Fast Correction of Eddy-Current and Susceptibility-Induced Distortions Using Rotation-Invariant Contrasts

Abstract

Diffusion MRI (dMRI) is typically time consuming as it involves acquiring a series of 3D volumes, each associated with a wave-vector in q-space that determines the diffusion direction and strength. The acquisition time is further increased when “blip-up blip-down” scans are acquired with opposite phase encoding directions (PEDs) to facilitate distortion correction. In this work, we show that geometric distortions can be corrected without acquiring with opposite PEDs for each wave-vector, and hence the acquisition time can be halved. Our method uses complimentary rotation-invariant contrasts across shells of different diffusion weightings. Distortion-free structural T1-/T2-weighted MRI is used as reference for nonlinear registration in correcting the distortions. Signal dropout and pileup are corrected with the help of spherical harmonics. To demonstrate that our method is robust to changes in image appearance, we show that distortion correction with good structural alignment can be achieved within minutes for dMRI data of infants between 1 to 24 months of age.

1. Introduction

Diffusion MRI (dMRI) is widely used in neuroimaging studies to investigate brain micro-architecture [11,14] and structural connectivity [6]. Diffusion-weighted (DW) images acquired using spin-echo echo-planar imaging (EPI) suffers from geometric distortions caused by gradient-switching-induced eddy currents and susceptibility-induced off-resonance fields associated with the low bandwidth in the phase-encode direction.

Existing distortion correction techniques are based on (i) B₀ field mapping [10,17], (ii) nonlinear spatial normalization of distorted images to undistorted structural T1-/T2-weighted (T1w/T2w) images [12,16], or (iii) unwarping of blip-up blip-down acquisitions to a common undistorted mid-point [1,5,13]. The first approach requires an accurate estimation of the field map, which can be challenging due to motion artifacts and phase errors. The second approach often results in poor alignment due to inter-modality contrast differences. The third approach improves the intensity uniformity of the distortion-corrected images, but has the following limitations: (i) The geometric distortions are corrected by warping opposing-PED non-DW (b = 0s/mm²) images to a hypothetical mid-point space that does not necessarily correspond with the undistorted space according to structural MRI (sMRI). This is problematic particularly when information from sMRI, such as cortical surface geometry, is needed for subsequent analyses. Additional registration and interpolation might cause errors and image degradation. (ii) The correction is dependent on PED-reversed scans, prolonging acquisition time. This can be prohibitive when imaging pediatric, elderly, or claustrophobic individuals. (iii) Typically only non-DW images are used for estimating the distortion-induced displacements. Irfanoglu et al. [9] proposed to include DW images to improve registration in homogeneous regions with large distortions. However, signal attenuation at high b-values results in low signal-to-noise ratio (SNR) and poor tissue contrast, impeding displacement field estimation. (iv) Correcting the distortions of a large number of DW images typically requires a long computation time.

To overcome these limitations, we show in this paper that eddy-current and susceptibility-induced geometric distortions can be corrected without acquiring with opposite PEDs for each wave-vector, and hence the acquisition time can be halved. Our method utilizes complimentary rotation-invariant contrasts (RICs) across shells of different diffusion weightings. Distortion-free structural T1-/T2-weighted MRI is used as reference for nonlinear registration in correcting the distortions. Signal dropout and pileup are corrected with the help of spherical harmonics. To demonstrate that our method is robust to changes in image appearance, we show that distortion correction with good structural alignment can be achieved within minutes for data of infants between 1 to 24 months of age.

2. Methods

The proposed post-acquisition distortion correction method consists of three main steps: (i) Correction of eddy-current-induced distortions; (ii) Multi-contrast correction of susceptibility-induced distortion; and (iii) Correction for signal intensity due to dropout and pileup.

2.1. Interleaved PEDs

Approaches based on “blip-up blip-down” scans require scanning for each wavevector in q-space using opposing PEDs, essentially doubling the acquisition time. We propose, in line with [4], to acquire each q-space sample with only one PED, but interlace q-space samples with different PEDs. In this paper, we will demonstrate with two PEDs, i.e., anterior-posterior (AP) or posterior-anterior (PA), but our method can be easily generalized to more than two PEDs. Unlike the constrained image reconstruction method proposed in [4], we will introduce a fast method for distortion correction using RICs, and hence avoiding the time-consuming process of explicitly correcting for each individual DW image. The appearances of DW images are direction dependent. The RICs remove directional dependency and allow DW images acquired with different PEDs to be registered efficiently. RICs, by agglomerating information from all directions, also improve SNR and facilitate registration of DW images with high b-values (above 1500 s/mm²), a problem that was reported in [3].

2.2. Distortion Correction

The eddy-current-induced distortions of images $I_p={\left\{I_p^n\right\}}_{n=1}^N$, acquired with PED p ∈ {AP, PA}, can be corrected by affine registration, followed by coarse nonlinear registration, of all DW images to their corresponding non-DW image. This is implemented using ANTs registration toolkit [2], resulting in affine transformation parameters ${\left\{{\mathcal{A}}_p^n\right\}}_{n=1}^N$ and nonlinear displacement fields ${\left\{{\varphi }_p^n\right\}}_{n=1}^N$.

