Mapping the impact of nonlinear gradient fields with noise on diffusion MRI

Abstract

In diffusion MRI, gradient nonlinearities cause spatial variations in the magnitude and direction of diffusion gradients. Studies have shown artifacts from these distortions can results in biased diffusion tensor information and tractography. Here, we investigate the impact of gradient nonlinearity correction in the presence of noise. We introduced empirically derived gradient nonlinear fields at different signal-to-noise ratio (SNR) levels in two experiments: tensor field simulation and simulation of the brain. For each experiment, this work compares two techniques empirically: voxel-wise gradient table correction and approximate correction by scaling the signal directly. The impact was assessed through diffusion metrics including mean diffusivity (MD), fractional anisotropy (FA), axial diffusivity (AD), radial diffusivity (RD), and principal eigen vector (V1). The study shows (1) the correction of gradient nonlinearities will not lead to substantively incorrect estimation of diffusion metrics in a linear system, (2) gradient nonlinearity correction does not interact adversely with noise, (3) nonlinearity correction suppresses the impact of nonlinearities in typical SNR data, (4) for SNR below 30, the performance of both the gradient nonlinearity correction techniques were similar, and (5) larger impacts are seen in regions where the gradient nonlinearities are distinct. Thus, this study suggests that there were greater beneficial effects than adverse effects due to the correction of nonlinearities. Additionally, correction of nonlinearities is recommended when region of interests are in areas with pronounced nonlinearities.

1. Introduction

Diffusion weighted magnetic resonance (MR) imaging (DW-MRI) provides image contrast determined by Brownian motion of water protons and has been increasingly used as biomarkers [1] to study acute ischemic stroke [2], brain trauma [3], multiple sclerosis [4], schizophrenia [5], or Alzheimer’s disease [6]. Hence, stable quantitative diffusion metrics are necessary to allow mapping across sites and populations to investigate and advance the science of neurodegenerative disorders [7,8]. However, since 1996 from the article by R. E. Hurd et al. [9] and in 2003 by Bammer et al. [10] it has been recognized that hardware limitations in scanner systems’ nonlinearities in gradient coils give rise to spatial variations in gradient amplitude and angular deviations of gradient orientation as in Fig. 1. Spatial variations that are introduced impact the spatial encoding of MR signals and cause impact in estimated diffusion measures. The effect from these sensitivities becomes more significant with distance from the magnet isocenter [10–12]. Further, with recent progress in gradient technology [13,14], clinical utilization of connectome scanners with ultra-strong gradient amplitudes up to 300 mT/m [15–17], deep learning reconstruction of DW-MRI [18], and enhanced signal-to-noise ratio (SNR) [19], it is essentially important to consider the bias caused due to gradient field distortions.

Fig. 1.

Diffusion weighted images (a) experience nonlinear magnetic fields (b). In theory, these nonlinearities are well controlled but in practice we may achieve b-value and gradients (shown in red in (c) and (d)) instead of the desired b-value and gradients (shown in green in (c) and (d)) that were provided as scan parameters.

Looking at the literature, we find many existing frameworks proposed to estimate nonlinear gradient coils including but not limited to (1) computing the spatially varying gradient coil tensor L(r) from the spherical harmonics expansion [10], (2) scaling the signal with the change in diffusion weighting [20], (3) polynomial interpolation [21], (4) rotation of L(r) into the diffusion gradient frame [11] and (5) velocity reconstruction using matrix formalization [22]. The first and second method are investigated in this work. Studies investigate and estimate the effect of gradient nonlinearities in DW-MRI to improve accuracy within and across sites and scanners [23–25]. Neglecting nonlinearity correction results in errors up to 10–30% in commonly used diffusion tensor imaging (DTI) metrics and 10% diffusion kurtosis imaging (DKI) metrics [26,27]. Furthermore, these studies show the change in significance of nonlinearity effect in group studies and its interaction with motion.

