtained from the same scanner. The empirical field maps are used to estimate the nonlinear gradient tensor L(r) as described¹⁰ and reported¹¹ by Rogers et al.
Pipeline: The raw diffusion data contain the achieved DW-MRI signal reconstructed from the scanner and desired gradients that were provided as scanner parameters. These desired gradients are mapped voxel-wise to the achieved gradients using the L(r) fields by the widely accepted Bammer et al correction technique^5,8,12. From the achieved DW-MRI signal and gradients, we approximate the desired signal with desired gradient table with two steps: 1) scaling the signal through gaussian approximation with the square of the length change in b-value (Eq.1,2) which is numerically equivalent to the magnitude change in the gradients, and 2) resampling the angular change via a spherical harmonics basis (Eq.3) which is numerically equivalent to the orientation change in the gradients (Figure 2). The approach assumes these two steps are separable⁵. The spherical harmonics fit is performed with the unique set of b-vectors for every voxel. The spherical harmonics basis functions described by Tournier et al¹³ were implemented using the DIPY library¹⁴ with no smoothing and the highest possible even order supported given the number of gradient directions. Finally, the desired signal was estimated with the spherical harmonics coefficients and desired (original) gradients (by rewriting Eq.3). Thus, we estimated a signal overcoming voxel-wise differences after correction for gradient nonlinear fields.
The achieved b-value $$$b^\prime$$$ is computed as⁸: $$b^\prime=b\ \left|g^\prime\right|^2 .....(1)$$
where $$$b$$$ is the desired b-value and $$$g^\prime$$$ is the achieved gradients. Substitute this in Stejskal-Tanner equation, $$S_{scaled}=\ S_o\ {e\ \ }^\frac{ln\ \ \ {\frac{S_i}{S_o}}}{\left|g^\prime\right|^2} .....(2)$$
where $$$S_i$$$ is diffusion signal at the i^th acquisition, $$$S_o$$$ is the reference signal, and $$$S_{scaled}$$$ is the scaled diffusion signal. The spherical harmonics coefficient $$$a_l^m$$$ can be expressed as¹⁴ $$a_l^m=\ \int_{S}^{\ }f\left(\theta,\phi\right)Y_l^m\ \left(\theta,\phi\right)\ ds .....(3)$$
where $$$l$$$ is the order, $$$m$$$ is the degree, $$$Y_l^m$$$ is spherical harmonics basis function, and $$$\left(\theta,\phi\right)$$$ represents the direction vector in spherical coordinates.
The proposed approximation was performed separately on each shell. We fit a NODDI model for Bammer correction and proposed approximation with University College of London NODDI Toolbox¹⁵. The model generate intra-cellular volume fraction (iVF), cerebrospinal fluid (CSF) volume fraction (cVF), and orientation dispersion index (ODI) representations. For the analysis, we used the white-matter and gray-matter mask from FSL¹⁶ for iVF and ODI, while for cVF we used the CSF mask from FSL¹⁶.

RESULTS

We show the absolute difference between the voxel-wise Bammer correction and the proposed approximation for iVF (Figure 3), cVF (Figure 4), and ODI (Figure 5) for five subjects. The mean absolute difference in all three NODDI scalar metrics are less than 10^-2 between the techniques. The Cohen’s d between the voxel-wise correction and proposed approximation was <0.2 for all subjects. This suggests the approximation does not change the DW-MRI information substantively from the standard Bammer correction technique.

CONCLUSION

With the emerging generation of MRI scanners¹⁷, the integration of a gradient nonlinearity correction step into contemporary diffusion preprocessing platforms has become necessary. In this work, we have proposed a simple approximation for the standard voxel-wise gradient nonlinearities correction in two steps: scaling the signal with the factor of b-value change from achieved to desired via gaussian approximation and resampling the rescaled signal based on b-vector change from achieved to desired via spherical harmonics approximation. We find that the proposed approximation resulted in negligible differences from the standard voxel-wise correction. Using our proposed approximation, the generally accepted voxel-wise gradient nonlinearity correction for diffusion encoding schemes can be easily incorporated with existing diffusion models. The key limitation was the assumption that the gradient field estimates of scanner are known.

Acknowledgements

This work was supported by the National Institutes of Health under award numbers R01EB017230, 1K01EB032898 and T32GM007347, and in part by the National Center for Research Resources, Grant UL1 RR024975-01. The content is solely the responsibility of the authors and does not necessarily represent the official views of the NIH.

References

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