DW images at high b-values exhibit low SNR, low tissues contrast, and therefore present insufficient structural details for accurate registration [3]. Existing methods that register only non-DW images for distortion correction do not necessarily guarantee alignment of high b-value DW images, which are necessary for sensitivity and specificity to tissue microstructure. Here, we propose to use the spherical mean images (SMIs) of DW images corrected for eddy-current distortions in a multi-contrast registration framework for correction of susceptibility-induced distortions. The SMIs—computed across gradient directions for each shell—exhibit higher SNR and are rotation invariant, facilitating the registration of DW images acquired with interleaved PEDs (see Fig. 4). For PED p, we denote the SMIs for M shells, in addition to b = 0s/mm², as $J_p={\left\{J_p^i\right\}}_{i=0}^M$. These SMIs are first jointly affine registered to aligned structural T1w and/or T2w images $S_{{\text{T}}_1}$ and $S_{{\text{T}}_2}$, producing ${\tilde{J}}_p={\left\{{\tilde{J}}_p^i\right\}}_{i=0}^M$ with affine transformation parameters ${\mathcal{A}}_p$. Multi-contrast nonlinear registration is then performed using symmetric normalization (SyN) [2] with a mutual information (MI) data matching term:

$${\zeta }_p=\int \sum _{i=0}^M\left(\text{MI}\left({\tilde{J}}_p^i\left({\varphi }_{p\to {\text{S}}_{\text{T}1}}\right),S_{\text{T}1}\right)+\text{MI}\left({\tilde{J}}_p^i\left({\varphi }_{p\to {\text{S}}_{\text{T}2}}\right),S_{\text{T}2}\right)\right)dx,$$ (1)

where ϕ_p→S is the displacement field that maps the SMIs to the structural reference space S. SyN multiplied each component of a displacement vector with a weight before updating the displacement field in the next iteration. The displacement field was weighted more towards the PED.

Fig. 4.

Spherical mean images for different b-values for a 12-month-old infant subject. Image brightness is adjusted for each shell for visibility.

DW images collected with two opposing PEDs typically exhibit distortions in opposite directions. To promote the inverse relationship between ϕ_AP→S and ϕ_PA→S, we first estimate the displacement fields between the SMIs for opposite PEDs as follows:

$${\varphi }_{\text{AP}\to \text{PA}}={\varphi }_{\text{AP}\to \text{S}}\circ {\varphi }_{\text{PA}\to \text{S}}^{-1},$$ (2)

$${\varphi }_{\text{PA}\to \text{AP}}={\varphi }_{\text{PA}\to \text{S}}\circ {\varphi }_{\text{AP}\to \text{S}}^{-1}.$$ (3)

The inverse displacement fields to the undistorted reference space are then given by

$${\varphi }_{\text{AP}\to \text{S}}\leftarrow 0.5\times {\varphi }_{\text{AP}\to \text{PA}},$$ (4)

$${\varphi }_{\text{PA}\to \text{S}}\leftarrow 0.5\times {\varphi }_{\text{PA}\to \text{AP}}.$$ (5)

This is illustrated in Fig. 1. In principle, ϕ_AP→S and ϕ_PA→S can be refined by using them to initialize the registration algorithm, and then recomputing Eqs. (2), (3), (4), and (5), but we found that the improvement is negligible for our data. The distortion-corrected DW images are then obtained by warping with the corresponding displacement field

$${\varphi }_p^n={\mathcal{A}}_p^n\circ {\varphi }_{\text{p}}^n\circ {\mathcal{A}}_p\circ {\varphi }_{\text{p}\to \text{S}}.$$ (6)

The undistorted images are given as ${\tilde{I}}_p^n=I_p^n\left({\varphi }_p^n\right)$. The signal pileup due to signal compression and signal dropout due to expansion are corrected by modulating the undistorted images ${\tilde{I}}_p^n$ with the Jacobian determinant of the displacement field ${\varphi }_p^n$ [7]:

$${\tilde{I}}_p^n\leftarrow {\tilde{I}}_p^n\times \mathcal{J}\left({\varphi }_p^n\right).$$ (7)

Fig. 1.

Displacement fields associated with two opposite PEDs.

Signal pileup might not always be completely rectifiable using images acquired with a single PED. Correction can be improved by using information from images collected with the opposing PED. Unlike blip-up blip-down approaches, our interleaved-PED acquisition scheme did not acquire each q-space sample with opposing PEDs. Therefore, we fitted the spherical harmonics (SHs) to the DW images acquired for a PED, and then estimated the DW images of the opposing PED. The signal was reoriented via the local Jacobian of the deformation field prior to SH fitting.