Though there are effective tools to address the spatial-temporal gradients and the b-field variations, the underlaying implications of how the correction of nonlinearities interact with noise has not been investigated in depth. To bridge this gap, we present an analysis of the effects of corruption and correction of gradient coil tensor L(r) on tensor simulation and synthetic imaging data with noise distortions introduced. We map the dependence of diffusion tensor metrics, 1) mean diffusivity (MD), 2) fractional anisotropy (FA), 3) axial diffusivity (AD), 4) radial diffusivity (RD), and 5) principal eigen vector (V1) on the L(r) field.

2. Materials and methods

As summarized in Fig. 2, we performed two simulation analyses: tensor and synthetic imaging. Briefly, we simulated ‘ground truth tensors’ for each of the procedures as described in section 2.1 and 2.2 below. Then, we reconstruct the signal from ground truth tensor model with complex Gaussian noise and/or gradient nonlinearity distortions introduced as described in section 2.3. The resulting signal was corrected for gradient nonlinearities with two correction techniques: 1) empirical correction presented by Bammer et al. [10] and 2) a recently presented approximate correction approach [20], this is discussed in section 2.4. We derived the diffusion metrics of fractional anisotropy (FA), mean diffusivity (MD), axial diffusivity (AD), radial diffusivity (RD), and principal eigenvector (V1) from the ground truth tensor, distorted tensor, and gradient nonlinearity corrected tensor to analyze the effect of nonlinearities. From the ground truth tensor, we reconstruct the synthetic signal.

Fig. 2.

From tensor simulation or synthetic DW-MRI signal, we obtained ground truth tensor (gray). This simulated signal from the ground truth tensor was corrupted with noise (brown) as well as noise and nonlinear fields L(r) (purple) to simulate the corrupt DW-MRI signals. These corrupt DW-MRI signals were corrected for gradient nonlinearities using the approximate correction (green) and empirical correction (yellow). Tensors were fit to the corrected signal and corrected gradients and b-value from approximate correction and empirical correction respectively. Derived indices were recomputed to compare the impact of nonlinear fields and correction techniques.

2.1. Tensor simulation

The tensor simulation experiment was adopted from [28]. The ‘ground truth’ diffusion tensors $D_n$ were simulated with 1) fifty evenly spaced FA values from 0.01 to 1 with Eq. (1), 2) constant MD = 0.8 × 10⁻³ mm²s⁻¹ since the MD of white matter (WM) and gray matter (GM) is approximately between 0.6 × 10⁻³ mm²s⁻¹ and 0.8 × 10⁻³ mm²s⁻¹ [30], 3) different orientations that were uniformly spaced with 100 steps at $\theta \in \left[0,\pi \right]$ and 100 steps at $\phi \in \left[0,2\pi \right]$ over a sphere using DIPY [32] with the electrostatic repulsion from Jones1999 [33], this resulted in 19,406 angles, 4) varied determinant of the expected L(r) fields by rank-ordering and choosing the first through 50th percentiles. Thus, there were 4.8515 × 10⁷ axially symmetric ‘ground truth’ diffusion tensors, $D$ with varying 19,406 angles on the sphere for orientation, 50 L(r) fields, 50 FA values. For these diffusion tensors, the gradient scheme with b = 1000 s/mm² and 30 gradient directions evenly distributed was generated using the sampling scheme proposed by Caruyer et al. [29]. Radial diffusivity and MD for axially symmetric are as follows:

$${\lambda }_{\perp }=MD\frac{1-FA}{\sqrt{3-2F{A}^2}}$$ (1)

and

$$MD=\frac{{\lambda }_{\perp }\times 2+{\lambda }_{\parallel }}{3}$$ (2)

Further simplified such that, the axial diffusivity was calculated based on radial diffusivity and MD by [31].

$${\lambda }_{\parallel }=3MD-2{\lambda }_{\perp }$$ (3)

The ground truth tensors would be computed as,

$$D_n=\left[\begin{array}{ccc}{\lambda }_{\perp }& 0& 0\\ 0& {\lambda }_{\perp }& 0\\ 0& 0& {\lambda }_{\parallel }\end{array}\right]$$ (4)

where $n$ refers to the number of the ground truth tensors from tensor simulation (here, 4.8515 × 10⁷), ${\lambda }_{\perp }$ is radial diffusivity and ${\lambda }_{\parallel }$ is the axial diffusivity.

