Efficient approximate signal reconstruction for correction of gradient nonlinearities in diffusion-weighted imaging

In practice, the magnetic field gradients are nonlinear given gradient coil designs and engineering constraints. These nonlinearities in magnetic field gradients give rise to spatial image warping [1]. 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], [3], [4]]. Thus, the achieved diffusion weighted (DW) encoding scheme varies from the one given to the scanner as input. There is a substantial bias in these gradient deviations in areas away from the isocenter at the edges of the imaging field of view [[5], [6], [7]].

Recently, with increased use of high-performance gradients, characterization of microstructural changes will experience higher gradients field distortions [[8], [9], [10]]. Neglecting correction of nonlinearities can impact diffusion scalar metrics [11], tractography [12], group-wise studies [11], and the reproducibility of apparent diffusion coefficient [13]. Hence, this correction step is important in all diffusion pipelines [11]. The distortion can be rectified given a scanner- and coil-specific estimate of nonlinear gradient field.

Prior research has proposed correction techniques for spatial variation due to gradient nonlinearity. Earlier on, in nuclear magnetic resonance (NMR), the issue of magnetic field gradient nonuniformity was addressed by determining the anticipated amount of pixel displacement caused by the nonuniformity of the gradient [5]. Later, Bammer et al. characterized and corrected gradient nonlinearity in diffusion tensor imaging (DTI) and has been considered as state-of-art (SOA) technique [14]. This technique involves calculation of gradient nonlinearity maps using spherical harmonics expansion, and then these maps are used to correct the spatial b-table [14]. This technique has been widely used [1,[15], [16], [17]]. The nonlinear gradient maps are obtained from phantom studies [[16], [17], [18]]. In phase contrast MRI, the similar correction approach was adapted [19]. Spherical harmonic expansion was the commonly used for approach for solving this problem [13,20]. There are several other studies that investigated the impact and proposed different variation of nonlinearity correction [6,12]. Attempts have been made for practical implementation of nonlinearity correction [7,11]. However, they can only address changes in magnitude component (change in b-value) and DWI intensities, and not suitable for DTI and fiber orientation distribution (FOD). We address this issue by using a spherical harmonic approximation to incorporate the angular change due to nonlinearities.

The SOA correction technique that was demonstrated by Bammer et al. [14] (referred here as empirical nonlinearity correction), despite its effectiveness, has not yet become standard practice: integration into pipelines and models was not straightforward due to the voxel-wise varying gradient table that results (Fig. 1). This work bridges this gap with a simple and efficient “two-step” approximation technique that estimates the desired corrected diffusion signal from the standard voxel-wise gradient. We note that the correction can be split into two separable parts [11] and the intuition was to approximate the two parts. First, the b-value correction is approximated by scaling the signal and then, the gradient direction change is approximated by resampling the sphere and interpolate where the signal should be measured.

In summary, we model the relationship between the desired (given as scanner parameters) and achieved (influenced by hardware designs) gradient field with the formulation for empirical nonlinearity correction. Then, we perform the proposed two-step approximation to reconstruct a corrected signal. We validate the proposed technique with a multi-compartment neurite orientation dispersion and density imaging (NODDI) model. We show our method allows nonlinearity-corrected DW-MRI to be seamlessly integrated in diffusion workflows.

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