The final corrected DW images are obtained by combining the DW images associated with the two PEDs. This is achieved by computing the harmonic mean [9] of the signals associated with the two PEDs.

3. Results and Discussion

The dataset consisted of dMRI and sMRI data of 54 infant subjects between 1 and 24 months of age, enrolled as part of the UNC/UMN Baby Connectome Project (BCP) [8]. The subjects were divided into three cohorts according to their first scheduled visits and were scanned every three months. Since not all subjects can be scanned every three months, each subject has a different number of scans, resulting in 68 scans in total. The sMRI data had 208 sagittal slices with 320 × 320 matrix size and (0.8 mm)³ voxel resolution. The dMRI data had 95 axial slices with 140 × 140 matrix size and (1.5 mm)³ voxel resolution. The diffusion data were acquired with b-values 500, 1000, 1500, 2000, 2500, and 3000 s/mm². The DW images were acquired with AP and PA PEDs, with a total of 144 DW images and six non-DW images for each PED. To demonstrate the effectiveness of our method, we removed the images associated with half of the gradient directions, resulting in 72 directions per PED. For comparison, the full number of gradients were used for FSL TOPUP+Eddy [15]. The parameters of TOPUP+Eddy were adjusted to obtain optimal results for infant dMRI.

We show the fractional anisotropy (FA) maps computed using the two methods in Fig. 2. The FA map given by TOPUP+Eddy shows structural discrepencies with respect to sMRI. Further evaluation was carried out by overlaying the white matter surface generated from the sMRI data onto the FA maps computed by the two methods. It is clear from Fig. 3 that the FA map generated by our method is well-aligned with sMRI at the interface between white matter and gray matter. Figure 4 shows the spherical mean images of the corrected dMRI data of 12-month-old infant, indicating that our method yields good correction outcomes comparable with TOPUP+Eddy.

Fig. 2.

FA maps and non-DW images of the corrected dMRI data of an 18-month-old infant.

Fig. 3.

White matter surface overlaid on the FA maps and T1w image of a 23-monthold infant.

We also qualitatively evaluated the Jacobian determinant maps of the deformation fields estimated by the two methods. Figure 5(a) shows that the deformation field generated by TOPUP+Eddy is not diffeomorphic as evident from the negative Jacobian determinant values. In contrast, our method gives positive Jacobian determinant values for all voxels. We validated the inverse relationship of the displacement fields given by our method. The displacement fields for the two opposing PEDs are shown in Fig. 5(b). The inverse consistency error (ICE) [18] map in Fig. 5(c) shows that the two displacement fields are the inverses of each other.

Fig. 5.

(a) Jacobian maps of the deformation fields given by TOPUP+Eddy and the proposed method. (b) Displacement fields for the two opposing PEDs estimated by the proposed method, and (c) the corresponding inverse consistency error (ICE) map.

Figure 6 shows the spherical mean images after geometric and intensity correction, indicating that signal inhomogeneity is reduced.

Fig. 6.

Spherical mean images after geometric and intensity correction of the dMRI data of a 12-months-old infant. Image brightness is adjusted for each shell for visibility.

In addition, we compared the FA values given by the two methods. The mean ± std FA values are 0.19 ± 0.14 and 0.24 ± 0.16 for TOPUP+Eddy and the proposed method (Fig. 7(a)), respectively. We quantified the similarity between the FA maps and the T1w images using MI. The boxplot for MI is given in Fig. 7(b), where the mean ± std MI values are 0.44 ± 0.07 and 0.47 ± 0.06 for TOPUP+Eddy and the proposed method, respectively. We also compared the two methods by evaluating the structural similarity (SSIM) between the reference T2w image and the corrected non-DW images for each subject. The mean ± std of SSIM is (28 ± 17)% for TOPUP+Eddy and (61 ± 7)% for the proposed method (Fig. 7(c)). We further validated the results by computing the PSNR of the spherical mean images for all the b-values obtained with TOPUP+Eddy and the proposed method, using the corresponding T1w as reference. Our method gives an average PSNR of 17.80, whereas TOPUP+Eddy gives an average of 13.27 (Fig. 7(d)). The results for all metrics are statistically significant (p < 0.01).

Fig. 7.

Quantitative comparison between distortion correction methods.

Our method is very time efficient and took 25 min per dMRI dataset, whereas TOPUP+Eddy took a few hours to complete.

4. Conclusion

In this paper, we proposed a distortion correction method for both eddy-current and susceptibility-induced distortions and demonstrated its effectiveness in infant brain dMRI. Our method does not require images of multiple phase-encoding directions to be acquired for each q-space sample, therefore reducing the acquisition time. Our method relies on high SNR spherical mean images for guiding registration. We have also shown that the commonly used TOPUP+Eddy does not fully remove distortions as compared to undistorted T1- and T2-weighted images, potentially causing problems when the different modalities need to be analyzed in a common space. The results show that our method outperforms TOPUP+Eddy both in terms of accuracy and speed.