2.2. Synthetic imaging data

For this experiment, the ‘ground truth’ tensors were obtained with a linear DTI fit on a whole brain DW-MRI data of one subject with b = 1000 s/mm² and 32 gradient directions from the MASiVar dataset [34]. These data were scanned on the same clinical scanner where the empirical field maps were obtained. Here, the number of ground truth tensors corresponds to the number of voxels.

2.3. Nonlinearity gradient distortions and noise distortions

The ground truth tensors from section 2.1 and 2.2 were distorted with empirically determined gradient nonlinearities computed on a clinical scanner using the conventions described by Rogers et al. [35,36]. The empirical field maps used in this study were estimated from a 3 T MRI scanner with a 24 L synthetic white oil phantom in a poly-propylene carboy with an approximate diameter of 290 mm and height of 500 mm [36,37]. The 3 T MRI scanner is a 94 cm bore Philips Intera Achieva MR whole-body system and has a gradient strength of 80 mT/m and 200 T/m/s slew-rate. These field maps were previously reported [37]. As described by Rogers et al., the gradient coil tensor L(r) was generated from the empirical field mapping procedure [35]. The L(r) tensor related the achieved magnetic gradient to the intended one. Stejskal-Tanner equation for a signal $S$ and diffusion gradient value $b$ was used as the signal model [38]:

$$S=S_o{e}^{-bD}$$ (5)

where $S_o$ is the MR signal at baseline and D is the diffusion coefficient. The normalized achieved gradients $g^{″}$ were calculated by [37].

$$g^{\prime }=Lg$$ (6)

$$g^{″}=\frac{g^{\prime }}{\left|g^{\prime }\right|}$$ (7)

where $g$ is the intended gradient, $g^{\prime }$ is the achieved gradient vector, and the normalized achieved gradients $g^{″}$. The achieved gradients were normalized with the length change in $g^{\prime }$. Then, the achieved b-value $b^{\prime }$ was the computed by adjusting the intended b-value, $b$ by the square of the length change.

$$b^{\prime }=b{\left|g^{\prime }\right|}^2$$ (8)

By rewriting Eq. (5) with $g^{\prime }$ and $b^{\prime }$, the signal ${\widehat{S}}_i$ for the i^th diffusion acquisition became:

$$\begin{gathered} {\widehat{S}}_i=S_o{e}^{-b{g^{\prime }}_i^{T}D{g^{\prime }}_i} \\ =S_o{e}^{-b^{\prime }}{g^{″}}_i^{T}D{g^{″}}_i \\ =S_o{e}^{-b{g}_i^T{L}^{T}DL{g}_i} \end{gathered}$$ (9)

Thus, the resulting distorted tensors $\widehat{D}$ contained the effect of the gradient nonlinearities field as.

$$\widehat{D}=\frac{\mathit{\text{ln}}\frac{{\widehat{S}}_i}{S_o}}{-b{g}_i^T{L}^{T}DL g_i}$$ (10)

The gradient nonlinearity field was applied to all the ground truth tensors. We computed the FA, MD, AD, RD, and V1 diffusion measurements and evaluated the metrics against the ground truth as shown in Fig. 2.

In addition to L(r) fields, we also introduced complex Gaussian noise in quadrature with standard deviation $\sigma =S_o/\text{SNR}$ where $S_o$ was the mean b0 signal in white matter and $\text{SNR}\in \left(\infty ,100,30,10\right)$. The signal was taken to be the absolute value of the resulting complex signal.

To fit a measured L(r) field as if there were perfectly linear gradients on each axis, L(r) would be an identity matrix for any image voxel. However, imperfections exist in the stability of gradient fields due scanner drift, variation in placement and the fitting process which leads to <0.5% variation in b-values and < 0.5° variation in gradients over the course of one month [36]. This instability was accounted in the identity L(r) matrix when complex Gaussian noise with SNR = 100 was introduced in quadrature.

2.4. Gradient nonlinearity correction techniques

The correction of L(r) fields induced in signal ${\widehat{S}}_i$ was implemented with two correction methods: 1) empirical method presented by Bammer et al. [10] (“empirical Bammer”) and 2) an approximate correction method [20,26]. The empirical Bammer approach estimates the actual gradient vectors and scalar b-value from intended gradient and intended b-value using Eqs. (6)–(10). The empirical Bammer correction approach corrected tensor, $D_{emp}$, was estimated with adjusted b-value $b^{\prime }$ and adjusted gradient vectors $g^{\prime }$ and corrupt signal ${\widehat{S}}_i$.

The approximate correction approach involved computing the corrected signal $S_{\mathit{\text{approx}}}$ (Eq. (12)) by scaling corrupt signal ${\widehat{S}}_i$ with the square length change in b-value. This was numerically equivalent to changing the b-value (Eq. (8)). The approximate corrected tensor $D_{\mathit{\text{approx}}}$, was estimated from corrected signal $S_{\mathit{\text{approx}}}$. Thus, we approximate the effect of L(r) by scaling the signal with ${\left|g^{\prime }\right|}^2$, so Eq. (9) was approximated as,

$${\widehat{S}}_i\cong S_o{e}^{-b{\left|g^{\prime }\right|}^2{g}_i^{T}D{g}_i}$$ (11)

which further simplified to,

$$S_{\mathit{\text{approx }}}=S_o{e}^{\frac{\mathit{\text{ln}}\frac{{\widehat{S}}_i}{S_0}}{{\left|g^{\prime }\right|}^2}}$$ (12)

We computed the FA, MD. AD, RD, and V1 measurements and evaluated these metrics against the ground truth and corruption.

2.5. Regional analysis

To provide anatomical context of the impact of gradient nonlinearities when the regional labels are needed for neuroanalysis, we used whole brain segmentation [40,41] with the BrainCOLOR protocol [42,43] for parcellation of gray matter (GM) regions. For parcellation of white matter (WM) regions we applied the JHU DTI-based white-matter atlas with 48 WM tract labels [44,45]. We register the atlas to the diffusion space with non-rigid registration [46]. The mean absolute percent error for diffusion scalar metrics and angular error for V1 was computed by the region of interest (ROI).

3. Evaluation metrics

3.1. Absolute percent error

We reported the error in terms of voxel wise absolute percent error (APE) (%) between diffusion scalar metrics of uncorrected tensor corrupted with L(r) fields and corrected tensor [26] as,

$${\mathit{\text{APE}}}_i\left(\mathit{\text{\%}}\right)=\left|\frac{M_{{\mathit{\text{corr}}}_i}-M_{{\mathit{\text{uncorr}}}_i}}{M_{{\mathit{\text{uncorr}}}_i}}\right|\times 100\mathit{\text{\%}}$$ (13)

where $M$ was the diffusion scalar metric, either FA, MD, AD, RD. $M_{{\mathit{\text{uncorr}}}_i}$ and $M_{{\mathit{\text{corr}}}_i}$ denoted without and with L(r) correction at the i^th voxel position.

3.2. Angular error

We reported voxel wise angular error (AE) (°) as the average angle between uncorrected V1 and corrected V1. We compute the chord length $C_i$ as follows,

$$C\left(V{1}_{\mathit{\text{corr}}},V{1}_{\mathit{\text{uncorr}}}\right)=\sqrt{\sum _{i=1}^n{\left(V{1}_{{\mathit{\text{corr}}}_i}-V{1}_{{\mathit{\text{uncorr}}}_i}\right)}^2}$$ (14)

Then angle was evaluated as,

$$AE=2\times \mathit{\text{arcsin}}\frac{C}{2}$$ (15)

where $V{1}_{\mathit{\text{uncorr}}}$ and $V{1}_{\mathit{\text{corr}}}$ denote without and with L(r) correction. As the principal eigenvectors are sign agnostic, and differences greater than $\pi /2$ radians are reoriented by multiplication by −1 such that the angle is $\le \pi /2$ [47,48]. AE was computed voxel-wise.

3.3. Statistical analysis

We reported the effect size measure $\mathit{\text{Cohen’s\hspace{0.17em}}}d$ to interpret the differences in uncorrected, empirical Bammer correction, and approximate correction technique. The $\mathit{\text{Cohen’s\hspace{0.17em}}}d$ was computed as,

$$\mathit{\text{Cohen's\hspace{0.17em}\hspace{0.17em}}}d=\frac{{\mathit{\text{mean}}}_{\mathit{\text{uncorr}}}-{\mathit{\text{mean}}}_{\mathit{\text{corr}}}}{s_{\mathit{\text{pooled}}}}$$ (16)

where ${\mathit{\text{mean}}}_{\mathit{\text{uncorr}}}$ and ${\mathit{\text{mean}}}_{\mathit{\text{corr}}}$ are mean of groups L(r) uncorrected and corrected respectively for FA, MD and V1 and $s_{\mathit{\text{pooled}}}$ was the pooled standard deviation of the groups. The effect size of <0.2 is considered small [49].

4. Results

4.1. Tensor simulation

We illustrate the interaction of L(r) corruption and L(r) correction with noise in tensor simulation on FA, MD, and V1 at $\text{SNR}=\infty$ and 30 in Fig. 3. We chose to show SNR = 30, since a typical image has a moderate SNR of 30. For these, we show 1) L(r) correction for a linear system (i.e., perfectly linear gradients where L(r) = I at every image voxel) (Fig. 3a), 2) additive effects of L(r) corruption and L(r) correction with noise (Fig. 3b), and 3) the separability of ground truth noise from L(r) corruption and L(r) correction (Fig. 3c). In a noisy linear system without L (r) corruption, we find median increases in APE and AE after L(r) correction: 0.88%, 0.11%, and 0.09° for FA, MD, and V1 respectively (Fig. 3a). This result was based on the uncertainty of the L(r) field [36]. When L(r) corruption was added with noise, we observe decreases in APE by 7.89% in FA, 3.2% in MD, and a decrease in AE by 1.48° in V1 after L(r) correction (Fig. 3b). When we subtract the ground truth noise from additive effects of L(r) and noise, we find L(r) correction produces near-zero APE and AE, respectively: 0.38% for FA, 0.07% for MD, and 0.28° for V1 (Fig. 3c).

Fig. 3.

The interaction of L(r) corruption and L(r) correction with noise in tensor simulation at FA = 0.25 averaged across orientation. a) A noisy linear system with L (r) correction (green) shows a 0.88%, 0.11% increase in median APE for FA, MD, and 0.09° increase in median AE for V1. b) The change in the derived metrics with noise and L(r) correction (green) shows decreases in median APE by 7.89% in FA, 3.2% in MD, and decrease in median AE by 1.48° in V1 from that of noise and L(r) corruption (blue). c) The effect of L(r) correction with ground truth noise subtracted (green) has median APE of 0.38% in FA, 0.07% in MD, and median AE of 0.28° in V1.

4.2. Synthetic imaging data

Analogously to Fig. 3, Fig. 4 illustrates the interaction of L(r) corruption and L(r) correction with noise, but for synthetic imaging data. In a noisy linear system without L(r) corruption, we find median increases in near-zero APE and AE after L(r) correction: 0.43%, 0.1%, and 0.28° for FA, MD, and V1 respectively (Fig. 4a). This result was based on the uncertainty of the L(r) field [36]. When noise was added to L(r) corruption, we observe decreases in APE by 0.55% in FA, 0.41% in MD, and a decrease in AE by 0.23° in V1 after L(r) correction at SNR = 30 (Fig. 4b). When we subtract the ground truth noise from additive effects of L(r) and noise, we find L(r) correction produces near-zero APE and AE, respectively: 0.13% for FA, 0.01% for MD, and 0.11° for V1 (Fig. 4c).

Fig. 4.

The interaction of L(r) corruption and L(r) correction with noise in synthetic imaging data. a) A noisy linear system L(r) correction (green) shows a 0.43%, 0.1%, increase in median APE for FA, MD, and 0.28° increase in median AE for V1. b) The change in the derived metrics with noise and L(r) correction (green) shows decrease in median APE by 0.55% in FA, 0.41% in MD, and decrease in median AE by 0.23° in V1 from that of noise and L(r) corruption (blue). c) The effect of L(r) correction with ground truth noise subtracted (green) has median APE of 0.13% in FA, 0.01% in MD, and median AE of 0.11° in V1.

4.3. Comparing correction techniques

To compare the effect of empirical L(r) correction and approximate L (r) correction, we tabulate the Cohen’s d between L(r) corruption and the L(r) correction techniques for both the experiments as shown in Table 1. In general, we notice the following trends: (1) Cohen’s d decreases as SNR increases for L(r) corruption and L(r) correction for the metrics, (2) In synthetic imaging data, we find Cohen’s d between the L(r) correction techniques was small (<0.2) for FA when SNR ≤ 100 and for MD and V1 when SNR ≤ 30, this suggest that both the correction techniques have nearly similar effect size when SNR ≤ 30, (3) In tensor simulation, we observe that Cohen’s d between L(r) corruption and L(r) correction was larger for FA, MD, and V1 when compare to Cohen’s d between the two correction techniques and L(r) corruption in syntenic imaging data, this was because the L(r) fields used in tensor simulation were within and outside the brain mask, and (4) In tensor simulation, Cohen’s d between the correction techniques was relatively small at SNR 30 and 10 for all three measures.

Table 1.

Cohen’s d for L(r) corruption and correction techniques for two experiments.

Approach	Measure	SNR	Corruption vs Emp. Correction	Corruption vs Approx. Correction	Emp. Correction vs Approx. Correction
Synthetic imaging data	FA	∞	0.69	0.67	1.2
		100	0.87	0.86	0.16
		30	0.84	0.84	0.013
		10	1.0	1.0	0.011
	MD	∞	1.5	1.44	0.28
		100	0.97	0.95	0.30
		30	0.08	0.07	0.19
		10	0.03	0.03	0.003
	V1	∞	0.75	0.67	1.51
		100	0.93	0.87	0.73
		30	1.26	1.25	0.20
		10	1.89	1.89	0.019
Tensor simulation	FA	∞	24.51	24.07	8.84
		100	28.82	28.59	6.98
		30	30.42	30.4	0.94
		10	28.83	28.85	1.82
	MD	∞	28.60	24.46	8.95
		100	30.11	25.74	8.56
		30	34.85	30.0	7.79
		10	74.06	68.14	5.1
	V1	∞	13.59	9.63	9.40
		100	15.90	11.79	8.78
		30	23.18	19.28	7.70
		10	28.97	27.56	4.79

The effect size of <0.2 was considered small (bold).

4.4. Regional impacts

The frontal regions are the most affected due to gradient nonlinear fields (Fig. 5). As outlined in Table Supp.1, the two highest impacted gray matter regions for FA and V1 were the right and left superior frontal gyrus, for MD and AD were the right and left postcentral gyrus medial segment, and for RD were the right postcentral gyrus medial segment and left precentral gyrus medial segment. As outlined in Table Supp.2, the two highest impacted white matter regions for FA, MD, AD, RD, and V1 were right and left cingulum.

Fig. 5.

Regional impacts of gradient nonlinearity for synthetic data. (a) GM parcellation of L(r) corruption and (b) WM parcellation of L(r) corruption. We show the APE and AE by region according to the BrainCOLOR and JHU DTI-based white matter atlas. Sagittal and axial views are shown for FA, MD, AD, RD, and V1. Large variation was observable in the superior regions of WM and GM.

5. Discussion

From the results, we interpret five conclusions. First, to determine the effects of L(r) correction in linear system, we investigated L(r) correction with only noise induced in the ground truth data shown in Figs. 3a and 4a. We find that when L(r) corruption is not present, L(r) correction does not produce dramatic increases in the estimation of diffusion metrics. To our best knowledge, this work represents one of the first in demonstration of L(r) correction in data assumed to have no gradient nonlinearities.

Second, studies have shown that although the smaller effect of L(r) corruption was immersed in larger effect of noise [26], L(r) correction is necessary to avoid bias in scientific findings due to the negative impact of ignoring L(r). The interaction of motion and gradient nonlinearities are shown by Umesh et al. [27], who reported that in the presence of motion accounting L(r) is necessary. In this study, to characterize the interaction of noise and L(r) correction we examined the effect of L(r) correction with noise in Figs. 3b and 4b. We discovered that L(r) correction does not interact adversely with noise.

Third, to determine that L(r) correction suppressed the impact of L(r) corruption including typical SNR data we show Figs. 3c and 4c. These results clearly demonstrate L(r) correction nearly completely suppressed the effects of L(r) corruption in the absence of ground truth noise. This confirms the results shown in multiple studies [26,27] that when L(r) correction was applied the effect size of L(r) corruption was decreased. The first three observations for tensor simulation lined up with synthetic imaging data and suggest that L(r) correction has a negligible negative impact in typical SNR data. Thus, L(r) correction is recommended for typical SNR data.

Fourth, we determine that approximate correction was acceptable for L(r) correction (Table 1). These findings agree with Mesri et al. [26], who reported the residual errors almost diminished in FA, MD, AD and RD after L(r) correction taking into account the magnitude component of deviations in actual and desired gradients amplitudes. In tensor simulation and synthetic data, the Cohen’s d between empirical correction and approximate correction was relatively small for FA, MD, and V1 at SNR = 30 and 10. This suggests that the distribution was close to each other, and the choice of correction technique will not matter at this noise level.

Finally, we find that regions in the anterior part of the brain are impacted most, shown in Fig. 5. These results emphasize that regions where the gradient nonlinearities are strong are affected the most as recently conveyed by Mesri et al. [26] and Guo et al. [50]. In general, when L(r) correction was neglected, they lead to consequential changes in diffusion metrics in regions where the gradient nonlinearities are strong. This suggests neurological studies with these regions of interests should consider L(r) correction.

In this work, we interpret the characteristic of interaction between noise and L(r) correction, how it alters and impacts the accuracy of diffusion tensor metrics. The L(r) field was derived from phantom-based study [35]. The other approaches to get the required L(r) field are by using manufacture specifications of the nonlinearity field, field camera, and smaller phantom method [51]. This study presumes field estimates are acquired with the data. The study is limited to the effect of noise and L(r) for n of 1. Associations with increasing gradient directions, other distortions such eddy currents, and underlying tissue characteristics are not investigated in this study. Future work could be investigating the L (r) effects in fiber orientations and tractography, its significance in group studies and implementation in diffusion preprocessing pipelines.

Availability of data and material

The data used for the analysis of this article are confidential due to privacy or other restrictions.